Heegner points on elliptic curves over the rational numbers¶

AUTHORS:

William Stein (August 2009)– most of the initial version
Robert Bradshaw (July 2009) – an early version of some specific code

EXAMPLES:

Sage

sage: E = EllipticCurve('433a')
sage: P = E.heegner_point(-8,3)
sage: z = P.point_exact(201); z
(-4/3 : 1/9*a : 1)
sage: parent(z)
Abelian group of points on Elliptic Curve defined by y^2 + x*y = x^3 + 1
 over Number Field in a with defining polynomial x^2 - 12*x + 111
sage: parent(z[0]).discriminant()
-3
sage: E.quadratic_twist(-3).rank()
1
sage: K.<a> = QuadraticField(-8)
sage: K.factor(3)
(Fractional ideal (1/2*a + 1)) * (Fractional ideal (-1/2*a + 1))

Python

>>> from sage.all import *
>>> E = EllipticCurve('433a')
>>> P = E.heegner_point(-Integer(8),Integer(3))
>>> z = P.point_exact(Integer(201)); z
(-4/3 : 1/9*a : 1)
>>> parent(z)
Abelian group of points on Elliptic Curve defined by y^2 + x*y = x^3 + 1
 over Number Field in a with defining polynomial x^2 - 12*x + 111
>>> parent(z[Integer(0)]).discriminant()
-3
>>> E.quadratic_twist(-Integer(3)).rank()
1
>>> K = QuadraticField(-Integer(8), names=('a',)); (a,) = K._first_ngens(1)
>>> K.factor(Integer(3))
(Fractional ideal (1/2*a + 1)) * (Fractional ideal (-1/2*a + 1))

Next try an inert prime:

Sage

sage: K.factor(5)
Fractional ideal (5)
sage: P = E.heegner_point(-8,5)
sage: z = P.point_exact(300)
sage: z[0].charpoly().factor()
(x^6 + x^5 - 1/4*x^4 + 19/10*x^3 + 31/20*x^2 - 7/10*x + 49/100)^2
sage: z[1].charpoly().factor()
x^12 - x^11 + 6/5*x^10 - 33/40*x^9 - 89/320*x^8 + 3287/800*x^7
 - 5273/1600*x^6 + 993/4000*x^5 + 823/320*x^4 - 2424/625*x^3
 + 12059/12500*x^2 + 3329/25000*x + 123251/250000
sage: f = P.x_poly_exact(300); f
x^6 + x^5 - 1/4*x^4 + 19/10*x^3 + 31/20*x^2 - 7/10*x + 49/100
sage: f.discriminant().factor()
-1 * 2^-9 * 5^-9 * 7^2 * 281^2 * 1021^2

Python

>>> from sage.all import *
>>> K.factor(Integer(5))
Fractional ideal (5)
>>> P = E.heegner_point(-Integer(8),Integer(5))
>>> z = P.point_exact(Integer(300))
>>> z[Integer(0)].charpoly().factor()
(x^6 + x^5 - 1/4*x^4 + 19/10*x^3 + 31/20*x^2 - 7/10*x + 49/100)^2
>>> z[Integer(1)].charpoly().factor()
x^12 - x^11 + 6/5*x^10 - 33/40*x^9 - 89/320*x^8 + 3287/800*x^7
 - 5273/1600*x^6 + 993/4000*x^5 + 823/320*x^4 - 2424/625*x^3
 + 12059/12500*x^2 + 3329/25000*x + 123251/250000
>>> f = P.x_poly_exact(Integer(300)); f
x^6 + x^5 - 1/4*x^4 + 19/10*x^3 + 31/20*x^2 - 7/10*x + 49/100
>>> f.discriminant().factor()
-1 * 2^-9 * 5^-9 * 7^2 * 281^2 * 1021^2

We find some Mordell-Weil generators in the rank 1 case using Heegner points:

Sage

sage: E = EllipticCurve('43a'); P = E.heegner_point(-7)
sage: P.x_poly_exact()
x
sage: z = P.point_exact(); z == E(0,0,1) or -z == E(0,0,1)
True

sage: E = EllipticCurve('997a')
sage: E.rank()
1
sage: E.heegner_discriminants_list(10)
[-19, -23, -31, -35, -39, -40, -52, -55, -56, -59]
sage: P = E.heegner_point(-19)
sage: P.x_poly_exact()
x - 141/49
sage: z = P.point_exact(); z == E(141/49, -162/343, 1)  or -z == E(141/49, -162/343, 1)
True

Python

>>> from sage.all import *
>>> E = EllipticCurve('43a'); P = E.heegner_point(-Integer(7))
>>> P.x_poly_exact()
x
>>> z = P.point_exact(); z == E(Integer(0),Integer(0),Integer(1)) or -z == E(Integer(0),Integer(0),Integer(1))
True

>>> E = EllipticCurve('997a')
>>> E.rank()
1
>>> E.heegner_discriminants_list(Integer(10))
[-19, -23, -31, -35, -39, -40, -52, -55, -56, -59]
>>> P = E.heegner_point(-Integer(19))
>>> P.x_poly_exact()
x - 141/49
>>> z = P.point_exact(); z == E(Integer(141)/Integer(49), -Integer(162)/Integer(343), Integer(1))  or -z == E(Integer(141)/Integer(49), -Integer(162)/Integer(343), Integer(1))
True

Here we find that the Heegner point generates a subgroup of index 3:

Sage

sage: E = EllipticCurve('92b1')
sage: E.heegner_discriminants_list(1)
[-7]
sage: P = E.heegner_point(-7)
sage: z = P.point_exact(); z == E(0, 1, 1)  or -z == E(0, 1, 1)
True
sage: E.regulator()
0.0498083972980648
sage: z.height()
0.448275575682583
sage: P = E(1,1); P # a generator
(1 : 1 : 1)
sage: -3*P
(0 : 1 : 1)
sage: E.tamagawa_product()
3

Python

>>> from sage.all import *
>>> E = EllipticCurve('92b1')
>>> E.heegner_discriminants_list(Integer(1))
[-7]
>>> P = E.heegner_point(-Integer(7))
>>> z = P.point_exact(); z == E(Integer(0), Integer(1), Integer(1))  or -z == E(Integer(0), Integer(1), Integer(1))
True
>>> E.regulator()
0.0498083972980648
>>> z.height()
0.448275575682583
>>> P = E(Integer(1),Integer(1)); P # a generator
(1 : 1 : 1)
>>> -Integer(3)*P
(0 : 1 : 1)
>>> E.tamagawa_product()
3

The above is consistent with the following analytic computation:

Sage

sage: E.heegner_index(-7)
3.0000?

Python

>>> from sage.all import *
>>> E.heegner_index(-Integer(7))
3.0000?

class sage.schemes.elliptic_curves.heegner.GaloisAutomorphism(parent)[source]¶

Bases: SageObject

An abstract automorphism of a ring class field.

Todo

make GaloisAutomorphism derive from GroupElement, so that one gets powers for free, etc.

domain()[source]¶

Return the domain of this automorphism.

EXAMPLES:

Sage

sage: E = EllipticCurve('389a')
sage: G = E.heegner_point(-7,5).ring_class_field().galois_group()
sage: s = G.complex_conjugation()
sage: s.domain()
Ring class field extension of QQ[sqrt(-7)] of conductor 5

Python

>>> from sage.all import *
>>> E = EllipticCurve('389a')
>>> G = E.heegner_point(-Integer(7),Integer(5)).ring_class_field().galois_group()
>>> s = G.complex_conjugation()
>>> s.domain()
Ring class field extension of QQ[sqrt(-7)] of conductor 5

parent()[source]¶

Return the parent of this automorphism, which is a Galois group of a ring class field.

EXAMPLES:

Sage

sage: E = EllipticCurve('389a')
sage: G = E.heegner_point(-7,5).ring_class_field().galois_group()
sage: s = G.complex_conjugation()
sage: s.parent()
Galois group of Ring class field extension of QQ[sqrt(-7)] of conductor 5

Python

>>> from sage.all import *
>>> E = EllipticCurve('389a')
>>> G = E.heegner_point(-Integer(7),Integer(5)).ring_class_field().galois_group()
>>> s = G.complex_conjugation()
>>> s.parent()
Galois group of Ring class field extension of QQ[sqrt(-7)] of conductor 5

class sage.schemes.elliptic_curves.heegner.GaloisAutomorphismComplexConjugation(parent)[source]¶

Bases: GaloisAutomorphism

The complex conjugation automorphism of a ring class field.

EXAMPLES:

Sage

sage: G = heegner_point(37,-7,5).ring_class_field().galois_group()
sage: conj = G.complex_conjugation()
sage: conj
Complex conjugation automorphism of Ring class field extension
 of QQ[sqrt(-7)] of conductor 5
sage: conj.domain()
Ring class field extension of QQ[sqrt(-7)] of conductor 5

Python

>>> from sage.all import *
>>> G = heegner_point(Integer(37),-Integer(7),Integer(5)).ring_class_field().galois_group()
>>> conj = G.complex_conjugation()
>>> conj
Complex conjugation automorphism of Ring class field extension
 of QQ[sqrt(-7)] of conductor 5
>>> conj.domain()
Ring class field extension of QQ[sqrt(-7)] of conductor 5

order()[source]¶

EXAMPLES:

Sage

sage: G = heegner_point(37,-7,5).ring_class_field().galois_group()
sage: conj = G.complex_conjugation()
sage: conj.order()
2

Python

>>> from sage.all import *
>>> G = heegner_point(Integer(37),-Integer(7),Integer(5)).ring_class_field().galois_group()
>>> conj = G.complex_conjugation()
>>> conj.order()
2

class sage.schemes.elliptic_curves.heegner.GaloisAutomorphismQuadraticForm(parent, quadratic_form, alpha=None)[source]¶

Bases: GaloisAutomorphism

An automorphism of a ring class field defined by a quadratic form.

EXAMPLES:

Sage

sage: H = heegner_points(389,-20,3)
sage: sigma = H.ring_class_field().galois_group(H.quadratic_field())[0]; sigma
Class field automorphism defined by x^2 + 45*y^2
sage: type(sigma)
<class 'sage.schemes.elliptic_curves.heegner.GaloisAutomorphismQuadraticForm'>
sage: loads(dumps(sigma)) == sigma
True

Python

>>> from sage.all import *
>>> H = heegner_points(Integer(389),-Integer(20),Integer(3))
>>> sigma = H.ring_class_field().galois_group(H.quadratic_field())[Integer(0)]; sigma
Class field automorphism defined by x^2 + 45*y^2
>>> type(sigma)
<class 'sage.schemes.elliptic_curves.heegner.GaloisAutomorphismQuadraticForm'>
>>> loads(dumps(sigma)) == sigma
True

alpha()[source]¶

Optional data that specified element corresponding element of \((\mathcal{O}_K / c\mathcal{O}_K)^* / (\ZZ/c\ZZ)^*\), via class field theory.

This is a generator of the ideal corresponding to this automorphism.

EXAMPLES:

Sage

sage: K3 = heegner_points(389,-52,3).ring_class_field()
sage: K1 = heegner_points(389,-52,1).ring_class_field()
sage: G = K3.galois_group(K1)
sage: orb = sorted([g.alpha() for g in G]); orb # random (the sign depends on the database being installed or not)
[1, 1/2*sqrt_minus_52 + 1, -1/2*sqrt_minus_52, 1/2*sqrt_minus_52 - 1]
sage: sorted([x^2 for x in orb]) # this is just for testing
[-13, -sqrt_minus_52 - 12, sqrt_minus_52 - 12, 1]

sage: K5 = heegner_points(389,-52,5).ring_class_field()
sage: K1 = heegner_points(389,-52,1).ring_class_field()
sage: G = K5.galois_group(K1)
sage: orb = sorted([g.alpha() for g in G]); orb # random (the sign depends on the database being installed or not)
[1, -1/2*sqrt_minus_52, 1/2*sqrt_minus_52 + 1, 1/2*sqrt_minus_52 - 1,
 1/2*sqrt_minus_52 - 2, -1/2*sqrt_minus_52 - 2]
sage: sorted([x^2 for x in orb]) # just for testing
[-13, -sqrt_minus_52 - 12, sqrt_minus_52 - 12,
 -2*sqrt_minus_52 - 9, 2*sqrt_minus_52 - 9, 1]

Python

>>> from sage.all import *
>>> K3 = heegner_points(Integer(389),-Integer(52),Integer(3)).ring_class_field()
>>> K1 = heegner_points(Integer(389),-Integer(52),Integer(1)).ring_class_field()
>>> G = K3.galois_group(K1)
>>> orb = sorted([g.alpha() for g in G]); orb # random (the sign depends on the database being installed or not)
[1, 1/2*sqrt_minus_52 + 1, -1/2*sqrt_minus_52, 1/2*sqrt_minus_52 - 1]
>>> sorted([x**Integer(2) for x in orb]) # this is just for testing
[-13, -sqrt_minus_52 - 12, sqrt_minus_52 - 12, 1]

>>> K5 = heegner_points(Integer(389),-Integer(52),Integer(5)).ring_class_field()
>>> K1 = heegner_points(Integer(389),-Integer(52),Integer(1)).ring_class_field()
>>> G = K5.galois_group(K1)
>>> orb = sorted([g.alpha() for g in G]); orb # random (the sign depends on the database being installed or not)
[1, -1/2*sqrt_minus_52, 1/2*sqrt_minus_52 + 1, 1/2*sqrt_minus_52 - 1,
 1/2*sqrt_minus_52 - 2, -1/2*sqrt_minus_52 - 2]
>>> sorted([x**Integer(2) for x in orb]) # just for testing
[-13, -sqrt_minus_52 - 12, sqrt_minus_52 - 12,
 -2*sqrt_minus_52 - 9, 2*sqrt_minus_52 - 9, 1]

ideal()[source]¶

Return ideal of ring of integers of quadratic imaginary field corresponding to this quadratic form. This is the ideal

\(I = \left(A, \frac{-B+ c\sqrt{D}}{2}\right) \mathcal{O}_K\).

EXAMPLES:

Sage

sage: E = EllipticCurve('389a'); F = E.heegner_point(-20,3).ring_class_field()
sage: G = F.galois_group(F.quadratic_field())
sage: G[1].ideal()
Fractional ideal (2, 1/2*sqrt_minus_20 + 1)
sage: [s.ideal().gens() for s in G]
[(1, 3/2*sqrt_minus_20), (2, 3/2*sqrt_minus_20 - 1),
 (5, 3/2*sqrt_minus_20), (7, 3/2*sqrt_minus_20 - 2)]

Python

>>> from sage.all import *
>>> E = EllipticCurve('389a'); F = E.heegner_point(-Integer(20),Integer(3)).ring_class_field()
>>> G = F.galois_group(F.quadratic_field())
>>> G[Integer(1)].ideal()
Fractional ideal (2, 1/2*sqrt_minus_20 + 1)
>>> [s.ideal().gens() for s in G]
[(1, 3/2*sqrt_minus_20), (2, 3/2*sqrt_minus_20 - 1),
 (5, 3/2*sqrt_minus_20), (7, 3/2*sqrt_minus_20 - 2)]

order()[source]¶

Return the multiplicative order of this Galois group automorphism.

EXAMPLES:

Sage

sage: K3 = heegner_points(389,-52,3).ring_class_field()
sage: K1 = heegner_points(389,-52,1).ring_class_field()
sage: G = K3.galois_group(K1)
sage: sorted([g.order() for g in G])
[1, 2, 4, 4]
sage: K5 = heegner_points(389,-52,5).ring_class_field()
sage: K1 = heegner_points(389,-52,1).ring_class_field()
sage: G = K5.galois_group(K1)
sage: sorted([g.order() for g in G])
[1, 2, 3, 3, 6, 6]

Python

>>> from sage.all import *
>>> K3 = heegner_points(Integer(389),-Integer(52),Integer(3)).ring_class_field()
>>> K1 = heegner_points(Integer(389),-Integer(52),Integer(1)).ring_class_field()
>>> G = K3.galois_group(K1)
>>> sorted([g.order() for g in G])
[1, 2, 4, 4]
>>> K5 = heegner_points(Integer(389),-Integer(52),Integer(5)).ring_class_field()
>>> K1 = heegner_points(Integer(389),-Integer(52),Integer(1)).ring_class_field()
>>> G = K5.galois_group(K1)
>>> sorted([g.order() for g in G])
[1, 2, 3, 3, 6, 6]

p1_element()[source]¶

Return element of the projective line corresponding to this automorphism.

This only makes sense if this automorphism is in the Galois group \(\textrm{Gal}(K_c/K_1)\).

EXAMPLES:

Sage

sage: K3 = heegner_points(389,-52,3).ring_class_field()
sage: K1 = heegner_points(389,-52,1).ring_class_field()
sage: G = K3.galois_group(K1)
sage: sorted([g.p1_element() for g in G])
[(0, 1), (1, 0), (1, 1), (1, 2)]

sage: K5 = heegner_points(389,-52,5).ring_class_field()
sage: K1 = heegner_points(389,-52,1).ring_class_field()
sage: G = K5.galois_group(K1)
sage: sorted([g.p1_element() for g in G])
[(0, 1), (1, 0), (1, 1), (1, 2), (1, 3), (1, 4)]

Python

>>> from sage.all import *
>>> K3 = heegner_points(Integer(389),-Integer(52),Integer(3)).ring_class_field()
>>> K1 = heegner_points(Integer(389),-Integer(52),Integer(1)).ring_class_field()
>>> G = K3.galois_group(K1)
>>> sorted([g.p1_element() for g in G])
[(0, 1), (1, 0), (1, 1), (1, 2)]

>>> K5 = heegner_points(Integer(389),-Integer(52),Integer(5)).ring_class_field()
>>> K1 = heegner_points(Integer(389),-Integer(52),Integer(1)).ring_class_field()
>>> G = K5.galois_group(K1)
>>> sorted([g.p1_element() for g in G])
[(0, 1), (1, 0), (1, 1), (1, 2), (1, 3), (1, 4)]

quadratic_form()[source]¶

Return reduced quadratic form corresponding to this Galois automorphism.

EXAMPLES:

Sage

sage: H = heegner_points(389,-20,3); s = H.ring_class_field().galois_group(H.quadratic_field())[0]
sage: s.quadratic_form()
x^2 + 45*y^2

Python

>>> from sage.all import *
>>> H = heegner_points(Integer(389),-Integer(20),Integer(3)); s = H.ring_class_field().galois_group(H.quadratic_field())[Integer(0)]
>>> s.quadratic_form()
x^2 + 45*y^2

class sage.schemes.elliptic_curves.heegner.GaloisGroup(field, base=Rational Field)[source]¶

Bases: SageObject

A Galois group of a ring class field.

EXAMPLES:

Sage

sage: E = EllipticCurve('389a')
sage: G = E.heegner_point(-7,5).ring_class_field().galois_group(); G
Galois group of Ring class field extension of QQ[sqrt(-7)] of conductor 5
sage: G.field()
Ring class field extension of QQ[sqrt(-7)] of conductor 5
sage: G.cardinality()
12
sage: G.complex_conjugation()
Complex conjugation automorphism of Ring class field extension of QQ[sqrt(-7)]
 of conductor 5

Python

>>> from sage.all import *
>>> E = EllipticCurve('389a')
>>> G = E.heegner_point(-Integer(7),Integer(5)).ring_class_field().galois_group(); G
Galois group of Ring class field extension of QQ[sqrt(-7)] of conductor 5
>>> G.field()
Ring class field extension of QQ[sqrt(-7)] of conductor 5
>>> G.cardinality()
12
>>> G.complex_conjugation()
Complex conjugation automorphism of Ring class field extension of QQ[sqrt(-7)]
 of conductor 5

base_field()[source]¶

Return the base field, which the field fixed by all the automorphisms in this Galois group.

EXAMPLES:

Sage

sage: x = heegner_point(37,-7,5)
sage: Kc = x.ring_class_field(); Kc
Ring class field extension of QQ[sqrt(-7)] of conductor 5
sage: K = x.quadratic_field()
sage: G = Kc.galois_group(); G
Galois group of Ring class field extension of QQ[sqrt(-7)] of conductor 5
sage: G.base_field()
Rational Field
sage: G.cardinality()
12
sage: Kc.absolute_degree()
12
sage: G = Kc.galois_group(K); G
Galois group of Ring class field extension of QQ[sqrt(-7)] of conductor 5
 over Number Field in sqrt_minus_7 with defining polynomial x^2 + 7
  with sqrt_minus_7 = 2.645751311064591?*I
sage: G.cardinality()
6
sage: G.base_field()
Number Field in sqrt_minus_7 with defining polynomial x^2 + 7
 with sqrt_minus_7 = 2.645751311064591?*I
sage: G = Kc.galois_group(Kc); G
Galois group of Ring class field extension of QQ[sqrt(-7)] of conductor 5
 over Ring class field extension of QQ[sqrt(-7)] of conductor 5
sage: G.cardinality()
1
sage: G.base_field()
Ring class field extension of QQ[sqrt(-7)] of conductor 5

Python

>>> from sage.all import *
>>> x = heegner_point(Integer(37),-Integer(7),Integer(5))
>>> Kc = x.ring_class_field(); Kc
Ring class field extension of QQ[sqrt(-7)] of conductor 5
>>> K = x.quadratic_field()
>>> G = Kc.galois_group(); G
Galois group of Ring class field extension of QQ[sqrt(-7)] of conductor 5
>>> G.base_field()
Rational Field
>>> G.cardinality()
12
>>> Kc.absolute_degree()
12
>>> G = Kc.galois_group(K); G
Galois group of Ring class field extension of QQ[sqrt(-7)] of conductor 5
 over Number Field in sqrt_minus_7 with defining polynomial x^2 + 7
  with sqrt_minus_7 = 2.645751311064591?*I
>>> G.cardinality()
6
>>> G.base_field()
Number Field in sqrt_minus_7 with defining polynomial x^2 + 7
 with sqrt_minus_7 = 2.645751311064591?*I
>>> G = Kc.galois_group(Kc); G
Galois group of Ring class field extension of QQ[sqrt(-7)] of conductor 5
 over Ring class field extension of QQ[sqrt(-7)] of conductor 5
>>> G.cardinality()
1
>>> G.base_field()
Ring class field extension of QQ[sqrt(-7)] of conductor 5

cardinality()[source]¶

Return the cardinality of this Galois group.

EXAMPLES:

Sage

sage: E = EllipticCurve('389a')
sage: G = E.heegner_point(-7,5).ring_class_field().galois_group(); G
Galois group of Ring class field extension of QQ[sqrt(-7)] of conductor 5
sage: G.cardinality()
12
sage: G = E.heegner_point(-7).ring_class_field().galois_group()
sage: G.cardinality()
2
sage: G = E.heegner_point(-7,55).ring_class_field().galois_group()
sage: G.cardinality()
120

Python

>>> from sage.all import *
>>> E = EllipticCurve('389a')
>>> G = E.heegner_point(-Integer(7),Integer(5)).ring_class_field().galois_group(); G
Galois group of Ring class field extension of QQ[sqrt(-7)] of conductor 5
>>> G.cardinality()
12
>>> G = E.heegner_point(-Integer(7)).ring_class_field().galois_group()
>>> G.cardinality()
2
>>> G = E.heegner_point(-Integer(7),Integer(55)).ring_class_field().galois_group()
>>> G.cardinality()
120

complex_conjugation()[source]¶

Return the automorphism of self determined by complex conjugation. The base field must be the rational numbers.

EXAMPLES:

Sage

sage: E = EllipticCurve('389a')
sage: G = E.heegner_point(-7,5).ring_class_field().galois_group()
sage: G.complex_conjugation()
Complex conjugation automorphism of Ring class field extension
 of QQ[sqrt(-7)] of conductor 5

Python

>>> from sage.all import *
>>> E = EllipticCurve('389a')
>>> G = E.heegner_point(-Integer(7),Integer(5)).ring_class_field().galois_group()
>>> G.complex_conjugation()
Complex conjugation automorphism of Ring class field extension
 of QQ[sqrt(-7)] of conductor 5

field()[source]¶

Return the ring class field that this Galois group acts on.

EXAMPLES:

Sage

sage: G = heegner_point(389,-7,5).ring_class_field().galois_group()
sage: G.field()
Ring class field extension of QQ[sqrt(-7)] of conductor 5

Python

>>> from sage.all import *
>>> G = heegner_point(Integer(389),-Integer(7),Integer(5)).ring_class_field().galois_group()
>>> G.field()
Ring class field extension of QQ[sqrt(-7)] of conductor 5

is_kolyvagin()[source]¶

Return True if conductor \(c\) is prime to the discriminant of the quadratic field, \(c\) is squarefree and each prime dividing \(c\) is inert.

EXAMPLES:

Sage

sage: K5 = heegner_points(389,-52,5).ring_class_field()
sage: K1 = heegner_points(389,-52,1).ring_class_field()
sage: K5.galois_group(K1).is_kolyvagin()
True
sage: K7 = heegner_points(389,-52,7).ring_class_field()
sage: K7.galois_group(K1).is_kolyvagin()
False
sage: K25 = heegner_points(389,-52,25).ring_class_field()
sage: K25.galois_group(K1).is_kolyvagin()
False

Python

>>> from sage.all import *
>>> K5 = heegner_points(Integer(389),-Integer(52),Integer(5)).ring_class_field()
>>> K1 = heegner_points(Integer(389),-Integer(52),Integer(1)).ring_class_field()
>>> K5.galois_group(K1).is_kolyvagin()
True
>>> K7 = heegner_points(Integer(389),-Integer(52),Integer(7)).ring_class_field()
>>> K7.galois_group(K1).is_kolyvagin()
False
>>> K25 = heegner_points(Integer(389),-Integer(52),Integer(25)).ring_class_field()
>>> K25.galois_group(K1).is_kolyvagin()
False

kolyvagin_generators()[source]¶

Assuming this Galois group \(G\) is of the form \(G=\textrm{Gal}(K_c/K_1)\), with \(c=p_1\dots p_n\) satisfying the Kolyvagin hypothesis, this function returns noncanonical choices of lifts of generators for each of the cyclic factors of \(G\) corresponding to the primes dividing \(c\). Thus the \(i\)-th returned valued is an element of \(G\) that maps to the identity element of \(\textrm{Gal}(K_p/K_1)\) for all \(p \neq p_i\) and to a choice of generator of \(\textrm{Gal}(K_{p_i}/K_1)\).

OUTPUT: list of elements of self

EXAMPLES:

Sage

sage: K3 = heegner_points(389,-52,3).ring_class_field()
sage: K1 = heegner_points(389,-52,1).ring_class_field()
sage: G = K3.galois_group(K1)
sage: G.kolyvagin_generators()
(Class field automorphism defined by 9*x^2 - 6*x*y + 14*y^2,)

sage: K5 = heegner_points(389,-52,5).ring_class_field()
sage: K1 = heegner_points(389,-52,1).ring_class_field()
sage: G = K5.galois_group(K1)
sage: G.kolyvagin_generators()
(Class field automorphism defined by 17*x^2 - 14*x*y + 22*y^2,)

Python

>>> from sage.all import *
>>> K3 = heegner_points(Integer(389),-Integer(52),Integer(3)).ring_class_field()
>>> K1 = heegner_points(Integer(389),-Integer(52),Integer(1)).ring_class_field()
>>> G = K3.galois_group(K1)
>>> G.kolyvagin_generators()
(Class field automorphism defined by 9*x^2 - 6*x*y + 14*y^2,)

>>> K5 = heegner_points(Integer(389),-Integer(52),Integer(5)).ring_class_field()
>>> K1 = heegner_points(Integer(389),-Integer(52),Integer(1)).ring_class_field()
>>> G = K5.galois_group(K1)
>>> G.kolyvagin_generators()
(Class field automorphism defined by 17*x^2 - 14*x*y + 22*y^2,)

lift_of_hilbert_class_field_galois_group()[source]¶

Assuming this Galois group \(G\) is of the form \(G=\textrm{Gal}(K_c/K)\), this function returns noncanonical choices of lifts of the elements of the quotient group \(\textrm{Gal}(K_1/K)\).

OUTPUT: tuple of elements of self

EXAMPLES:

Sage

sage: K5 = heegner_points(389,-52,5).ring_class_field()
sage: G = K5.galois_group(K5.quadratic_field())
sage: G.lift_of_hilbert_class_field_galois_group()
(Class field automorphism defined by x^2 + 325*y^2,
 Class field automorphism defined by 2*x^2 + 2*x*y + 163*y^2)
sage: G.cardinality()
12
sage: K5.quadratic_field().class_number()
2

Python

>>> from sage.all import *
>>> K5 = heegner_points(Integer(389),-Integer(52),Integer(5)).ring_class_field()
>>> G = K5.galois_group(K5.quadratic_field())
>>> G.lift_of_hilbert_class_field_galois_group()
(Class field automorphism defined by x^2 + 325*y^2,
 Class field automorphism defined by 2*x^2 + 2*x*y + 163*y^2)
>>> G.cardinality()
12
>>> K5.quadratic_field().class_number()
2

class sage.schemes.elliptic_curves.heegner.HeegnerPoint(N, D, c)[source]¶

Bases: SageObject

A Heegner point of level \(N\), discriminant \(D\) and conductor \(c\) is any point on a modular curve or elliptic curve that is concocted in some way from a quadratic imaginary \(\tau\) in the upper half plane with \(\Delta(\tau) = D c = \Delta(N \tau)\).

EXAMPLES:

Sage

sage: x = sage.schemes.elliptic_curves.heegner.HeegnerPoint(389,-7,13); x
Heegner point of level 389, discriminant -7, and conductor 13
sage: type(x)
<class 'sage.schemes.elliptic_curves.heegner.HeegnerPoint'>
sage: loads(dumps(x)) == x
True

Python

>>> from sage.all import *
>>> x = sage.schemes.elliptic_curves.heegner.HeegnerPoint(Integer(389),-Integer(7),Integer(13)); x
Heegner point of level 389, discriminant -7, and conductor 13
>>> type(x)
<class 'sage.schemes.elliptic_curves.heegner.HeegnerPoint'>
>>> loads(dumps(x)) == x
True

conductor()[source]¶

Return the conductor of this Heegner point.

EXAMPLES:

Sage

sage: heegner_point(389,-7,5).conductor()
5
sage: E = EllipticCurve('37a1'); P = E.kolyvagin_point(-67,7); P
Kolyvagin point of discriminant -67 and conductor 7
 on elliptic curve of conductor 37
sage: P.conductor()
7
sage: E = EllipticCurve('389a'); P = E.heegner_point(-7, 5); P.conductor()
5

Python

>>> from sage.all import *
>>> heegner_point(Integer(389),-Integer(7),Integer(5)).conductor()
5
>>> E = EllipticCurve('37a1'); P = E.kolyvagin_point(-Integer(67),Integer(7)); P
Kolyvagin point of discriminant -67 and conductor 7
 on elliptic curve of conductor 37
>>> P.conductor()
7
>>> E = EllipticCurve('389a'); P = E.heegner_point(-Integer(7), Integer(5)); P.conductor()
5

discriminant()[source]¶

Return the discriminant of the quadratic imaginary field associated to this Heegner point.

EXAMPLES:

Sage

sage: heegner_point(389,-7,5).discriminant()
-7
sage: E = EllipticCurve('37a1'); P = E.kolyvagin_point(-67,7); P
Kolyvagin point of discriminant -67 and conductor 7
 on elliptic curve of conductor 37
sage: P.discriminant()
-67
sage: E = EllipticCurve('389a'); P = E.heegner_point(-7, 5); P.discriminant()
-7

Python

>>> from sage.all import *
>>> heegner_point(Integer(389),-Integer(7),Integer(5)).discriminant()
-7
>>> E = EllipticCurve('37a1'); P = E.kolyvagin_point(-Integer(67),Integer(7)); P
Kolyvagin point of discriminant -67 and conductor 7
 on elliptic curve of conductor 37
>>> P.discriminant()
-67
>>> E = EllipticCurve('389a'); P = E.heegner_point(-Integer(7), Integer(5)); P.discriminant()
-7

level()[source]¶

Return the level of this Heegner point, which is the level of the modular curve \(X_0(N)\) on which this is a Heegner point.

EXAMPLES:

Sage

sage: heegner_point(389,-7,5).level()
389

Python

>>> from sage.all import *
>>> heegner_point(Integer(389),-Integer(7),Integer(5)).level()
389

quadratic_field()[source]¶

Return the quadratic number field of discriminant \(D\).

EXAMPLES:

Sage

sage: x = heegner_point(37,-7,5)
sage: x.quadratic_field()
Number Field in sqrt_minus_7 with defining polynomial x^2 + 7
 with sqrt_minus_7 = 2.645751311064591?*I

sage: E = EllipticCurve('37a'); P = E.heegner_point(-40)
sage: P.quadratic_field()
Number Field in sqrt_minus_40 with defining polynomial x^2 + 40
 with sqrt_minus_40 = 6.324555320336759?*I
sage: P.quadratic_field() is P.quadratic_field()
True
sage: type(P.quadratic_field())
<class 'sage.rings.number_field.number_field.NumberField_quadratic_with_category'>

Python

>>> from sage.all import *
>>> x = heegner_point(Integer(37),-Integer(7),Integer(5))
>>> x.quadratic_field()
Number Field in sqrt_minus_7 with defining polynomial x^2 + 7
 with sqrt_minus_7 = 2.645751311064591?*I

>>> E = EllipticCurve('37a'); P = E.heegner_point(-Integer(40))
>>> P.quadratic_field()
Number Field in sqrt_minus_40 with defining polynomial x^2 + 40
 with sqrt_minus_40 = 6.324555320336759?*I
>>> P.quadratic_field() is P.quadratic_field()
True
>>> type(P.quadratic_field())
<class 'sage.rings.number_field.number_field.NumberField_quadratic_with_category'>

quadratic_order()[source]¶

Return the order in the quadratic imaginary field of conductor \(c\), where \(c\) is the conductor of this Heegner point.

EXAMPLES:

Sage

sage: heegner_point(389,-7,5).quadratic_order()
Order of conductor 10 generated by 5*sqrt_minus_7
 in Number Field in sqrt_minus_7 with defining polynomial x^2 + 7
 with sqrt_minus_7 = 2.645751311064591?*I
sage: heegner_point(389,-7,5).quadratic_order().basis()
[1, 5*sqrt_minus_7]

sage: E = EllipticCurve('37a'); P = E.heegner_point(-40,11)
sage: P.quadratic_order()
Order of conductor 22 generated by 11*sqrt_minus_40
 in Number Field in sqrt_minus_40 with defining polynomial x^2 + 40
 with sqrt_minus_40 = 6.324555320336759?*I
sage: P.quadratic_order().basis()
[1, 11*sqrt_minus_40]

Python

>>> from sage.all import *
>>> heegner_point(Integer(389),-Integer(7),Integer(5)).quadratic_order()
Order of conductor 10 generated by 5*sqrt_minus_7
 in Number Field in sqrt_minus_7 with defining polynomial x^2 + 7
 with sqrt_minus_7 = 2.645751311064591?*I
>>> heegner_point(Integer(389),-Integer(7),Integer(5)).quadratic_order().basis()
[1, 5*sqrt_minus_7]

>>> E = EllipticCurve('37a'); P = E.heegner_point(-Integer(40),Integer(11))
>>> P.quadratic_order()
Order of conductor 22 generated by 11*sqrt_minus_40
 in Number Field in sqrt_minus_40 with defining polynomial x^2 + 40
 with sqrt_minus_40 = 6.324555320336759?*I
>>> P.quadratic_order().basis()
[1, 11*sqrt_minus_40]

ring_class_field()[source]¶

Return the ring class field associated to this Heegner point. This is an extension \(K_c\) over \(K\), where \(K\) is the quadratic imaginary field and \(c\) is the conductor associated to this Heegner point. This Heegner point is defined over \(K_c\) and the Galois group \(Gal(K_c/K)\) acts transitively on the Galois conjugates of this Heegner point.

EXAMPLES:

Sage

sage: E = EllipticCurve('389a'); K.<a> = QuadraticField(-5)
sage: len(K.factor(5))
1
sage: len(K.factor(23))
2
sage: E.heegner_point(-7, 5).ring_class_field().degree_over_K()
6
sage: E.heegner_point(-7, 23).ring_class_field().degree_over_K()
22
sage: E.heegner_point(-7, 5*23).ring_class_field().degree_over_K()
132
sage: E.heegner_point(-7, 5^2).ring_class_field().degree_over_K()
30
sage: E.heegner_point(-7, 7).ring_class_field().degree_over_K()
7

Python

>>> from sage.all import *
>>> E = EllipticCurve('389a'); K = QuadraticField(-Integer(5), names=('a',)); (a,) = K._first_ngens(1)
>>> len(K.factor(Integer(5)))
1
>>> len(K.factor(Integer(23)))
2
>>> E.heegner_point(-Integer(7), Integer(5)).ring_class_field().degree_over_K()
6
>>> E.heegner_point(-Integer(7), Integer(23)).ring_class_field().degree_over_K()
22
>>> E.heegner_point(-Integer(7), Integer(5)*Integer(23)).ring_class_field().degree_over_K()
132
>>> E.heegner_point(-Integer(7), Integer(5)**Integer(2)).ring_class_field().degree_over_K()
30
>>> E.heegner_point(-Integer(7), Integer(7)).ring_class_field().degree_over_K()
7

class sage.schemes.elliptic_curves.heegner.HeegnerPointOnEllipticCurve(E, x, check=True)[source]¶

Bases: HeegnerPoint

A Heegner point on a curve associated to an order in a quadratic imaginary field.

EXAMPLES:

Sage

sage: E = EllipticCurve('37a'); P = E.heegner_point(-7,5); P
Heegner point of discriminant -7 and conductor 5 on elliptic curve of conductor 37
sage: type(P)
<class 'sage.schemes.elliptic_curves.heegner.HeegnerPointOnEllipticCurve'>

Python

>>> from sage.all import *
>>> E = EllipticCurve('37a'); P = E.heegner_point(-Integer(7),Integer(5)); P
Heegner point of discriminant -7 and conductor 5 on elliptic curve of conductor 37
>>> type(P)
<class 'sage.schemes.elliptic_curves.heegner.HeegnerPointOnEllipticCurve'>

conjugates_over_K()[source]¶

Return the \(Gal(K_c/K)\) conjugates of this Heegner point.

EXAMPLES:

Sage

sage: E = EllipticCurve('77a')
sage: y = E.heegner_point(-52,5); y
Heegner point of discriminant -52 and conductor 5
 on elliptic curve of conductor 77
sage: print([z.quadratic_form() for z in y.conjugates_over_K()])
[77*x^2 + 52*x*y + 13*y^2, 154*x^2 + 206*x*y + 71*y^2, 539*x^2 + 822*x*y + 314*y^2,
 847*x^2 + 1284*x*y + 487*y^2, 1001*x^2 + 52*x*y + y^2, 1078*x^2 + 822*x*y + 157*y^2,
 1309*x^2 + 360*x*y + 25*y^2, 1309*x^2 + 2054*x*y + 806*y^2,
 1463*x^2 + 976*x*y + 163*y^2, 2233*x^2 + 2824*x*y + 893*y^2,
 2387*x^2 + 2054*x*y + 442*y^2, 3619*x^2 + 3286*x*y + 746*y^2]
sage: y.quadratic_form()
77*x^2 + 52*x*y + 13*y^2

Python

>>> from sage.all import *
>>> E = EllipticCurve('77a')
>>> y = E.heegner_point(-Integer(52),Integer(5)); y
Heegner point of discriminant -52 and conductor 5
 on elliptic curve of conductor 77
>>> print([z.quadratic_form() for z in y.conjugates_over_K()])
[77*x^2 + 52*x*y + 13*y^2, 154*x^2 + 206*x*y + 71*y^2, 539*x^2 + 822*x*y + 314*y^2,
 847*x^2 + 1284*x*y + 487*y^2, 1001*x^2 + 52*x*y + y^2, 1078*x^2 + 822*x*y + 157*y^2,
 1309*x^2 + 360*x*y + 25*y^2, 1309*x^2 + 2054*x*y + 806*y^2,
 1463*x^2 + 976*x*y + 163*y^2, 2233*x^2 + 2824*x*y + 893*y^2,
 2387*x^2 + 2054*x*y + 442*y^2, 3619*x^2 + 3286*x*y + 746*y^2]
>>> y.quadratic_form()
77*x^2 + 52*x*y + 13*y^2

curve()[source]¶

Return the elliptic curve on which this is a Heegner point.

EXAMPLES:

Sage

sage: E = EllipticCurve('389a'); P = E.heegner_point(-7, 5)
sage: P.curve()
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field
sage: P.curve() is E
True

Python

>>> from sage.all import *
>>> E = EllipticCurve('389a'); P = E.heegner_point(-Integer(7), Integer(5))
>>> P.curve()
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field
>>> P.curve() is E
True

heegner_point_on_X0N()[source]¶

Return Heegner point on \(X_0(N)\) that maps to this Heegner point on \(E\).

EXAMPLES:

Sage

sage: E = EllipticCurve('37a'); P = E.heegner_point(-7,5); P
Heegner point of discriminant -7 and conductor 5 on elliptic curve
 of conductor 37
sage: P.heegner_point_on_X0N()
Heegner point 5/74*sqrt(-7) - 11/74 of discriminant -7 and conductor 5
 on X_0(37)

Python

>>> from sage.all import *
>>> E = EllipticCurve('37a'); P = E.heegner_point(-Integer(7),Integer(5)); P
Heegner point of discriminant -7 and conductor 5 on elliptic curve
 of conductor 37
>>> P.heegner_point_on_X0N()
Heegner point 5/74*sqrt(-7) - 11/74 of discriminant -7 and conductor 5
 on X_0(37)

kolyvagin_cohomology_class(n=None)[source]¶

Return the Kolyvagin class associated to this Heegner point.

INPUT:

n – positive integer that divides the gcd of \(a_p\) and \(p+1\) for all \(p\) dividing the conductor. If \(n\) is None, choose the largest valid \(n\).

EXAMPLES:

Sage

sage: y = EllipticCurve('389a').heegner_point(-7,5)
sage: y.kolyvagin_cohomology_class(3)
Kolyvagin cohomology class c(5) in H^1(K,E[3])

Python

>>> from sage.all import *
>>> y = EllipticCurve('389a').heegner_point(-Integer(7),Integer(5))
>>> y.kolyvagin_cohomology_class(Integer(3))
Kolyvagin cohomology class c(5) in H^1(K,E[3])

kolyvagin_point()[source]¶

Return the Kolyvagin point corresponding to this Heegner point.

This is the point obtained by applying the Kolyvagin operator \(J_c I_c\) in the group ring of the Galois group to this Heegner point. It is a point that defines an element of \(H^1(K, E[n])\), under certain hypotheses on \(n\).

EXAMPLES:

Sage

sage: E = EllipticCurve('37a1'); y = E.heegner_point(-7); y
Heegner point of discriminant -7 on elliptic curve of conductor 37
sage: P = y.kolyvagin_point(); P
Kolyvagin point of discriminant -7 on elliptic curve of conductor 37
sage: P.numerical_approx()  # abs tol 1e-15
(-3.36910401903861e-16 - 2.22076195576076e-16*I
 : 3.33066907387547e-16 + 2.22076195576075e-16*I : 1.00000000000000)

Python

>>> from sage.all import *
>>> E = EllipticCurve('37a1'); y = E.heegner_point(-Integer(7)); y
Heegner point of discriminant -7 on elliptic curve of conductor 37
>>> P = y.kolyvagin_point(); P
Kolyvagin point of discriminant -7 on elliptic curve of conductor 37
>>> P.numerical_approx()  # abs tol 1e-15
(-3.36910401903861e-16 - 2.22076195576076e-16*I
 : 3.33066907387547e-16 + 2.22076195576075e-16*I : 1.00000000000000)

map_to_complex_numbers(prec=53)[source]¶

Return the point in the subfield \(M\) of the complex numbers (well defined only modulo the period lattice) corresponding to this Heegner point.

EXAMPLES:

We compute a nonzero Heegner point over a ring class field on a curve of rank 2:

Sage

sage: E = EllipticCurve('389a'); y = E.heegner_point(-7,5)
sage: y.map_to_complex_numbers()
1.49979679635196 + 0.369156204821526*I
sage: y.map_to_complex_numbers(100)
1.4997967963519640592142411892 + 0.36915620482152626830089145962*I
sage: y.map_to_complex_numbers(10)
1.5 + 0.37*I

Python

>>> from sage.all import *
>>> E = EllipticCurve('389a'); y = E.heegner_point(-Integer(7),Integer(5))
>>> y.map_to_complex_numbers()
1.49979679635196 + 0.369156204821526*I
>>> y.map_to_complex_numbers(Integer(100))
1.4997967963519640592142411892 + 0.36915620482152626830089145962*I
>>> y.map_to_complex_numbers(Integer(10))
1.5 + 0.37*I

Here we see that the Heegner point is 0 since it lies in the lattice:

Sage

sage: E = EllipticCurve('389a'); y = E.heegner_point(-7)
sage: y.map_to_complex_numbers(10)
0.0034 - 3.9*I
sage: y.map_to_complex_numbers()
4.71844785465692e-15 - 3.94347540310330*I
sage: E.period_lattice().basis()
(2.49021256085505, 1.97173770155165*I)
sage: 2*E.period_lattice().basis()[1]
3.94347540310330*I

Python

>>> from sage.all import *
>>> E = EllipticCurve('389a'); y = E.heegner_point(-Integer(7))
>>> y.map_to_complex_numbers(Integer(10))
0.0034 - 3.9*I
>>> y.map_to_complex_numbers()
4.71844785465692e-15 - 3.94347540310330*I
>>> E.period_lattice().basis()
(2.49021256085505, 1.97173770155165*I)
>>> Integer(2)*E.period_lattice().basis()[Integer(1)]
3.94347540310330*I

You can also directly coerce to the complex field:

Sage

sage: E = EllipticCurve('389a'); y = E.heegner_point(-7)
sage: z = ComplexField(100)(y); z # real part approx. 0
-... - 3.9434754031032964088448153963*I
sage: E.period_lattice().elliptic_exponential(z)
(0.00000000000000000000000000000 : 1.0000000000000000000000000000 : 0.00000000000000000000000000000)

Python

>>> from sage.all import *
>>> E = EllipticCurve('389a'); y = E.heegner_point(-Integer(7))
>>> z = ComplexField(Integer(100))(y); z # real part approx. 0
-... - 3.9434754031032964088448153963*I
>>> E.period_lattice().elliptic_exponential(z)
(0.00000000000000000000000000000 : 1.0000000000000000000000000000 : 0.00000000000000000000000000000)

numerical_approx(prec=53, algorithm=None)[source]¶

Return a numerical approximation to this Heegner point computed using a working precision of prec bits.

Warning

The answer is not provably correct to prec bits! A priori, due to rounding and other errors, it is possible that not a single digit is correct.

INPUT:

prec – (default: None) the working precision

EXAMPLES:

Sage

sage: E = EllipticCurve('37a'); P = E.heegner_point(-7); P
Heegner point of discriminant -7 on elliptic curve of conductor 37
sage: P.numerical_approx()  # abs tol 1e-15
(-3.36910401903861e-16 - 2.22076195576076e-16*I
 : 3.33066907387547e-16 + 2.22076195576075e-16*I : 1.00000000000000)
sage: P.numerical_approx(10)  # expect random digits
(0.0030 - 0.0028*I : -0.0030 + 0.0028*I : 1.0)
sage: P.numerical_approx(100)[0]  # expect random digits
8.4...e-31 + 6.0...e-31*I
sage: E = EllipticCurve('37a'); P = E.heegner_point(-40); P
Heegner point of discriminant -40 on elliptic curve of conductor 37
sage: P.numerical_approx()  # abs tol 1e-14
(-3.15940603400359e-16 + 1.41421356237309*I
 : 1.00000000000000 - 1.41421356237309*I : 1.00000000000000)

Python

>>> from sage.all import *
>>> E = EllipticCurve('37a'); P = E.heegner_point(-Integer(7)); P
Heegner point of discriminant -7 on elliptic curve of conductor 37
>>> P.numerical_approx()  # abs tol 1e-15
(-3.36910401903861e-16 - 2.22076195576076e-16*I
 : 3.33066907387547e-16 + 2.22076195576075e-16*I : 1.00000000000000)
>>> P.numerical_approx(Integer(10))  # expect random digits
(0.0030 - 0.0028*I : -0.0030 + 0.0028*I : 1.0)
>>> P.numerical_approx(Integer(100))[Integer(0)]  # expect random digits
8.4...e-31 + 6.0...e-31*I
>>> E = EllipticCurve('37a'); P = E.heegner_point(-Integer(40)); P
Heegner point of discriminant -40 on elliptic curve of conductor 37
>>> P.numerical_approx()  # abs tol 1e-14
(-3.15940603400359e-16 + 1.41421356237309*I
 : 1.00000000000000 - 1.41421356237309*I : 1.00000000000000)

A rank 2 curve, where all Heegner points of conductor 1 are 0:

Sage

sage: E = EllipticCurve('389a'); E.rank()
2
sage: P = E.heegner_point(-7); P
Heegner point of discriminant -7 on elliptic curve of conductor 389
sage: P.numerical_approx()
(0.000000000000000 : 1.00000000000000 : 0.000000000000000)

Python

>>> from sage.all import *
>>> E = EllipticCurve('389a'); E.rank()
2
>>> P = E.heegner_point(-Integer(7)); P
Heegner point of discriminant -7 on elliptic curve of conductor 389
>>> P.numerical_approx()
(0.000000000000000 : 1.00000000000000 : 0.000000000000000)

However, Heegner points of bigger conductor are often nonzero:

Sage

sage: E = EllipticCurve('389a'); P = E.heegner_point(-7, 5); P
Heegner point of discriminant -7 and conductor 5 on elliptic curve
 of conductor 389
sage: numerical_approx(P)
(0.675507556926807 + 0.344749649302635*I
 : -0.377142931401887 + 0.843366227137146*I : 1.00000000000000)
sage: P.numerical_approx()
(0.6755075569268... + 0.3447496493026...*I
 : -0.3771429314018... + 0.8433662271371...*I : 1.00000000000000)
sage: E.heegner_point(-7, 11).numerical_approx()
(0.1795583794118... + 0.02035501750912...*I
 : -0.5573941377055... + 0.2738940831635...*I : 1.00000000000000)
sage: E.heegner_point(-7, 13).numerical_approx()
(1.034302915374... - 3.302744319777...*I
 : 1.323937875767... + 6.908264226850...*I : 1.00000000000000)

Python

>>> from sage.all import *
>>> E = EllipticCurve('389a'); P = E.heegner_point(-Integer(7), Integer(5)); P
Heegner point of discriminant -7 and conductor 5 on elliptic curve
 of conductor 389
>>> numerical_approx(P)
(0.675507556926807 + 0.344749649302635*I
 : -0.377142931401887 + 0.843366227137146*I : 1.00000000000000)
>>> P.numerical_approx()
(0.6755075569268... + 0.3447496493026...*I
 : -0.3771429314018... + 0.8433662271371...*I : 1.00000000000000)
>>> E.heegner_point(-Integer(7), Integer(11)).numerical_approx()
(0.1795583794118... + 0.02035501750912...*I
 : -0.5573941377055... + 0.2738940831635...*I : 1.00000000000000)
>>> E.heegner_point(-Integer(7), Integer(13)).numerical_approx()
(1.034302915374... - 3.302744319777...*I
 : 1.323937875767... + 6.908264226850...*I : 1.00000000000000)

We find (probably) the defining polynomial of the \(x\)-coordinate of \(P\), which defines a class field. The shape of the discriminant below is strong confirmation – but not proof – that this polynomial is correct:

Sage

sage: f = P.numerical_approx(70)[0].algdep(6); f
1225*x^6 + 1750*x^5 - 21675*x^4 - 380*x^3 + 110180*x^2 - 129720*x + 48771
sage: f.discriminant().factor()
2^6 * 3^2 * 5^11 * 7^4 * 13^2 * 19^6 * 199^2 * 719^2 * 26161^2

Python

>>> from sage.all import *
>>> f = P.numerical_approx(Integer(70))[Integer(0)].algdep(Integer(6)); f
1225*x^6 + 1750*x^5 - 21675*x^4 - 380*x^3 + 110180*x^2 - 129720*x + 48771
>>> f.discriminant().factor()
2^6 * 3^2 * 5^11 * 7^4 * 13^2 * 19^6 * 199^2 * 719^2 * 26161^2

point_exact(prec=53, algorithm='lll', var='a', optimize=False)[source]¶

Return exact point on the elliptic curve over a number field defined by computing this Heegner point to the given number of bits of precision. A ValueError is raised if the precision is clearly insignificant to define a point on the curve.

Warning

It is in theory possible for this function to not raise a ValueError, find a point on the curve, but via some very unlikely coincidence that point is not actually this Heegner point.

Warning

Currently we make an arbitrary choice of \(y\)-coordinate for the lift of the \(x\)-coordinate.

INPUT:

prec – integer (default: 53)
algorithm – see the description of the algorithm parameter for the x_poly_exact method
var – string (default: 'a')
optimize – boolean (default: False); if True, try to optimize defining polynomial for the number field that the point is defined over. Off by default, since this can be very expensive.

EXAMPLES:

Sage

sage: E = EllipticCurve('389a'); P = E.heegner_point(-7, 5); P
Heegner point of discriminant -7 and conductor 5
 on elliptic curve of conductor 389
sage: z = P.point_exact(200, optimize=True)
sage: z[1].charpoly()
x^12 + 6*x^11 + 90089/1715*x^10 + 71224/343*x^9 + 52563964/588245*x^8 - 483814934/588245*x^7 - 156744579/16807*x^6 - 2041518032/84035*x^5 + 1259355443184/14706125*x^4 + 3094420220918/14706125*x^3 + 123060442043827/367653125*x^2 + 82963044474852/367653125*x + 211679465261391/1838265625
sage: f = P.numerical_approx(500)[1].algdep(12); f / f.leading_coefficient()
x^12 + 6*x^11 + 90089/1715*x^10 + 71224/343*x^9 + 52563964/588245*x^8 - 483814934/588245*x^7 - 156744579/16807*x^6 - 2041518032/84035*x^5 + 1259355443184/14706125*x^4 + 3094420220918/14706125*x^3 + 123060442043827/367653125*x^2 + 82963044474852/367653125*x + 211679465261391/1838265625

sage: E = EllipticCurve('5077a')
sage: P = E.heegner_point(-7)
sage: P.point_exact(prec=100)
(0 : 1 : 0)

Python

>>> from sage.all import *
>>> E = EllipticCurve('389a'); P = E.heegner_point(-Integer(7), Integer(5)); P
Heegner point of discriminant -7 and conductor 5
 on elliptic curve of conductor 389
>>> z = P.point_exact(Integer(200), optimize=True)
>>> z[Integer(1)].charpoly()
x^12 + 6*x^11 + 90089/1715*x^10 + 71224/343*x^9 + 52563964/588245*x^8 - 483814934/588245*x^7 - 156744579/16807*x^6 - 2041518032/84035*x^5 + 1259355443184/14706125*x^4 + 3094420220918/14706125*x^3 + 123060442043827/367653125*x^2 + 82963044474852/367653125*x + 211679465261391/1838265625
>>> f = P.numerical_approx(Integer(500))[Integer(1)].algdep(Integer(12)); f / f.leading_coefficient()
x^12 + 6*x^11 + 90089/1715*x^10 + 71224/343*x^9 + 52563964/588245*x^8 - 483814934/588245*x^7 - 156744579/16807*x^6 - 2041518032/84035*x^5 + 1259355443184/14706125*x^4 + 3094420220918/14706125*x^3 + 123060442043827/367653125*x^2 + 82963044474852/367653125*x + 211679465261391/1838265625

>>> E = EllipticCurve('5077a')
>>> P = E.heegner_point(-Integer(7))
>>> P.point_exact(prec=Integer(100))
(0 : 1 : 0)

quadratic_form()[source]¶

Return the integral primitive positive definite binary quadratic form associated to this Heegner point.

EXAMPLES:

Sage

sage: EllipticCurve('389a').heegner_point(-7, 5).quadratic_form()
389*x^2 + 147*x*y + 14*y^2

sage: P = EllipticCurve('389a').heegner_point(-7, 5, (778,925,275)); P
Heegner point of discriminant -7 and conductor 5 on elliptic curve
 of conductor 389
sage: P.quadratic_form()
778*x^2 + 925*x*y + 275*y^2

Python

>>> from sage.all import *
>>> EllipticCurve('389a').heegner_point(-Integer(7), Integer(5)).quadratic_form()
389*x^2 + 147*x*y + 14*y^2

>>> P = EllipticCurve('389a').heegner_point(-Integer(7), Integer(5), (Integer(778),Integer(925),Integer(275))); P
Heegner point of discriminant -7 and conductor 5 on elliptic curve
 of conductor 389
>>> P.quadratic_form()
778*x^2 + 925*x*y + 275*y^2

satisfies_kolyvagin_hypothesis(n=None)[source]¶

Return True if this Heegner point and \(n\) satisfy the Kolyvagin hypothesis, i.e., that each prime dividing the conductor \(c\) of self is inert in K and coprime to \(ND\). Moreover, if \(n\) is not None, also check that for each prime \(p\) dividing \(c\) we have that \(n | \gcd(a_p(E), p+1)\).

INPUT:

n – positive integer

EXAMPLES:

Sage

sage: EllipticCurve('389a').heegner_point(-7).satisfies_kolyvagin_hypothesis()
True
sage: EllipticCurve('389a').heegner_point(-7, 5).satisfies_kolyvagin_hypothesis()
True
sage: EllipticCurve('389a').heegner_point(-7, 11).satisfies_kolyvagin_hypothesis()
False

Python

>>> from sage.all import *
>>> EllipticCurve('389a').heegner_point(-Integer(7)).satisfies_kolyvagin_hypothesis()
True
>>> EllipticCurve('389a').heegner_point(-Integer(7), Integer(5)).satisfies_kolyvagin_hypothesis()
True
>>> EllipticCurve('389a').heegner_point(-Integer(7), Integer(11)).satisfies_kolyvagin_hypothesis()
False

tau()[source]¶

Return \(\tau\) in the upper half plane that maps via the modular parametrization to this Heegner point.

EXAMPLES:

Sage

sage: E = EllipticCurve('389a'); P = E.heegner_point(-7, 5)
sage: P.tau()
5/778*sqrt_minus_7 - 147/778

Python

>>> from sage.all import *
>>> E = EllipticCurve('389a'); P = E.heegner_point(-Integer(7), Integer(5))
>>> P.tau()
5/778*sqrt_minus_7 - 147/778

x_poly_exact(prec=53, algorithm='lll')[source]¶

Return irreducible polynomial over the rational numbers satisfied by the \(x\) coordinate of this Heegner point. A ValueError is raised if the precision is clearly insignificant to define a point on the curve.

Warning

It is in theory possible for this function to not raise a ValueError, find a polynomial, but via some very unlikely coincidence that point is not actually this Heegner point.

INPUT:

prec – integer (default: 53)
algorithm – ‘conjugates’ or ‘lll’ (default); if ‘conjugates’, compute numerically all the conjugates y[i] of the Heegner point and construct the characteristic polynomial as the product \(f(X)=(X-y[i])\). If ‘lll’, compute only one of the conjugates y[0], then uses the LLL algorithm to guess \(f(X)\).

EXAMPLES:

We compute some \(x\)-coordinate polynomials of some conductor 1 Heegner points:

Sage

sage: E = EllipticCurve('37a')
sage: v = E.heegner_discriminants_list(10)
sage: [E.heegner_point(D).x_poly_exact() for D in v]
[x, x, x^2 + 2, x^5 - x^4 + x^3 + x^2 - 2*x + 1, x - 6,
 x^7 - 2*x^6 + 9*x^5 - 10*x^4 - x^3 + 8*x^2 - 5*x + 1,
 x^3 + 5*x^2 + 10*x + 4, x^4 - 10*x^3 + 10*x^2 + 12*x - 12,
 x^8 - 5*x^7 + 7*x^6 + 13*x^5 - 10*x^4 - 4*x^3 + x^2 - 5*x + 7,
 x^6 - 2*x^5 + 11*x^4 - 24*x^3 + 30*x^2 - 16*x + 4]

Python

>>> from sage.all import *
>>> E = EllipticCurve('37a')
>>> v = E.heegner_discriminants_list(Integer(10))
>>> [E.heegner_point(D).x_poly_exact() for D in v]
[x, x, x^2 + 2, x^5 - x^4 + x^3 + x^2 - 2*x + 1, x - 6,
 x^7 - 2*x^6 + 9*x^5 - 10*x^4 - x^3 + 8*x^2 - 5*x + 1,
 x^3 + 5*x^2 + 10*x + 4, x^4 - 10*x^3 + 10*x^2 + 12*x - 12,
 x^8 - 5*x^7 + 7*x^6 + 13*x^5 - 10*x^4 - 4*x^3 + x^2 - 5*x + 7,
 x^6 - 2*x^5 + 11*x^4 - 24*x^3 + 30*x^2 - 16*x + 4]

We compute \(x\)-coordinate polynomials for some Heegner points of conductor bigger than 1 on a rank 2 curve:

Sage

sage: E = EllipticCurve('389a'); P = E.heegner_point(-7, 5); P
Heegner point of discriminant -7 and conductor 5
 on elliptic curve of conductor 389
sage: P.x_poly_exact()
Traceback (most recent call last):
...
ValueError: insufficient precision to determine Heegner point
(fails discriminant test)
sage: P.x_poly_exact(120)
x^6 + 10/7*x^5 - 867/49*x^4 - 76/245*x^3 + 3148/35*x^2 - 25944/245*x + 48771/1225
sage: E.heegner_point(-7, 11).x_poly_exact(500)
x^10 + 282527/52441*x^9 + 27049007420/2750058481*x^8 - 22058564794/2750058481*x^7
 - 140054237301/2750058481*x^6 + 696429998952/30250643291*x^5
 + 2791387923058/30250643291*x^4 - 3148473886134/30250643291*x^3
 + 1359454055022/30250643291*x^2 - 250620385365/30250643291*x
 + 181599685425/332757076201

Python

>>> from sage.all import *
>>> E = EllipticCurve('389a'); P = E.heegner_point(-Integer(7), Integer(5)); P
Heegner point of discriminant -7 and conductor 5
 on elliptic curve of conductor 389
>>> P.x_poly_exact()
Traceback (most recent call last):
...
ValueError: insufficient precision to determine Heegner point
(fails discriminant test)
>>> P.x_poly_exact(Integer(120))
x^6 + 10/7*x^5 - 867/49*x^4 - 76/245*x^3 + 3148/35*x^2 - 25944/245*x + 48771/1225
>>> E.heegner_point(-Integer(7), Integer(11)).x_poly_exact(Integer(500))
x^10 + 282527/52441*x^9 + 27049007420/2750058481*x^8 - 22058564794/2750058481*x^7
 - 140054237301/2750058481*x^6 + 696429998952/30250643291*x^5
 + 2791387923058/30250643291*x^4 - 3148473886134/30250643291*x^3
 + 1359454055022/30250643291*x^2 - 250620385365/30250643291*x
 + 181599685425/332757076201

Here we compute a Heegner point of conductor 5 on a rank 3 curve:

Sage

sage: E = EllipticCurve('5077a'); P = E.heegner_point(-7,5); P
Heegner point of discriminant -7 and conductor 5
 on elliptic curve of conductor 5077
sage: P.x_poly_exact(500)
x^6 + 1108754853727159228/72351048803252547*x^5 + 88875505551184048168/1953478317687818769*x^4 - 2216200271166098662132/3255797196146364615*x^3 + 14941627504168839449851/9767391588439093845*x^2 - 3456417460183342963918/3255797196146364615*x + 1306572835857500500459/5426328660243941025

Python

>>> from sage.all import *
>>> E = EllipticCurve('5077a'); P = E.heegner_point(-Integer(7),Integer(5)); P
Heegner point of discriminant -7 and conductor 5
 on elliptic curve of conductor 5077
>>> P.x_poly_exact(Integer(500))
x^6 + 1108754853727159228/72351048803252547*x^5 + 88875505551184048168/1953478317687818769*x^4 - 2216200271166098662132/3255797196146364615*x^3 + 14941627504168839449851/9767391588439093845*x^2 - 3456417460183342963918/3255797196146364615*x + 1306572835857500500459/5426328660243941025

See Issue #34121:

Sage

sage: E = EllipticCurve('11a1')
sage: P = E.heegner_point(-7)
sage: PE = P.point_exact()
sage: PE
(a : -4*a + 3 : 1)
sage: all(c.parent().disc() == -7 for c in PE)
True

Python

>>> from sage.all import *
>>> E = EllipticCurve('11a1')
>>> P = E.heegner_point(-Integer(7))
>>> PE = P.point_exact()
>>> PE
(a : -4*a + 3 : 1)
>>> all(c.parent().disc() == -Integer(7) for c in PE)
True

class sage.schemes.elliptic_curves.heegner.HeegnerPointOnX0N(N, D, c=1, f=None, check=True)[source]¶

Bases: HeegnerPoint

A Heegner point as a point on the modular curve \(X_0(N)\), which we view as the upper half plane modulo the action of \(\Gamma_0(N)\).

EXAMPLES:

Sage

sage: x = heegner_point(37, -7, 5); x
Heegner point 5/74*sqrt(-7) - 11/74 of discriminant -7 and conductor 5 on X_0(37)
sage: type(x)
<class 'sage.schemes.elliptic_curves.heegner.HeegnerPointOnX0N'>
sage: x.level()
37
sage: x.conductor()
5
sage: x.discriminant()
-7
sage: x.quadratic_field()
Number Field in sqrt_minus_7 with defining polynomial x^2 + 7
 with sqrt_minus_7 = 2.645751311064591?*I
sage: x.quadratic_form()
37*x^2 + 11*x*y + 2*y^2
sage: x.quadratic_order()
Order of conductor 10 generated by 5*sqrt_minus_7
 in Number Field in sqrt_minus_7 with defining polynomial x^2 + 7
 with sqrt_minus_7 = 2.645751311064591?*I
sage: x.tau()
5/74*sqrt_minus_7 - 11/74
sage: loads(dumps(x)) == x
True

Python

>>> from sage.all import *
>>> x = heegner_point(Integer(37), -Integer(7), Integer(5)); x
Heegner point 5/74*sqrt(-7) - 11/74 of discriminant -7 and conductor 5 on X_0(37)
>>> type(x)
<class 'sage.schemes.elliptic_curves.heegner.HeegnerPointOnX0N'>
>>> x.level()
37
>>> x.conductor()
5
>>> x.discriminant()
-7
>>> x.quadratic_field()
Number Field in sqrt_minus_7 with defining polynomial x^2 + 7
 with sqrt_minus_7 = 2.645751311064591?*I
>>> x.quadratic_form()
37*x^2 + 11*x*y + 2*y^2
>>> x.quadratic_order()
Order of conductor 10 generated by 5*sqrt_minus_7
 in Number Field in sqrt_minus_7 with defining polynomial x^2 + 7
 with sqrt_minus_7 = 2.645751311064591?*I
>>> x.tau()
5/74*sqrt_minus_7 - 11/74
>>> loads(dumps(x)) == x
True

atkin_lehner_act(Q=None)[source]¶

Given an integer Q dividing the level N such that \(\gcd(Q, N/Q) = 1\), return the image of this Heegner point under the Atkin-Lehner operator \(W_Q\).

INPUT:

Q – positive divisor of \(N\); if not given, default to \(N\)

EXAMPLES:

Sage

sage: x = heegner_point(389, -7, 5)
sage: x.atkin_lehner_act()
Heegner point 5/199168*sqrt(-7) - 631/199168 of discriminant -7
 and conductor 5 on X_0(389)

sage: x = heegner_point(45, D=-11, c=1); x
Heegner point 1/90*sqrt(-11) - 13/90 of discriminant -11 on X_0(45)
sage: x.atkin_lehner_act(5)
Heegner point 1/90*sqrt(-11) + 23/90 of discriminant -11 on X_0(45)
sage: y = x.atkin_lehner_act(9); y
Heegner point 1/90*sqrt(-11) - 23/90 of discriminant -11 on X_0(45)
sage: z = y.atkin_lehner_act(9); z
Heegner point 1/90*sqrt(-11) - 13/90 of discriminant -11 on X_0(45)
sage: z == x
True

Python

>>> from sage.all import *
>>> x = heegner_point(Integer(389), -Integer(7), Integer(5))
>>> x.atkin_lehner_act()
Heegner point 5/199168*sqrt(-7) - 631/199168 of discriminant -7
 and conductor 5 on X_0(389)

>>> x = heegner_point(Integer(45), D=-Integer(11), c=Integer(1)); x
Heegner point 1/90*sqrt(-11) - 13/90 of discriminant -11 on X_0(45)
>>> x.atkin_lehner_act(Integer(5))
Heegner point 1/90*sqrt(-11) + 23/90 of discriminant -11 on X_0(45)
>>> y = x.atkin_lehner_act(Integer(9)); y
Heegner point 1/90*sqrt(-11) - 23/90 of discriminant -11 on X_0(45)
>>> z = y.atkin_lehner_act(Integer(9)); z
Heegner point 1/90*sqrt(-11) - 13/90 of discriminant -11 on X_0(45)
>>> z == x
True

galois_orbit_over_K()[source]¶

Return the \(Gal(K_c/K)\)-orbit of this Heegner point.

EXAMPLES:

Sage

sage: x = heegner_point(389, -7, 3); x
Heegner point 3/778*sqrt(-7) - 223/778 of discriminant -7
 and conductor 3 on X_0(389)
sage: x.galois_orbit_over_K()
[Heegner point 3/778*sqrt(-7) - 223/778 of discriminant -7 and conductor 3 on X_0(389),
 Heegner point 3/1556*sqrt(-7) - 223/1556 of discriminant -7 and conductor 3 on X_0(389),
 Heegner point 3/1556*sqrt(-7) - 1001/1556 of discriminant -7 and conductor 3 on X_0(389),
 Heegner point 3/3112*sqrt(-7) - 223/3112 of discriminant -7 and conductor 3 on X_0(389)]

Python

>>> from sage.all import *
>>> x = heegner_point(Integer(389), -Integer(7), Integer(3)); x
Heegner point 3/778*sqrt(-7) - 223/778 of discriminant -7
 and conductor 3 on X_0(389)
>>> x.galois_orbit_over_K()
[Heegner point 3/778*sqrt(-7) - 223/778 of discriminant -7 and conductor 3 on X_0(389),
 Heegner point 3/1556*sqrt(-7) - 223/1556 of discriminant -7 and conductor 3 on X_0(389),
 Heegner point 3/1556*sqrt(-7) - 1001/1556 of discriminant -7 and conductor 3 on X_0(389),
 Heegner point 3/3112*sqrt(-7) - 223/3112 of discriminant -7 and conductor 3 on X_0(389)]

map_to_curve(E)[source]¶

Return the image of this Heegner point on the elliptic curve \(E\), which must also have conductor \(N\), where \(N\) is the level of self.

EXAMPLES:

Sage

sage: x = heegner_point(389, -7, 5); x
Heegner point 5/778*sqrt(-7) - 147/778 of discriminant -7
 and conductor 5 on X_0(389)
sage: y = x.map_to_curve(EllipticCurve('389a')); y
Heegner point of discriminant -7 and conductor 5
 on elliptic curve of conductor 389
sage: y.curve().cremona_label()
'389a1'
sage: y.heegner_point_on_X0N()
Heegner point 5/778*sqrt(-7) - 147/778 of discriminant -7
 and conductor 5 on X_0(389)

Python

>>> from sage.all import *
>>> x = heegner_point(Integer(389), -Integer(7), Integer(5)); x
Heegner point 5/778*sqrt(-7) - 147/778 of discriminant -7
 and conductor 5 on X_0(389)
>>> y = x.map_to_curve(EllipticCurve('389a')); y
Heegner point of discriminant -7 and conductor 5
 on elliptic curve of conductor 389
>>> y.curve().cremona_label()
'389a1'
>>> y.heegner_point_on_X0N()
Heegner point 5/778*sqrt(-7) - 147/778 of discriminant -7
 and conductor 5 on X_0(389)

You can also directly apply the modular parametrization of the elliptic curve:

Sage

sage: x = heegner_point(37,-7); x
Heegner point 1/74*sqrt(-7) - 17/74 of discriminant -7 on X_0(37)
sage: E = EllipticCurve('37a'); phi = E.modular_parametrization()
sage: phi(x)
Heegner point of discriminant -7 on elliptic curve of conductor 37

Python

>>> from sage.all import *
>>> x = heegner_point(Integer(37),-Integer(7)); x
Heegner point 1/74*sqrt(-7) - 17/74 of discriminant -7 on X_0(37)
>>> E = EllipticCurve('37a'); phi = E.modular_parametrization()
>>> phi(x)
Heegner point of discriminant -7 on elliptic curve of conductor 37

plot(**kwds)[source]¶

Draw a point at \((x,y)\) where this Heegner point is represented by the point \(\tau = x + i y\) in the upper half plane.

The kwds get passed onto the point plotting command.

EXAMPLES:

Sage

sage: heegner_point(389,-7,1).plot(pointsize=50)
Graphics object consisting of 1 graphics primitive

Python

>>> from sage.all import *
>>> heegner_point(Integer(389),-Integer(7),Integer(1)).plot(pointsize=Integer(50))
Graphics object consisting of 1 graphics primitive

quadratic_form()[source]¶

Return the integral primitive positive-definite binary quadratic form associated to this Heegner point.

EXAMPLES:

Sage

sage: heegner_point(389, -7, 5).quadratic_form()
389*x^2 + 147*x*y + 14*y^2

Python

>>> from sage.all import *
>>> heegner_point(Integer(389), -Integer(7), Integer(5)).quadratic_form()
389*x^2 + 147*x*y + 14*y^2

reduced_quadratic_form()[source]¶

Return reduced binary quadratic corresponding to this Heegner point.

EXAMPLES:

Sage

sage: x = heegner_point(389, -7, 5)
sage: x.quadratic_form()
389*x^2 + 147*x*y + 14*y^2
sage: x.reduced_quadratic_form()
4*x^2 - x*y + 11*y^2

Python

>>> from sage.all import *
>>> x = heegner_point(Integer(389), -Integer(7), Integer(5))
>>> x.quadratic_form()
389*x^2 + 147*x*y + 14*y^2
>>> x.reduced_quadratic_form()
4*x^2 - x*y + 11*y^2

tau()[source]¶

Return an element tau in the upper half plane that corresponds to this particular Heegner point.

Actually, tau is in the quadratic imaginary field K associated to this Heegner point.

EXAMPLES:

Sage

sage: x = heegner_point(37, -7, 5); tau = x.tau(); tau
5/74*sqrt_minus_7 - 11/74
sage: 37 * tau.minpoly()
37*x^2 + 11*x + 2
sage: x.quadratic_form()
37*x^2 + 11*x*y + 2*y^2

Python

>>> from sage.all import *
>>> x = heegner_point(Integer(37), -Integer(7), Integer(5)); tau = x.tau(); tau
5/74*sqrt_minus_7 - 11/74
>>> Integer(37) * tau.minpoly()
37*x^2 + 11*x + 2
>>> x.quadratic_form()
37*x^2 + 11*x*y + 2*y^2

class sage.schemes.elliptic_curves.heegner.HeegnerPoints(N)[source]¶

Bases: SageObject

The set of Heegner points with given parameters.

EXAMPLES:

Sage

sage: H = heegner_points(389); H
Set of all Heegner points on X_0(389)
sage: type(H)
<class 'sage.schemes.elliptic_curves.heegner.HeegnerPoints_level'>
sage: isinstance(H, sage.schemes.elliptic_curves.heegner.HeegnerPoints)
True

Python

>>> from sage.all import *
>>> H = heegner_points(Integer(389)); H
Set of all Heegner points on X_0(389)
>>> type(H)
<class 'sage.schemes.elliptic_curves.heegner.HeegnerPoints_level'>
>>> isinstance(H, sage.schemes.elliptic_curves.heegner.HeegnerPoints)
True

level()[source]¶

Return the level \(N\) of the modular curve \(X_0(N)\).

EXAMPLES:

Sage

sage: heegner_points(389).level()
389

Python

>>> from sage.all import *
>>> heegner_points(Integer(389)).level()
389

class sage.schemes.elliptic_curves.heegner.HeegnerPoints_level(N)[source]¶

Bases: HeegnerPoints

Return the infinite set of all Heegner points on \(X_0(N)\) for all quadratic imaginary fields.

EXAMPLES:

Sage

sage: H = heegner_points(11); H
Set of all Heegner points on X_0(11)
sage: type(H)
<class 'sage.schemes.elliptic_curves.heegner.HeegnerPoints_level'>
sage: loads(dumps(H)) == H
True

Python

>>> from sage.all import *
>>> H = heegner_points(Integer(11)); H
Set of all Heegner points on X_0(11)
>>> type(H)
<class 'sage.schemes.elliptic_curves.heegner.HeegnerPoints_level'>
>>> loads(dumps(H)) == H
True

discriminants(n=10, weak=False)[source]¶

Return the first \(n\) quadratic imaginary discriminants that satisfy the Heegner hypothesis for \(N\).

INPUT:

n – nonnegative integer
weak – boolean (default: False); if True only require weak Heegner hypothesis, which is the same as usual but without the condition that \(\gcd(D,N)=1\).

EXAMPLES:

Sage

sage: X = heegner_points(37)
sage: X.discriminants(5)
[-7, -11, -40, -47, -67]

Python

>>> from sage.all import *
>>> X = heegner_points(Integer(37))
>>> X.discriminants(Integer(5))
[-7, -11, -40, -47, -67]

The default is 10:

Sage

sage: X.discriminants()
[-7, -11, -40, -47, -67, -71, -83, -84, -95, -104]
sage: X.discriminants(15)
[-7, -11, -40, -47, -67, -71, -83, -84, -95, -104, -107, -115, -120, -123, -127]

Python

>>> from sage.all import *
>>> X.discriminants()
[-7, -11, -40, -47, -67, -71, -83, -84, -95, -104]
>>> X.discriminants(Integer(15))
[-7, -11, -40, -47, -67, -71, -83, -84, -95, -104, -107, -115, -120, -123, -127]

The discriminant -111 satisfies only the weak Heegner hypothesis, since it is divisible by 37:

Sage

sage: X.discriminants(15, weak=True)
[-7, -11, -40, -47, -67, -71, -83, -84, -95, -104, -107, -111, -115, -120, -123]

Python

>>> from sage.all import *
>>> X.discriminants(Integer(15), weak=True)
[-7, -11, -40, -47, -67, -71, -83, -84, -95, -104, -107, -111, -115, -120, -123]

reduce_mod(ell)[source]¶

Return object that allows for computation with Heegner points of level \(N\) modulo the prime \(\ell\), represented using quaternion algebras.

INPUT:

\(\ell\) – prime

EXAMPLES:

Sage

sage: heegner_points(389).reduce_mod(7).quaternion_algebra()
Quaternion Algebra (-1, -7) with base ring Rational Field

Python

>>> from sage.all import *
>>> heegner_points(Integer(389)).reduce_mod(Integer(7)).quaternion_algebra()
Quaternion Algebra (-1, -7) with base ring Rational Field

class sage.schemes.elliptic_curves.heegner.HeegnerPoints_level_disc(N, D)[source]¶

Bases: HeegnerPoints

Set of Heegner points of given level and all conductors associated to a quadratic imaginary field.

EXAMPLES:

Sage

sage: H = heegner_points(389,-7); H
Set of all Heegner points on X_0(389) associated to QQ[sqrt(-7)]
sage: type(H)
<class 'sage.schemes.elliptic_curves.heegner.HeegnerPoints_level_disc'>
sage: H._repr_()
'Set of all Heegner points on X_0(389) associated to QQ[sqrt(-7)]'
sage: H.discriminant()
-7
sage: H.quadratic_field()
Number Field in sqrt_minus_7 with defining polynomial x^2 + 7
 with sqrt_minus_7 = 2.645751311064591?*I
sage: H.kolyvagin_conductors()
[1, 3, 5, 13, 15, 17, 19, 31, 39, 41]

sage: loads(dumps(H)) == H
True

Python

>>> from sage.all import *
>>> H = heegner_points(Integer(389),-Integer(7)); H
Set of all Heegner points on X_0(389) associated to QQ[sqrt(-7)]
>>> type(H)
<class 'sage.schemes.elliptic_curves.heegner.HeegnerPoints_level_disc'>
>>> H._repr_()
'Set of all Heegner points on X_0(389) associated to QQ[sqrt(-7)]'
>>> H.discriminant()
-7
>>> H.quadratic_field()
Number Field in sqrt_minus_7 with defining polynomial x^2 + 7
 with sqrt_minus_7 = 2.645751311064591?*I
>>> H.kolyvagin_conductors()
[1, 3, 5, 13, 15, 17, 19, 31, 39, 41]

>>> loads(dumps(H)) == H
True

discriminant()[source]¶

Return the discriminant of the quadratic imaginary extension \(K\).

EXAMPLES:

Sage

sage: heegner_points(389,-7).discriminant()
-7

Python

>>> from sage.all import *
>>> heegner_points(Integer(389),-Integer(7)).discriminant()
-7

kolyvagin_conductors(r=None, n=10, E=None, m=None)[source]¶

Return the first \(n\) conductors that are squarefree products of distinct primes inert in the quadratic imaginary field \(K = \QQ(\sqrt{D})\). If \(r\) is specified, return only conductors that are a product of \(r\) distinct primes all inert in \(K\). If \(r = 0\), always return the list [1], no matter what.

If the optional elliptic curve \(E\) and integer \(m\) are given, then only include conductors \(c\) such that for each prime divisor \(p\) of \(c\) we have \(m \mid \gcd(a_p(E), p+1)\).

INPUT:

r – (default: None) nonnegative integer or None
n – positive integer
E – an elliptic curve
m – positive integer

EXAMPLES:

Sage

sage: H = heegner_points(389, -7)
sage: H.kolyvagin_conductors(0)
[1]
sage: H.kolyvagin_conductors(1)
[3, 5, 13, 17, 19, 31, 41, 47, 59, 61]
sage: H.kolyvagin_conductors(1,15)
[3, 5, 13, 17, 19, 31, 41, 47, 59, 61, 73, 83, 89, 97, 101]
sage: H.kolyvagin_conductors(1,5)
[3, 5, 13, 17, 19]
sage: H.kolyvagin_conductors(1, 5, EllipticCurve('389a'), 3)
[5, 17, 41, 59, 83]
sage: H.kolyvagin_conductors(2, 5, EllipticCurve('389a'), 3)
[85, 205, 295, 415, 697]

Python

>>> from sage.all import *
>>> H = heegner_points(Integer(389), -Integer(7))
>>> H.kolyvagin_conductors(Integer(0))
[1]
>>> H.kolyvagin_conductors(Integer(1))
[3, 5, 13, 17, 19, 31, 41, 47, 59, 61]
>>> H.kolyvagin_conductors(Integer(1),Integer(15))
[3, 5, 13, 17, 19, 31, 41, 47, 59, 61, 73, 83, 89, 97, 101]
>>> H.kolyvagin_conductors(Integer(1),Integer(5))
[3, 5, 13, 17, 19]
>>> H.kolyvagin_conductors(Integer(1), Integer(5), EllipticCurve('389a'), Integer(3))
[5, 17, 41, 59, 83]
>>> H.kolyvagin_conductors(Integer(2), Integer(5), EllipticCurve('389a'), Integer(3))
[85, 205, 295, 415, 697]

quadratic_field()[source]¶

Return the quadratic imaginary field \(K = \QQ(\sqrt{D})\).

EXAMPLES:

Sage

sage: E = EllipticCurve('389a'); K = E.heegner_point(-7,5).ring_class_field()
sage: K.quadratic_field()
Number Field in sqrt_minus_7 with defining polynomial x^2 + 7
 with sqrt_minus_7 = 2.645751311064591?*I

Python

>>> from sage.all import *
>>> E = EllipticCurve('389a'); K = E.heegner_point(-Integer(7),Integer(5)).ring_class_field()
>>> K.quadratic_field()
Number Field in sqrt_minus_7 with defining polynomial x^2 + 7
 with sqrt_minus_7 = 2.645751311064591?*I

class sage.schemes.elliptic_curves.heegner.HeegnerPoints_level_disc_cond(N, D, c=1)[source]¶

Bases: HeegnerPoints_level, HeegnerPoints_level_disc

The set of Heegner points of given level, discriminant, and conductor.

EXAMPLES:

Sage

sage: H = heegner_points(389,-7,5); H
All Heegner points of conductor 5 on X_0(389) associated to QQ[sqrt(-7)]
sage: type(H)
<class 'sage.schemes.elliptic_curves.heegner.HeegnerPoints_level_disc_cond'>
sage: H.discriminant()
-7
sage: H.level()
389

sage: len(H.points())
12
sage: H.points()[0]
Heegner point 5/778*sqrt(-7) - 147/778 of discriminant -7 and conductor 5 on X_0(389)
sage: H.betas()
(147, 631)

sage: H.quadratic_field()
Number Field in sqrt_minus_7 with defining polynomial x^2 + 7
 with sqrt_minus_7 = 2.645751311064591?*I
sage: H.ring_class_field()
Ring class field extension of QQ[sqrt(-7)] of conductor 5

sage: H.kolyvagin_conductors()
[1, 3, 5, 13, 15, 17, 19, 31, 39, 41]
sage: H.satisfies_kolyvagin_hypothesis()
True

sage: H = heegner_points(389,-7,5)
sage: loads(dumps(H)) == H
True

Python

>>> from sage.all import *
>>> H = heegner_points(Integer(389),-Integer(7),Integer(5)); H
All Heegner points of conductor 5 on X_0(389) associated to QQ[sqrt(-7)]
>>> type(H)
<class 'sage.schemes.elliptic_curves.heegner.HeegnerPoints_level_disc_cond'>
>>> H.discriminant()
-7
>>> H.level()
389

>>> len(H.points())
12
>>> H.points()[Integer(0)]
Heegner point 5/778*sqrt(-7) - 147/778 of discriminant -7 and conductor 5 on X_0(389)
>>> H.betas()
(147, 631)

>>> H.quadratic_field()
Number Field in sqrt_minus_7 with defining polynomial x^2 + 7
 with sqrt_minus_7 = 2.645751311064591?*I
>>> H.ring_class_field()
Ring class field extension of QQ[sqrt(-7)] of conductor 5

>>> H.kolyvagin_conductors()
[1, 3, 5, 13, 15, 17, 19, 31, 39, 41]
>>> H.satisfies_kolyvagin_hypothesis()
True

>>> H = heegner_points(Integer(389),-Integer(7),Integer(5))
>>> loads(dumps(H)) == H
True

betas()[source]¶

Return the square roots of \(D c^2\) modulo \(4 N\) all reduced mod \(2 N\), without multiplicity.

EXAMPLES:

Sage

sage: X = heegner_points(45,-11,1); X
All Heegner points of conductor 1 on X_0(45) associated to QQ[sqrt(-11)]
sage: [x.quadratic_form() for x in X]
[45*x^2 + 13*x*y + y^2,
 45*x^2 + 23*x*y + 3*y^2,
 45*x^2 + 67*x*y + 25*y^2,
 45*x^2 + 77*x*y + 33*y^2]
sage: X.betas()
(13, 23, 67, 77)
sage: X.points(13)
(Heegner point 1/90*sqrt(-11) - 13/90 of discriminant -11 on X_0(45),)
sage: [x.quadratic_form() for x in X.points(13)]
[45*x^2 + 13*x*y + y^2]

Python

>>> from sage.all import *
>>> X = heegner_points(Integer(45),-Integer(11),Integer(1)); X
All Heegner points of conductor 1 on X_0(45) associated to QQ[sqrt(-11)]
>>> [x.quadratic_form() for x in X]
[45*x^2 + 13*x*y + y^2,
 45*x^2 + 23*x*y + 3*y^2,
 45*x^2 + 67*x*y + 25*y^2,
 45*x^2 + 77*x*y + 33*y^2]
>>> X.betas()
(13, 23, 67, 77)
>>> X.points(Integer(13))
(Heegner point 1/90*sqrt(-11) - 13/90 of discriminant -11 on X_0(45),)
>>> [x.quadratic_form() for x in X.points(Integer(13))]
[45*x^2 + 13*x*y + y^2]

conductor()[source]¶

Return the level of the conductor.

EXAMPLES:

Sage

sage: heegner_points(389,-7,5).conductor()
5

Python

>>> from sage.all import *
>>> heegner_points(Integer(389),-Integer(7),Integer(5)).conductor()
5

plot(*args, **kwds)[source]¶

Return plot of all the representatives in the upper half plane of the Heegner points in this set of Heegner points.

The inputs to this function get passed onto the point command.

EXAMPLES:

Sage

sage: heegner_points(389,-7,5).plot(pointsize=50, rgbcolor='red')           # needs sage.plot
Graphics object consisting of 12 graphics primitives
sage: heegner_points(53,-7,15).plot(pointsize=50, rgbcolor='purple')        # needs sage.plot
Graphics object consisting of 48 graphics primitives

Python

>>> from sage.all import *
>>> heegner_points(Integer(389),-Integer(7),Integer(5)).plot(pointsize=Integer(50), rgbcolor='red')           # needs sage.plot
Graphics object consisting of 12 graphics primitives
>>> heegner_points(Integer(53),-Integer(7),Integer(15)).plot(pointsize=Integer(50), rgbcolor='purple')        # needs sage.plot
Graphics object consisting of 48 graphics primitives

points(beta=None)[source]¶

Return the Heegner points in self. If \(\beta\) is given, return only those Heegner points with given \(\beta\), i.e., whose quadratic form has \(B\) congruent to \(\beta\) modulo \(2 N\).

Use self.beta() to get a list of betas.

EXAMPLES:

Sage

sage: H = heegner_points(389, -7, 5); H
All Heegner points of conductor 5 on X_0(389) associated to QQ[sqrt(-7)]
sage: H.points()
(Heegner point 5/778*sqrt(-7) - 147/778 of discriminant -7 and conductor 5
  on X_0(389),
 ...,
 Heegner point 5/5446*sqrt(-7) - 757/778 of discriminant -7 and conductor 5
  on X_0(389))
sage: H.betas()
(147, 631)
sage: [x.tau() for x in H.points(147)]
[5/778*sqrt_minus_7 - 147/778, 5/1556*sqrt_minus_7 - 147/1556,
 5/1556*sqrt_minus_7 - 925/1556, 5/3112*sqrt_minus_7 - 1703/3112,
 5/3112*sqrt_minus_7 - 2481/3112, 5/5446*sqrt_minus_7 - 21/778]

sage: [x.tau() for x in H.points(631)]
[5/778*sqrt_minus_7 - 631/778, 5/1556*sqrt_minus_7 - 631/1556,
 5/1556*sqrt_minus_7 - 1409/1556, 5/3112*sqrt_minus_7 - 631/3112,
 5/3112*sqrt_minus_7 - 1409/3112, 5/5446*sqrt_minus_7 - 757/778]

Python

>>> from sage.all import *
>>> H = heegner_points(Integer(389), -Integer(7), Integer(5)); H
All Heegner points of conductor 5 on X_0(389) associated to QQ[sqrt(-7)]
>>> H.points()
(Heegner point 5/778*sqrt(-7) - 147/778 of discriminant -7 and conductor 5
  on X_0(389),
 ...,
 Heegner point 5/5446*sqrt(-7) - 757/778 of discriminant -7 and conductor 5
  on X_0(389))
>>> H.betas()
(147, 631)
>>> [x.tau() for x in H.points(Integer(147))]
[5/778*sqrt_minus_7 - 147/778, 5/1556*sqrt_minus_7 - 147/1556,
 5/1556*sqrt_minus_7 - 925/1556, 5/3112*sqrt_minus_7 - 1703/3112,
 5/3112*sqrt_minus_7 - 2481/3112, 5/5446*sqrt_minus_7 - 21/778]

>>> [x.tau() for x in H.points(Integer(631))]
[5/778*sqrt_minus_7 - 631/778, 5/1556*sqrt_minus_7 - 631/1556,
 5/1556*sqrt_minus_7 - 1409/1556, 5/3112*sqrt_minus_7 - 631/3112,
 5/3112*sqrt_minus_7 - 1409/3112, 5/5446*sqrt_minus_7 - 757/778]

The result is cached and is a tuple (since it is immutable):

Sage

sage: H.points() is H.points()
True
sage: type(H.points())
<... 'tuple'>

Python

>>> from sage.all import *
>>> H.points() is H.points()
True
>>> type(H.points())
<... 'tuple'>

ring_class_field()[source]¶

Return the ring class field associated to this set of Heegner points. This is an extension \(K_c\) over \(K\), where \(K\) is the quadratic imaginary field and \(c\) the conductor associated to this Heegner point. This Heegner point is defined over \(K_c\) and the Galois group \(Gal(K_c/K)\) acts transitively on the Galois conjugates of this Heegner point.

EXAMPLES:

Sage

sage: heegner_points(389,-7,5).ring_class_field()
Ring class field extension of QQ[sqrt(-7)] of conductor 5

Python

>>> from sage.all import *
>>> heegner_points(Integer(389),-Integer(7),Integer(5)).ring_class_field()
Ring class field extension of QQ[sqrt(-7)] of conductor 5

satisfies_kolyvagin_hypothesis()[source]¶

Return True if self satisfies the Kolyvagin hypothesis, i.e., that each prime dividing the conductor \(c\) of self is inert in \(K\) and coprime to \(ND\).

EXAMPLES:

The prime 5 is inert, but the prime 11 is not:

Sage

sage: heegner_points(389,-7,5).satisfies_kolyvagin_hypothesis()
True
sage: heegner_points(389,-7,11).satisfies_kolyvagin_hypothesis()
False

Python

>>> from sage.all import *
>>> heegner_points(Integer(389),-Integer(7),Integer(5)).satisfies_kolyvagin_hypothesis()
True
>>> heegner_points(Integer(389),-Integer(7),Integer(11)).satisfies_kolyvagin_hypothesis()
False

class sage.schemes.elliptic_curves.heegner.HeegnerQuatAlg(level, ell)[source]¶

Bases: SageObject

Heegner points viewed as supersingular points on the modular curve \(X_0(N)/\mathbf{F}_{\ell}\).

EXAMPLES:

Sage

sage: H = heegner_points(11).reduce_mod(13); H
Heegner points on X_0(11) over F_13
sage: type(H)
<class 'sage.schemes.elliptic_curves.heegner.HeegnerQuatAlg'>
sage: loads(dumps(H)) == H
True

Python

>>> from sage.all import *
>>> H = heegner_points(Integer(11)).reduce_mod(Integer(13)); H
Heegner points on X_0(11) over F_13
>>> type(H)
<class 'sage.schemes.elliptic_curves.heegner.HeegnerQuatAlg'>
>>> loads(dumps(H)) == H
True

brandt_module()[source]¶

Return the Brandt module of right ideal classes that we used to represent the set of supersingular points on the modular curve.

EXAMPLES:

Sage

sage: heegner_points(11).reduce_mod(3).brandt_module()
Brandt module of dimension 2 of level 3*11 of weight 2 over Rational Field

Python

>>> from sage.all import *
>>> heegner_points(Integer(11)).reduce_mod(Integer(3)).brandt_module()
Brandt module of dimension 2 of level 3*11 of weight 2 over Rational Field

cyclic_subideal_p1(I, c)[source]¶

Compute dictionary mapping 2-tuples that defined normalized elements of \(P^1(\ZZ/c\ZZ)\)

INPUT:

I – right ideal of Eichler order or in quaternion algebra
c – square free integer (currently must be odd prime and coprime to level, discriminant, characteristic, etc.

OUTPUT:

dictionary mapping 2-tuples (u,v) to ideals

EXAMPLES:

Sage

sage: H = heegner_points(11).reduce_mod(7)
sage: I = H.brandt_module().right_ideals()[0]
sage: sorted(H.cyclic_subideal_p1(I, 3).items())
[((0, 1), Fractional ideal (12, 132*i, 10 + 76*i + 2*j, 4 + 86*i + 2*k)),
 ((1, 0), Fractional ideal (12, 132*i, 2 + 32*i + 2*j, 8 + 130*i + 2*k)),
 ((1, 1), Fractional ideal (12, 132*i, 10 + 32*i + 2*j, 8 + 86*i + 2*k)),
 ((1, 2), Fractional ideal (12, 132*i, 2 + 76*i + 2*j, 4 + 130*i + 2*k))]
sage: len(H.cyclic_subideal_p1(I, 17))
18

Python

>>> from sage.all import *
>>> H = heegner_points(Integer(11)).reduce_mod(Integer(7))
>>> I = H.brandt_module().right_ideals()[Integer(0)]
>>> sorted(H.cyclic_subideal_p1(I, Integer(3)).items())
[((0, 1), Fractional ideal (12, 132*i, 10 + 76*i + 2*j, 4 + 86*i + 2*k)),
 ((1, 0), Fractional ideal (12, 132*i, 2 + 32*i + 2*j, 8 + 130*i + 2*k)),
 ((1, 1), Fractional ideal (12, 132*i, 10 + 32*i + 2*j, 8 + 86*i + 2*k)),
 ((1, 2), Fractional ideal (12, 132*i, 2 + 76*i + 2*j, 4 + 130*i + 2*k))]
>>> len(H.cyclic_subideal_p1(I, Integer(17)))
18

ell()[source]¶

Return the prime \(\ell\) modulo which we are working.

EXAMPLES:

Sage

sage: heegner_points(11).reduce_mod(3).ell()
3

Python

>>> from sage.all import *
>>> heegner_points(Integer(11)).reduce_mod(Integer(3)).ell()
3

galois_group_over_hilbert_class_field(D, c)[source]¶

Return the Galois group of the extension of ring class fields \(K_c\) over the Hilbert class field \(K_{1}\) of the quadratic imaginary field of discriminant \(D\).

INPUT:

D – fundamental discriminant
c – conductor (square-free integer)

EXAMPLES:

Sage

sage: N = 37; D = -7; ell = 17; c = 41; p = 3
sage: H = heegner_points(N).reduce_mod(ell)
sage: H.galois_group_over_hilbert_class_field(D, c)
Galois group of Ring class field extension of QQ[sqrt(-7)] of conductor 41
 over Hilbert class field of QQ[sqrt(-7)]

Python

>>> from sage.all import *
>>> N = Integer(37); D = -Integer(7); ell = Integer(17); c = Integer(41); p = Integer(3)
>>> H = heegner_points(N).reduce_mod(ell)
>>> H.galois_group_over_hilbert_class_field(D, c)
Galois group of Ring class field extension of QQ[sqrt(-7)] of conductor 41
 over Hilbert class field of QQ[sqrt(-7)]

galois_group_over_quadratic_field(D, c)[source]¶

Return the Galois group of the extension of ring class fields \(K_c\) over the quadratic imaginary field \(K\) of discriminant \(D\).

INPUT:

D – fundamental discriminant
c – conductor (square-free integer)

EXAMPLES:

Sage

sage: N = 37; D = -7; ell = 17; c = 41; p = 3
sage: H = heegner_points(N).reduce_mod(ell)
sage: H.galois_group_over_quadratic_field(D, c)
Galois group of Ring class field extension of QQ[sqrt(-7)] of conductor 41
 over Number Field in sqrt_minus_7 with defining polynomial x^2 + 7
  with sqrt_minus_7 = 2.645751311064591?*I

Python

>>> from sage.all import *
>>> N = Integer(37); D = -Integer(7); ell = Integer(17); c = Integer(41); p = Integer(3)
>>> H = heegner_points(N).reduce_mod(ell)
>>> H.galois_group_over_quadratic_field(D, c)
Galois group of Ring class field extension of QQ[sqrt(-7)] of conductor 41
 over Number Field in sqrt_minus_7 with defining polynomial x^2 + 7
  with sqrt_minus_7 = 2.645751311064591?*I

heegner_conductors(D, n=5)[source]¶

Return the first \(n\) negative fundamental discriminants coprime to \(N\ell\) such that \(\ell\) is inert in the corresponding quadratic imaginary field and that field satisfies the Heegner hypothesis.

INPUT:

D – negative integer; a fundamental Heegner discriminant
n – positive integer (default: 5)

OUTPUT: list

EXAMPLES:

Sage

sage: H = heegner_points(11).reduce_mod(3)
sage: H.heegner_conductors(-7)
[1, 2, 4, 5, 8]
sage: H.heegner_conductors(-7, 10)
[1, 2, 4, 5, 8, 10, 13, 16, 17, 19]

Python

>>> from sage.all import *
>>> H = heegner_points(Integer(11)).reduce_mod(Integer(3))
>>> H.heegner_conductors(-Integer(7))
[1, 2, 4, 5, 8]
>>> H.heegner_conductors(-Integer(7), Integer(10))
[1, 2, 4, 5, 8, 10, 13, 16, 17, 19]

heegner_discriminants(n=5)[source]¶

INPUT:

n – positive integer (default: 5)

OUTPUT: list

EXAMPLES:

Sage

sage: H = heegner_points(11).reduce_mod(3)
sage: H.heegner_discriminants()
[-7, -19, -40, -43, -52]
sage: H.heegner_discriminants(10)
[-7, -19, -40, -43, -52, -79, -127, -139, -151, -184]

Python

>>> from sage.all import *
>>> H = heegner_points(Integer(11)).reduce_mod(Integer(3))
>>> H.heegner_discriminants()
[-7, -19, -40, -43, -52]
>>> H.heegner_discriminants(Integer(10))
[-7, -19, -40, -43, -52, -79, -127, -139, -151, -184]

heegner_divisor(D, c=1)[source]¶

Return Heegner divisor as an element of the Brandt module corresponding to the discriminant \(D\) and conductor \(c\), which both must be coprime to \(N\ell\).

More precisely, we compute the sum of the reductions of the \(\textrm{Gal}(K_1/K)\)-conjugates of each choice of \(y_1\), where the choice comes from choosing the ideal \(\mathcal{N}\). Then we apply the Hecke operator \(T_c\) to this sum.

INPUT:

D – discriminant (negative integer)
c – conductor (positive integer)

OUTPUT: a Brandt module element

EXAMPLES:

Sage

sage: H = heegner_points(11).reduce_mod(7)
sage: H.heegner_discriminants()
[-8, -39, -43, -51, -79]
sage: H.heegner_divisor(-8)
(1, 0, 0, 1, 0, 0)
sage: H.heegner_divisor(-39)
(1, 2, 2, 1, 2, 0)
sage: H.heegner_divisor(-43)
(1, 0, 0, 1, 0, 0)
sage: H.heegner_divisor(-51)
(1, 0, 0, 1, 0, 2)
sage: H.heegner_divisor(-79)
(3, 2, 2, 3, 0, 0)

sage: sum(H.heegner_divisor(-39).element())
8
sage: QuadraticField(-39,'a').class_number()
4

Python

>>> from sage.all import *
>>> H = heegner_points(Integer(11)).reduce_mod(Integer(7))
>>> H.heegner_discriminants()
[-8, -39, -43, -51, -79]
>>> H.heegner_divisor(-Integer(8))
(1, 0, 0, 1, 0, 0)
>>> H.heegner_divisor(-Integer(39))
(1, 2, 2, 1, 2, 0)
>>> H.heegner_divisor(-Integer(43))
(1, 0, 0, 1, 0, 0)
>>> H.heegner_divisor(-Integer(51))
(1, 0, 0, 1, 0, 2)
>>> H.heegner_divisor(-Integer(79))
(3, 2, 2, 3, 0, 0)

>>> sum(H.heegner_divisor(-Integer(39)).element())
8
>>> QuadraticField(-Integer(39),'a').class_number()
4

kolyvagin_cyclic_subideals(I, p, alpha_quaternion)[source]¶

Return list of pairs \((J, n)\) where \(J\) runs through the cyclic subideals of \(I\) of index \((\ZZ/p\ZZ)^2\), and \(J \sim \alpha^n(J_0)\) for some fixed choice of cyclic subideal \(J_0\).

INPUT:

I – right ideal of the quaternion algebra
p – prime number
alpha_quaternion – image in the quaternion algebra of generator \(\alpha\) for \((\mathcal{O}_K / c\mathcal{O}_K)^* / (\ZZ/c\ZZ)^*\)

OUTPUT: list of 2-tuples

EXAMPLES:

Sage

sage: N = 37; D = -7; ell = 17; c = 5
sage: H = heegner_points(N).reduce_mod(ell)
sage: I = H.brandt_module().right_ideals()[49]
sage: f = H.optimal_embeddings(D, 1, I.left_order())[1]
sage: g = H.kolyvagin_generators(f.domain().number_field(), c)
sage: alpha_quaternion = f(g[0]); alpha_quaternion
1 - 77/192*i - 5/128*j - 137/384*k
sage: H.kolyvagin_cyclic_subideals(I, 5, alpha_quaternion)
[(Fractional ideal (2560, 1280 + 47360*i, 1146 + 37678*i + 4*j, 212 + 54664/3*i + 2*j + 2/3*k), 0),
 (Fractional ideal (2560, 1280 + 47360*i, 2426 + 9262*i + 4*j, 2004 + 83080/3*i + 2*j + 2/3*k), 1),
 (Fractional ideal (2560, 1280 + 47360*i, 1914 + 9262*i + 4*j, 1748 + 111496/3*i + 2*j + 2/3*k), 2),
 (Fractional ideal (2560, 1280 + 47360*i, 2170 + 18734*i + 4*j, 212 + 111496/3*i + 2*j + 2/3*k), 3),
 (Fractional ideal (2560, 1280 + 47360*i, 890 + 28206*i + 4*j, 1748 + 54664/3*i + 2*j + 2/3*k), 4),
 (Fractional ideal (2560, 1280 + 47360*i, 634 + 37678*i + 4*j, 2516 + 83080/3*i + 2*j + 2/3*k), 5)]

Python

>>> from sage.all import *
>>> N = Integer(37); D = -Integer(7); ell = Integer(17); c = Integer(5)
>>> H = heegner_points(N).reduce_mod(ell)
>>> I = H.brandt_module().right_ideals()[Integer(49)]
>>> f = H.optimal_embeddings(D, Integer(1), I.left_order())[Integer(1)]
>>> g = H.kolyvagin_generators(f.domain().number_field(), c)
>>> alpha_quaternion = f(g[Integer(0)]); alpha_quaternion
1 - 77/192*i - 5/128*j - 137/384*k
>>> H.kolyvagin_cyclic_subideals(I, Integer(5), alpha_quaternion)
[(Fractional ideal (2560, 1280 + 47360*i, 1146 + 37678*i + 4*j, 212 + 54664/3*i + 2*j + 2/3*k), 0),
 (Fractional ideal (2560, 1280 + 47360*i, 2426 + 9262*i + 4*j, 2004 + 83080/3*i + 2*j + 2/3*k), 1),
 (Fractional ideal (2560, 1280 + 47360*i, 1914 + 9262*i + 4*j, 1748 + 111496/3*i + 2*j + 2/3*k), 2),
 (Fractional ideal (2560, 1280 + 47360*i, 2170 + 18734*i + 4*j, 212 + 111496/3*i + 2*j + 2/3*k), 3),
 (Fractional ideal (2560, 1280 + 47360*i, 890 + 28206*i + 4*j, 1748 + 54664/3*i + 2*j + 2/3*k), 4),
 (Fractional ideal (2560, 1280 + 47360*i, 634 + 37678*i + 4*j, 2516 + 83080/3*i + 2*j + 2/3*k), 5)]

kolyvagin_generator(K, p)[source]¶

Return element in \(K\) that maps to the multiplicative generator for the quotient group

\((\mathcal{O}_K / p \mathcal{O}_K)^* / (\ZZ/p\ZZ)^*\)

of the form \(\sqrt{D}+n\) with \(n\geq 1\) minimal.

INPUT:

K – quadratic imaginary field
p – inert prime

EXAMPLES:

Sage

sage: N = 37; D = -7; ell = 17; p = 5
sage: H = heegner_points(N).reduce_mod(ell)
sage: I = H.brandt_module().right_ideals()[49]
sage: f = H.optimal_embeddings(D, 1, I.left_order())[0]
sage: H.kolyvagin_generator(f.domain().number_field(), 5)
a + 1

Python

>>> from sage.all import *
>>> N = Integer(37); D = -Integer(7); ell = Integer(17); p = Integer(5)
>>> H = heegner_points(N).reduce_mod(ell)
>>> I = H.brandt_module().right_ideals()[Integer(49)]
>>> f = H.optimal_embeddings(D, Integer(1), I.left_order())[Integer(0)]
>>> H.kolyvagin_generator(f.domain().number_field(), Integer(5))
a + 1

This function requires that \(p\) be prime, but kolyvagin_generators works in general:

Sage

sage: H.kolyvagin_generator(f.domain().number_field(), 5*17)
Traceback (most recent call last):
...
NotImplementedError: p must be prime
sage: H.kolyvagin_generators(f.domain().number_field(), 5*17)
[-34*a + 1, 35*a + 106]

Python

>>> from sage.all import *
>>> H.kolyvagin_generator(f.domain().number_field(), Integer(5)*Integer(17))
Traceback (most recent call last):
...
NotImplementedError: p must be prime
>>> H.kolyvagin_generators(f.domain().number_field(), Integer(5)*Integer(17))
[-34*a + 1, 35*a + 106]

kolyvagin_generators(K, c)[source]¶

Return elements in \(\mathcal{O}_K\) that map to multiplicative generators for the factors of the quotient group

\((\mathcal{O}_K / c \mathcal{O}_K)^* / (\ZZ/c\ZZ)^*\)

corresponding to the prime divisors of c. Each generator is of the form \(\sqrt{D}+n\) with \(n\geq 1\) minimal.

INPUT:

K – quadratic imaginary field
c – square free product of inert prime

EXAMPLES:

Sage

sage: N = 37; D = -7; ell = 17; p = 5
sage: H = heegner_points(N).reduce_mod(ell)
sage: I = H.brandt_module().right_ideals()[49]
sage: f = H.optimal_embeddings(D, 1, I.left_order())[0]
sage: H.kolyvagin_generators(f.domain().number_field(), 5*17)
[-34*a + 1, 35*a + 106]

Python

>>> from sage.all import *
>>> N = Integer(37); D = -Integer(7); ell = Integer(17); p = Integer(5)
>>> H = heegner_points(N).reduce_mod(ell)
>>> I = H.brandt_module().right_ideals()[Integer(49)]
>>> f = H.optimal_embeddings(D, Integer(1), I.left_order())[Integer(0)]
>>> H.kolyvagin_generators(f.domain().number_field(), Integer(5)*Integer(17))
[-34*a + 1, 35*a + 106]

kolyvagin_point_on_curve(D, c, E, p, bound=10)[source]¶

Compute image of the Kolyvagin divisor \(P_c\) in \(E(\GF{\ell^2}) / p E(\GF{\ell^2})\).

Note that this image is by definition only well defined up to scalars. However, doing multiple computations will always yield the same result, and working modulo different \(\ell\) is compatible (since we always choose the same generator for \(\textrm{Gal}(K_c/K_1)\)).

INPUT:

D – fundamental negative discriminant
c – conductor
E – elliptic curve of conductor the level of self
p – odd prime number such that we consider image in \(E(\GF{\ell^2}) / p E(\GF{\ell^2})\)
bound – integer (default: 10)

EXAMPLES:

Sage

sage: N = 37; D = -7; ell = 17; c = 41; p = 3
sage: H = heegner_points(N).reduce_mod(ell)
sage: H.kolyvagin_point_on_curve(D, c, EllipticCurve('37a'), p)
[1, 1]

Python

>>> from sage.all import *
>>> N = Integer(37); D = -Integer(7); ell = Integer(17); c = Integer(41); p = Integer(3)
>>> H = heegner_points(N).reduce_mod(ell)
>>> H.kolyvagin_point_on_curve(D, c, EllipticCurve('37a'), p)
[1, 1]

kolyvagin_sigma_operator(D, c, r, bound=None)[source]¶

Return the action of the Kolyvagin sigma operator on the \(r\)-th basis vector.

INPUT:

D – fundamental discriminant
c – conductor (square-free integer, need not be prime)
r – nonnegative integer
bound – (default: None), if given, controls precision of computation of theta series, which could impact performance, but does not impact correctness

EXAMPLES:

We first try to verify Kolyvagin’s conjecture for a rank 2 curve by working modulo 5, but we are unlucky with \(c=17\):

Sage

sage: N = 389; D = -7; ell = 5; c = 17; q = 3
sage: H = heegner_points(N).reduce_mod(ell)
sage: E = EllipticCurve('389a')
sage: V = H.modp_dual_elliptic_curve_factor(E, q, 5)  # long time (4s on sage.math, 2012)
sage: k118 = H.kolyvagin_sigma_operator(D, c, 118)
sage: k104 = H.kolyvagin_sigma_operator(D, c, 104)
sage: [b.dot_product(k104.element().change_ring(GF(3)))  # long time
....:  for b in V.basis()]
[0, 0]
sage: [b.dot_product(k118.element().change_ring(GF(3)))  # long time
....:  for b in V.basis()]
[0, 0]

Python

>>> from sage.all import *
>>> N = Integer(389); D = -Integer(7); ell = Integer(5); c = Integer(17); q = Integer(3)
>>> H = heegner_points(N).reduce_mod(ell)
>>> E = EllipticCurve('389a')
>>> V = H.modp_dual_elliptic_curve_factor(E, q, Integer(5))  # long time (4s on sage.math, 2012)
>>> k118 = H.kolyvagin_sigma_operator(D, c, Integer(118))
>>> k104 = H.kolyvagin_sigma_operator(D, c, Integer(104))
>>> [b.dot_product(k104.element().change_ring(GF(Integer(3))))  # long time
...  for b in V.basis()]
[0, 0]
>>> [b.dot_product(k118.element().change_ring(GF(Integer(3))))  # long time
...  for b in V.basis()]
[0, 0]

Next we try again with \(c=41\) and this does work, in that we get something nonzero, when dotting with V:

Sage

sage: c = 41
sage: k118 = H.kolyvagin_sigma_operator(D, c, 118)
sage: k104 = H.kolyvagin_sigma_operator(D, c, 104)
sage: [b.dot_product(k118.element().change_ring(GF(3)))  # long time
....:  for b in V.basis()]
[2, 0]
sage: [b.dot_product(k104.element().change_ring(GF(3)))  # long time
....:  for b in V.basis()]
[1, 0]

Python

>>> from sage.all import *
>>> c = Integer(41)
>>> k118 = H.kolyvagin_sigma_operator(D, c, Integer(118))
>>> k104 = H.kolyvagin_sigma_operator(D, c, Integer(104))
>>> [b.dot_product(k118.element().change_ring(GF(Integer(3))))  # long time
...  for b in V.basis()]
[2, 0]
>>> [b.dot_product(k104.element().change_ring(GF(Integer(3))))  # long time
...  for b in V.basis()]
[1, 0]

By the way, the above is the first ever provable verification of Kolyvagin’s conjecture for any curve of rank at least 2.

Another example, but where the curve has rank 1:

Sage

sage: N = 37; D = -7; ell = 17; c = 41; q = 3
sage: H = heegner_points(N).reduce_mod(ell)
sage: H.heegner_divisor(D,1).element().nonzero_positions()
[49, 51]
sage: k49 = H.kolyvagin_sigma_operator(D, c, 49); k49
(79, 32, 31, 11, 53, 37, 1, 23, 15, 7, 0, 0, 0, 64, 32, 34, 53, 0, 27, 27, 0, 0, 0, 26, 0, 0, 18, 0, 22, 0, 53, 19, 27, 10, 0, 0, 0, 30, 35, 38, 0, 0, 0, 53, 0, 0, 4, 0, 0, 0, 0, 0)
sage: k51 = H.kolyvagin_sigma_operator(D, c, 51); k51
(20, 12, 57, 0, 0, 0, 0, 52, 23, 15, 0, 7, 0, 0, 19, 4, 0, 73, 11, 0, 104, 31, 0, 38, 31, 0, 0, 31, 5, 47, 0, 27, 35, 0, 57, 32, 24, 10, 0, 8, 0, 31, 41, 0, 0, 0, 16, 0, 0, 0, 0, 0)
sage: V = H.modp_dual_elliptic_curve_factor(EllipticCurve('37a'), q, 5); V
Vector space of degree 52 and dimension 2 over Ring of integers modulo 3
Basis matrix:
2 x 52 dense matrix over Ring of integers modulo 3
sage: [b.dot_product(k49.element().change_ring(GF(q))) for b in V.basis()]
[1, 1]
sage: [b.dot_product(k51.element().change_ring(GF(q))) for b in V.basis()]
[1, 1]

Python

>>> from sage.all import *
>>> N = Integer(37); D = -Integer(7); ell = Integer(17); c = Integer(41); q = Integer(3)
>>> H = heegner_points(N).reduce_mod(ell)
>>> H.heegner_divisor(D,Integer(1)).element().nonzero_positions()
[49, 51]
>>> k49 = H.kolyvagin_sigma_operator(D, c, Integer(49)); k49
(79, 32, 31, 11, 53, 37, 1, 23, 15, 7, 0, 0, 0, 64, 32, 34, 53, 0, 27, 27, 0, 0, 0, 26, 0, 0, 18, 0, 22, 0, 53, 19, 27, 10, 0, 0, 0, 30, 35, 38, 0, 0, 0, 53, 0, 0, 4, 0, 0, 0, 0, 0)
>>> k51 = H.kolyvagin_sigma_operator(D, c, Integer(51)); k51
(20, 12, 57, 0, 0, 0, 0, 52, 23, 15, 0, 7, 0, 0, 19, 4, 0, 73, 11, 0, 104, 31, 0, 38, 31, 0, 0, 31, 5, 47, 0, 27, 35, 0, 57, 32, 24, 10, 0, 8, 0, 31, 41, 0, 0, 0, 16, 0, 0, 0, 0, 0)
>>> V = H.modp_dual_elliptic_curve_factor(EllipticCurve('37a'), q, Integer(5)); V
Vector space of degree 52 and dimension 2 over Ring of integers modulo 3
Basis matrix:
2 x 52 dense matrix over Ring of integers modulo 3
>>> [b.dot_product(k49.element().change_ring(GF(q))) for b in V.basis()]
[1, 1]
>>> [b.dot_product(k51.element().change_ring(GF(q))) for b in V.basis()]
[1, 1]

An example with \(c\) a product of two primes:

Sage

sage: N = 389; D = -7; ell = 5; q = 3
sage: H = heegner_points(N).reduce_mod(ell)
sage: V = H.modp_dual_elliptic_curve_factor(EllipticCurve('389a'), q, 5)
sage: k = H.kolyvagin_sigma_operator(D, 17*41, 104)     # long time
sage: k                                                 # long time
(990, 656, 219, ..., 246, 534, 1254)
sage: [b.dot_product(k.element().change_ring(GF(3))) for b in V.basis()]   # long time (but only because depends on something slow)
[0, 0]

Python

>>> from sage.all import *
>>> N = Integer(389); D = -Integer(7); ell = Integer(5); q = Integer(3)
>>> H = heegner_points(N).reduce_mod(ell)
>>> V = H.modp_dual_elliptic_curve_factor(EllipticCurve('389a'), q, Integer(5))
>>> k = H.kolyvagin_sigma_operator(D, Integer(17)*Integer(41), Integer(104))     # long time
>>> k                                                 # long time
(990, 656, 219, ..., 246, 534, 1254)
>>> [b.dot_product(k.element().change_ring(GF(Integer(3)))) for b in V.basis()]   # long time (but only because depends on something slow)
[0, 0]

left_orders()[source]¶

Return the left orders associated to the representative right ideals in the Brandt module.

EXAMPLES:

Sage

sage: heegner_points(11).reduce_mod(3).left_orders()
[Order of Quaternion Algebra (-1, -3) with base ring Rational Field
  with basis (1/2 + 1/2*j + 7*k, 1/2*i + 13/2*k, j + 3*k, 11*k),
 Order of Quaternion Algebra (-1, -3) with base ring Rational Field
  with basis (1/2 + 1/2*j + 7*k, 1/4*i + 1/2*j + 63/4*k, j + 14*k, 22*k)]

Python

>>> from sage.all import *
>>> heegner_points(Integer(11)).reduce_mod(Integer(3)).left_orders()
[Order of Quaternion Algebra (-1, -3) with base ring Rational Field
  with basis (1/2 + 1/2*j + 7*k, 1/2*i + 13/2*k, j + 3*k, 11*k),
 Order of Quaternion Algebra (-1, -3) with base ring Rational Field
  with basis (1/2 + 1/2*j + 7*k, 1/4*i + 1/2*j + 63/4*k, j + 14*k, 22*k)]

level()[source]¶

Return the level.

EXAMPLES:

Sage

sage: heegner_points(11).reduce_mod(3).level()
11

Python

>>> from sage.all import *
>>> heegner_points(Integer(11)).reduce_mod(Integer(3)).level()
11

modp_dual_elliptic_curve_factor(E, p, bound=10)[source]¶

Return the factor of the Brandt module space modulo \(p\) corresponding to the elliptic curve \(E\), cut out using Hecke operators up to bound.

INPUT:

E – elliptic curve of conductor equal to the level of self
p – prime number
bound – positive integer (default: 10)

EXAMPLES:

Sage

sage: N = 37; D = -7; ell = 17; c = 41; q = 3
sage: H = heegner_points(N).reduce_mod(ell)
sage: V = H.modp_dual_elliptic_curve_factor(EllipticCurve('37a'), q, 5); V
Vector space of degree 52 and dimension 2 over Ring of integers modulo 3
 Basis matrix: 2 x 52 dense matrix over Ring of integers modulo 3

Python

>>> from sage.all import *
>>> N = Integer(37); D = -Integer(7); ell = Integer(17); c = Integer(41); q = Integer(3)
>>> H = heegner_points(N).reduce_mod(ell)
>>> V = H.modp_dual_elliptic_curve_factor(EllipticCurve('37a'), q, Integer(5)); V
Vector space of degree 52 and dimension 2 over Ring of integers modulo 3
 Basis matrix: 2 x 52 dense matrix over Ring of integers modulo 3

modp_splitting_data(p)[source]¶

Return mod \(p\) splitting data for the quaternion algebra at the unramified prime \(p\). This is a pair of \(2\times 2\) matrices \(A\), \(B\) over the finite field \(\GF{p}\) such that if the quaternion algebra has generators \(i, j, k\), then the homomorphism sending \(i\) to \(A\) and \(j\) to \(B\) maps any maximal order homomorphically onto the ring of \(2\times 2\) matrices.

Because of how the homomorphism is defined, we must assume that the prime \(p\) is odd.

INPUT:

p – unramified odd prime

OUTPUT: a 2-tuple of matrices over finite field

EXAMPLES:

Sage

sage: H = heegner_points(11).reduce_mod(7)
sage: H.quaternion_algebra()
Quaternion Algebra (-1, -7) with base ring Rational Field
sage: I, J = H.modp_splitting_data(13)
sage: I
[ 0 12]
[ 1  0]
sage: J
[7 3]
[3 6]
sage: I^2
[12  0]
[ 0 12]
sage: J^2
[6 0]
[0 6]
sage: I*J == -J*I
True

Python

>>> from sage.all import *
>>> H = heegner_points(Integer(11)).reduce_mod(Integer(7))
>>> H.quaternion_algebra()
Quaternion Algebra (-1, -7) with base ring Rational Field
>>> I, J = H.modp_splitting_data(Integer(13))
>>> I
[ 0 12]
[ 1  0]
>>> J
[7 3]
[3 6]
>>> I**Integer(2)
[12  0]
[ 0 12]
>>> J**Integer(2)
[6 0]
[0 6]
>>> I*J == -J*I
True

The following is a good test because of the asserts in the code:

Sage

sage: v = [H.modp_splitting_data(p) for p in primes(13,200)]

Python

>>> from sage.all import *
>>> v = [H.modp_splitting_data(p) for p in primes(Integer(13),Integer(200))]

Some edge cases:

Sage

sage: H.modp_splitting_data(11)
(
[ 0 10]  [6 1]
[ 1  0], [1 5]
)

Python

>>> from sage.all import *
>>> H.modp_splitting_data(Integer(11))
(
[ 0 10]  [6 1]
[ 1  0], [1 5]
)

Proper error handling:

Sage

sage: H.modp_splitting_data(7)
Traceback (most recent call last):
...
ValueError: p (=7) must be an unramified prime

sage: H.modp_splitting_data(2)
Traceback (most recent call last):
...
ValueError: p must be odd

Python

>>> from sage.all import *
>>> H.modp_splitting_data(Integer(7))
Traceback (most recent call last):
...
ValueError: p (=7) must be an unramified prime

>>> H.modp_splitting_data(Integer(2))
Traceback (most recent call last):
...
ValueError: p must be odd

modp_splitting_map(p)[source]¶

Return (algebra) map from the (\(p\)-integral) quaternion algebra to the set of \(2\times 2\) matrices over \(\GF{p}\).

INPUT:

p – prime number

EXAMPLES:

Sage

sage: H = heegner_points(11).reduce_mod(7)
sage: f = H.modp_splitting_map(13)
sage: B = H.quaternion_algebra(); B
Quaternion Algebra (-1, -7) with base ring Rational Field
sage: i, j, k = H.quaternion_algebra().gens()
sage: a = 2 + i - j + 3*k; b = 7 + 2*i - 4*j + k
sage: f(a*b)
[12  3]
[10  5]
sage: f(a)*f(b)
[12  3]
[10  5]

Python

>>> from sage.all import *
>>> H = heegner_points(Integer(11)).reduce_mod(Integer(7))
>>> f = H.modp_splitting_map(Integer(13))
>>> B = H.quaternion_algebra(); B
Quaternion Algebra (-1, -7) with base ring Rational Field
>>> i, j, k = H.quaternion_algebra().gens()
>>> a = Integer(2) + i - j + Integer(3)*k; b = Integer(7) + Integer(2)*i - Integer(4)*j + k
>>> f(a*b)
[12  3]
[10  5]
>>> f(a)*f(b)
[12  3]
[10  5]

optimal_embeddings(D, c, R)[source]¶

INPUT:

D – negative fundamental discriminant
c – integer coprime
R – Eichler order

EXAMPLES:

Sage

sage: H = heegner_points(11).reduce_mod(3)
sage: R = H.left_orders()[0]
sage: H.optimal_embeddings(-7, 1, R)
[Embedding sending sqrt(-7) to i - j - k,
 Embedding sending sqrt(-7) to -i + j + k]
sage: H.optimal_embeddings(-7, 2, R)
[Embedding sending 2*sqrt(-7) to 5*i - k,
 Embedding sending 2*sqrt(-7) to -5*i + k,
 Embedding sending 2*sqrt(-7) to 2*i - 2*j - 2*k,
 Embedding sending 2*sqrt(-7) to -2*i + 2*j + 2*k]

Python

>>> from sage.all import *
>>> H = heegner_points(Integer(11)).reduce_mod(Integer(3))
>>> R = H.left_orders()[Integer(0)]
>>> H.optimal_embeddings(-Integer(7), Integer(1), R)
[Embedding sending sqrt(-7) to i - j - k,
 Embedding sending sqrt(-7) to -i + j + k]
>>> H.optimal_embeddings(-Integer(7), Integer(2), R)
[Embedding sending 2*sqrt(-7) to 5*i - k,
 Embedding sending 2*sqrt(-7) to -5*i + k,
 Embedding sending 2*sqrt(-7) to 2*i - 2*j - 2*k,
 Embedding sending 2*sqrt(-7) to -2*i + 2*j + 2*k]

quadratic_field(D)[source]¶

Return our fixed choice of quadratic imaginary field of discriminant \(D\).

INPUT:

D – fundamental discriminant

OUTPUT: a quadratic number field

EXAMPLES:

Sage

sage: H = heegner_points(389).reduce_mod(5)
sage: H.quadratic_field(-7)
Number Field in sqrt_minus_7 with defining polynomial x^2 + 7
 with sqrt_minus_7 = 2.645751311064591?*I

Python

>>> from sage.all import *
>>> H = heegner_points(Integer(389)).reduce_mod(Integer(5))
>>> H.quadratic_field(-Integer(7))
Number Field in sqrt_minus_7 with defining polynomial x^2 + 7
 with sqrt_minus_7 = 2.645751311064591?*I

quaternion_algebra()[source]¶

Return the rational quaternion algebra used to implement self.

EXAMPLES:

Sage

sage: heegner_points(389).reduce_mod(7).quaternion_algebra()
Quaternion Algebra (-1, -7) with base ring Rational Field

Python

>>> from sage.all import *
>>> heegner_points(Integer(389)).reduce_mod(Integer(7)).quaternion_algebra()
Quaternion Algebra (-1, -7) with base ring Rational Field

rational_kolyvagin_divisor(D, c)[source]¶

Return the Kolyvagin divisor as an element of the Brandt module corresponding to the discriminant \(D\) and conductor \(c\), which both must be coprime to \(N\ell\).

INPUT:

D – discriminant (negative integer)
c – conductor (positive integer)

OUTPUT: Brandt module element (or tuple of them)

EXAMPLES:

Sage

sage: N = 389; D = -7; ell = 5; c = 17; q = 3
sage: H = heegner_points(N).reduce_mod(ell)
sage: k = H.rational_kolyvagin_divisor(D, c); k  # long time (5s on sage.math, 2013)
(2, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 4, 0, 0, 9, 11, 0, 6, 0, 0, 7, 0, 0, 0, 0, 14, 12, 13, 15, 17, 0, 0, 0, 0, 8, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
sage: V = H.modp_dual_elliptic_curve_factor(EllipticCurve('389a'), q, 2)
sage: [b.dot_product(k.element().change_ring(GF(q))) for b in V.basis()]  # long time
[0, 0]
sage: k = H.rational_kolyvagin_divisor(D, 59)
sage: [b.dot_product(k.element().change_ring(GF(q))) for b in V.basis()]
[2, 0]

Python

>>> from sage.all import *
>>> N = Integer(389); D = -Integer(7); ell = Integer(5); c = Integer(17); q = Integer(3)
>>> H = heegner_points(N).reduce_mod(ell)
>>> k = H.rational_kolyvagin_divisor(D, c); k  # long time (5s on sage.math, 2013)
(2, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 4, 0, 0, 9, 11, 0, 6, 0, 0, 7, 0, 0, 0, 0, 14, 12, 13, 15, 17, 0, 0, 0, 0, 8, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
>>> V = H.modp_dual_elliptic_curve_factor(EllipticCurve('389a'), q, Integer(2))
>>> [b.dot_product(k.element().change_ring(GF(q))) for b in V.basis()]  # long time
[0, 0]
>>> k = H.rational_kolyvagin_divisor(D, Integer(59))
>>> [b.dot_product(k.element().change_ring(GF(q))) for b in V.basis()]
[2, 0]

right_ideals()[source]¶

Return representative right ideals in the Brandt module.

EXAMPLES:

Sage

sage: heegner_points(11).reduce_mod(3).right_ideals()
(Fractional ideal (4, 44*i, 2 + 8*i + 2*j, 34*i + 2*k),
 Fractional ideal (8, 88*i, 2 + 52*i + 2*j, 4 + 78*i + 2*k))

Python

>>> from sage.all import *
>>> heegner_points(Integer(11)).reduce_mod(Integer(3)).right_ideals()
(Fractional ideal (4, 44*i, 2 + 8*i + 2*j, 34*i + 2*k),
 Fractional ideal (8, 88*i, 2 + 52*i + 2*j, 4 + 78*i + 2*k))

satisfies_heegner_hypothesis(D, c=1)[source]¶

The fundamental discriminant \(D\) must be coprime to \(N\ell\), and must define a quadratic imaginary field \(K\) in which \(\ell\) is inert. Also, all primes dividing \(N\) must split in \(K\), and \(c\) must be squarefree and coprime to \(ND\ell\).

INPUT:

D – negative integer
c – positive integer (default: 1)

OUTPUT: boolean

EXAMPLES:

Sage

sage: H = heegner_points(11).reduce_mod(7)
sage: H.satisfies_heegner_hypothesis(-5)
False
sage: H.satisfies_heegner_hypothesis(-7)
False
sage: H.satisfies_heegner_hypothesis(-8)
True
sage: [D for D in [-1,-2..-100] if H.satisfies_heegner_hypothesis(D)]
[-8, -39, -43, -51, -79, -95]

Python

>>> from sage.all import *
>>> H = heegner_points(Integer(11)).reduce_mod(Integer(7))
>>> H.satisfies_heegner_hypothesis(-Integer(5))
False
>>> H.satisfies_heegner_hypothesis(-Integer(7))
False
>>> H.satisfies_heegner_hypothesis(-Integer(8))
True
>>> [D for D in (ellipsis_range(-Integer(1),-Integer(2),Ellipsis,-Integer(100))) if H.satisfies_heegner_hypothesis(D)]
[-8, -39, -43, -51, -79, -95]

class sage.schemes.elliptic_curves.heegner.HeegnerQuatAlgEmbedding(D, c, R, beta)[source]¶

Bases: SageObject

The homomorphism \(\mathcal{O} \to R\), where \(\mathcal{O}\) is the order of conductor \(c\) in the quadratic field of discriminant \(D\), and \(R\) is an Eichler order in a quaternion algebra.

EXAMPLES:

Sage

sage: H = heegner_points(11).reduce_mod(3); R = H.left_orders()[0]
sage: f = H.optimal_embeddings(-7, 2, R)[1]; f
Embedding sending 2*sqrt(-7) to -5*i + k
sage: type(f)
<class 'sage.schemes.elliptic_curves.heegner.HeegnerQuatAlgEmbedding'>
sage: loads(dumps(f)) == f
True

Python

>>> from sage.all import *
>>> H = heegner_points(Integer(11)).reduce_mod(Integer(3)); R = H.left_orders()[Integer(0)]
>>> f = H.optimal_embeddings(-Integer(7), Integer(2), R)[Integer(1)]; f
Embedding sending 2*sqrt(-7) to -5*i + k
>>> type(f)
<class 'sage.schemes.elliptic_curves.heegner.HeegnerQuatAlgEmbedding'>
>>> loads(dumps(f)) == f
True

beta()[source]¶

Return the element \(\beta\) in the quaternion algebra order that \(c\sqrt{D}\) maps to.

EXAMPLES:

Sage

sage: H = heegner_points(11).reduce_mod(3); R = H.left_orders()[0]
sage: H.optimal_embeddings(-7, 2, R)[1].beta()
-5*i + k

Python

>>> from sage.all import *
>>> H = heegner_points(Integer(11)).reduce_mod(Integer(3)); R = H.left_orders()[Integer(0)]
>>> H.optimal_embeddings(-Integer(7), Integer(2), R)[Integer(1)].beta()
-5*i + k

codomain()[source]¶

Return the codomain of this embedding.

EXAMPLES:

Sage

sage: H = heegner_points(11).reduce_mod(3); R = H.left_orders()[0]
sage: H.optimal_embeddings(-7, 2, R)[0].codomain()
Order of Quaternion Algebra (-1, -3) with base ring Rational Field
 with basis (1/2 + 1/2*j + 7*k, 1/2*i + 13/2*k, j + 3*k, 11*k)

Python

>>> from sage.all import *
>>> H = heegner_points(Integer(11)).reduce_mod(Integer(3)); R = H.left_orders()[Integer(0)]
>>> H.optimal_embeddings(-Integer(7), Integer(2), R)[Integer(0)].codomain()
Order of Quaternion Algebra (-1, -3) with base ring Rational Field
 with basis (1/2 + 1/2*j + 7*k, 1/2*i + 13/2*k, j + 3*k, 11*k)

conjugate()[source]¶

Return the conjugate of this embedding, which is also an embedding.

EXAMPLES:

Sage

sage: H = heegner_points(11).reduce_mod(3); R = H.left_orders()[0]
sage: f = H.optimal_embeddings(-7, 2, R)[1]
sage: f.conjugate()
Embedding sending 2*sqrt(-7) to 5*i - k
sage: f
Embedding sending 2*sqrt(-7) to -5*i + k

Python

>>> from sage.all import *
>>> H = heegner_points(Integer(11)).reduce_mod(Integer(3)); R = H.left_orders()[Integer(0)]
>>> f = H.optimal_embeddings(-Integer(7), Integer(2), R)[Integer(1)]
>>> f.conjugate()
Embedding sending 2*sqrt(-7) to 5*i - k
>>> f
Embedding sending 2*sqrt(-7) to -5*i + k

domain()[source]¶

Return the domain of this embedding.

EXAMPLES:

Sage

sage: H = heegner_points(11).reduce_mod(3); R = H.left_orders()[0]
sage: H.optimal_embeddings(-7, 2, R)[0].domain()
Order of conductor 4 generated by 2*a
 in Number Field in a with defining polynomial x^2 + 7
 with a = 2.645751311064591?*I

Python

>>> from sage.all import *
>>> H = heegner_points(Integer(11)).reduce_mod(Integer(3)); R = H.left_orders()[Integer(0)]
>>> H.optimal_embeddings(-Integer(7), Integer(2), R)[Integer(0)].domain()
Order of conductor 4 generated by 2*a
 in Number Field in a with defining polynomial x^2 + 7
 with a = 2.645751311064591?*I

domain_conductor()[source]¶

Return the conductor of the domain.

EXAMPLES:

Sage

sage: H = heegner_points(11).reduce_mod(3); R = H.left_orders()[0]
sage: H.optimal_embeddings(-7, 2, R)[0].domain_conductor()
2

Python

>>> from sage.all import *
>>> H = heegner_points(Integer(11)).reduce_mod(Integer(3)); R = H.left_orders()[Integer(0)]
>>> H.optimal_embeddings(-Integer(7), Integer(2), R)[Integer(0)].domain_conductor()
2

domain_gen()[source]¶

Return the specific generator \(c \sqrt{D}\) for the domain order.

EXAMPLES:

Sage

sage: H = heegner_points(11).reduce_mod(3); R = H.left_orders()[0]
sage: f = H.optimal_embeddings(-7, 2, R)[0]
sage: f.domain_gen()
2*a
sage: f.domain_gen()^2
-28

Python

>>> from sage.all import *
>>> H = heegner_points(Integer(11)).reduce_mod(Integer(3)); R = H.left_orders()[Integer(0)]
>>> f = H.optimal_embeddings(-Integer(7), Integer(2), R)[Integer(0)]
>>> f.domain_gen()
2*a
>>> f.domain_gen()**Integer(2)
-28

matrix()[source]¶

Return matrix over \(\QQ\) of this morphism, with respect to the basis 1, \(c\sqrt{D}\) of the domain and the basis \(1,i,j,k\) of the ambient rational quaternion algebra (which contains the domain).

EXAMPLES:

Sage

sage: H = heegner_points(11).reduce_mod(3); R = H.left_orders()[0]
sage: f = H.optimal_embeddings(-7, 1, R)[1]; f
Embedding sending sqrt(-7) to -i + j + k
sage: f.matrix()
[ 1  0  0  0]
[ 0 -1  1  1]
sage: f.conjugate().matrix()
[ 1  0  0  0]
[ 0  1 -1 -1]

Python

>>> from sage.all import *
>>> H = heegner_points(Integer(11)).reduce_mod(Integer(3)); R = H.left_orders()[Integer(0)]
>>> f = H.optimal_embeddings(-Integer(7), Integer(1), R)[Integer(1)]; f
Embedding sending sqrt(-7) to -i + j + k
>>> f.matrix()
[ 1  0  0  0]
[ 0 -1  1  1]
>>> f.conjugate().matrix()
[ 1  0  0  0]
[ 0  1 -1 -1]

class sage.schemes.elliptic_curves.heegner.KolyvaginCohomologyClass(kolyvagin_point, n)[source]¶

Bases: SageObject

A Kolyvagin cohomology class in \(H^1(K,E[n])\) or \(H^1(K,E)[n]\) attached to a Heegner point.

EXAMPLES:

Sage

sage: y = EllipticCurve('37a').heegner_point(-7)
sage: c = y.kolyvagin_cohomology_class(3); c
Kolyvagin cohomology class c(1) in H^1(K,E[3])
sage: type(c)
<class 'sage.schemes.elliptic_curves.heegner.KolyvaginCohomologyClassEn'>
sage: loads(dumps(c)) == c
True
sage: y.kolyvagin_cohomology_class(5)
Kolyvagin cohomology class c(1) in H^1(K,E[5])

Python

>>> from sage.all import *
>>> y = EllipticCurve('37a').heegner_point(-Integer(7))
>>> c = y.kolyvagin_cohomology_class(Integer(3)); c
Kolyvagin cohomology class c(1) in H^1(K,E[3])
>>> type(c)
<class 'sage.schemes.elliptic_curves.heegner.KolyvaginCohomologyClassEn'>
>>> loads(dumps(c)) == c
True
>>> y.kolyvagin_cohomology_class(Integer(5))
Kolyvagin cohomology class c(1) in H^1(K,E[5])

conductor()[source]¶

Return the integer \(c\) such that this cohomology class is associated to the Heegner point \(y_c\).

EXAMPLES:

Sage

sage: y = EllipticCurve('37a').heegner_point(-7, 5)
sage: t = y.kolyvagin_cohomology_class()
sage: t.conductor()
5

Python

>>> from sage.all import *
>>> y = EllipticCurve('37a').heegner_point(-Integer(7), Integer(5))
>>> t = y.kolyvagin_cohomology_class()
>>> t.conductor()
5

heegner_point()[source]¶

Return the Heegner point \(y_c\) to which this cohomology class is associated.

EXAMPLES:

Sage

sage: y = EllipticCurve('37a').heegner_point(-7, 5)
sage: t = y.kolyvagin_cohomology_class()
sage: t.heegner_point()
Heegner point of discriminant -7 and conductor 5
 on elliptic curve of conductor 37

Python

>>> from sage.all import *
>>> y = EllipticCurve('37a').heegner_point(-Integer(7), Integer(5))
>>> t = y.kolyvagin_cohomology_class()
>>> t.heegner_point()
Heegner point of discriminant -7 and conductor 5
 on elliptic curve of conductor 37

kolyvagin_point()[source]¶

Return the Kolyvagin point \(P_c\) to which this cohomology class is associated.

EXAMPLES:

Sage

sage: y = EllipticCurve('37a').heegner_point(-7, 5)
sage: t = y.kolyvagin_cohomology_class()
sage: t.kolyvagin_point()
Kolyvagin point of discriminant -7 and conductor 5
 on elliptic curve of conductor 37

Python

>>> from sage.all import *
>>> y = EllipticCurve('37a').heegner_point(-Integer(7), Integer(5))
>>> t = y.kolyvagin_cohomology_class()
>>> t.kolyvagin_point()
Kolyvagin point of discriminant -7 and conductor 5
 on elliptic curve of conductor 37

n()[source]¶

Return the integer \(n\) so that this is a cohomology class in \(H^1(K,E[n])\) or \(H^1(K,E)[n]\).

EXAMPLES:

Sage

sage: y = EllipticCurve('37a').heegner_point(-7)
sage: t = y.kolyvagin_cohomology_class(3); t
Kolyvagin cohomology class c(1) in H^1(K,E[3])
sage: t.n()
3

Python

>>> from sage.all import *
>>> y = EllipticCurve('37a').heegner_point(-Integer(7))
>>> t = y.kolyvagin_cohomology_class(Integer(3)); t
Kolyvagin cohomology class c(1) in H^1(K,E[3])
>>> t.n()
3

class sage.schemes.elliptic_curves.heegner.KolyvaginCohomologyClassEn(kolyvagin_point, n)[source]¶: Bases: KolyvaginCohomologyClass

class sage.schemes.elliptic_curves.heegner.KolyvaginPoint(heegner_point)[source]¶

Bases: HeegnerPoint

A Kolyvagin point.

EXAMPLES:

We create a few Kolyvagin points:

Sage

sage: EllipticCurve('11a1').kolyvagin_point(-7)
Kolyvagin point of discriminant -7 on elliptic curve of conductor 11
sage: EllipticCurve('37a1').kolyvagin_point(-7)
Kolyvagin point of discriminant -7 on elliptic curve of conductor 37
sage: EllipticCurve('37a1').kolyvagin_point(-67)
Kolyvagin point of discriminant -67 on elliptic curve of conductor 37
sage: EllipticCurve('389a1').kolyvagin_point(-7, 5)
Kolyvagin point of discriminant -7 and conductor 5
 on elliptic curve of conductor 389

Python

>>> from sage.all import *
>>> EllipticCurve('11a1').kolyvagin_point(-Integer(7))
Kolyvagin point of discriminant -7 on elliptic curve of conductor 11
>>> EllipticCurve('37a1').kolyvagin_point(-Integer(7))
Kolyvagin point of discriminant -7 on elliptic curve of conductor 37
>>> EllipticCurve('37a1').kolyvagin_point(-Integer(67))
Kolyvagin point of discriminant -67 on elliptic curve of conductor 37
>>> EllipticCurve('389a1').kolyvagin_point(-Integer(7), Integer(5))
Kolyvagin point of discriminant -7 and conductor 5
 on elliptic curve of conductor 389

One can also associated a Kolyvagin point to a Heegner point:

Sage

sage: y = EllipticCurve('37a1').heegner_point(-7); y
Heegner point of discriminant -7 on elliptic curve of conductor 37
sage: y.kolyvagin_point()
Kolyvagin point of discriminant -7 on elliptic curve of conductor 37

Python

>>> from sage.all import *
>>> y = EllipticCurve('37a1').heegner_point(-Integer(7)); y
Heegner point of discriminant -7 on elliptic curve of conductor 37
>>> y.kolyvagin_point()
Kolyvagin point of discriminant -7 on elliptic curve of conductor 37

curve()[source]¶

Return the elliptic curve over \(\QQ\) on which this Kolyvagin point sits.

EXAMPLES:

Sage

sage: E = EllipticCurve('37a1'); P = E.kolyvagin_point(-67, 3)
sage: P.curve()
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field

Python

>>> from sage.all import *
>>> E = EllipticCurve('37a1'); P = E.kolyvagin_point(-Integer(67), Integer(3))
>>> P.curve()
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field

heegner_point()[source]¶

This Kolyvagin point \(P_c\) is associated to some Heegner point \(y_c\) via Kolyvagin’s construction. This function returns that point \(y_c\).

EXAMPLES:

Sage

sage: E = EllipticCurve('37a1')
sage: P = E.kolyvagin_point(-67); P
Kolyvagin point of discriminant -67 on elliptic curve of conductor 37
sage: y = P.heegner_point(); y
Heegner point of discriminant -67 on elliptic curve of conductor 37
sage: y.kolyvagin_point() is P
True

Python

>>> from sage.all import *
>>> E = EllipticCurve('37a1')
>>> P = E.kolyvagin_point(-Integer(67)); P
Kolyvagin point of discriminant -67 on elliptic curve of conductor 37
>>> y = P.heegner_point(); y
Heegner point of discriminant -67 on elliptic curve of conductor 37
>>> y.kolyvagin_point() is P
True

index(*args, **kwds)[source]¶

Return index of this Kolyvagin point in the full group of \(K_c\) rational points on \(E\).

When the conductor is 1, this is computed numerically using the Gross-Zagier formula and explicit point search, and it may be off by \(2\). See the documentation for E.heegner_index, where \(E\) is the curve attached to self.

EXAMPLES:

Sage

sage: E = EllipticCurve('37a1'); P = E.kolyvagin_point(-67); P.index()
6

Python

>>> from sage.all import *
>>> E = EllipticCurve('37a1'); P = E.kolyvagin_point(-Integer(67)); P.index()
6

kolyvagin_cohomology_class(n=None)[source]¶

INPUT:

n – positive integer that divides the gcd of \(a_p\) and \(p+1\) for all \(p\) dividing the conductor. If \(n\) is None, choose the largest valid \(n\).

EXAMPLES:

Sage

sage: y = EllipticCurve('389a').heegner_point(-7, 5)
sage: P = y.kolyvagin_point()
sage: P.kolyvagin_cohomology_class(3)
Kolyvagin cohomology class c(5) in H^1(K,E[3])

sage: y = EllipticCurve('37a').heegner_point(-7, 5).kolyvagin_point()
sage: y.kolyvagin_cohomology_class()
Kolyvagin cohomology class c(5) in H^1(K,E[2])

Python

>>> from sage.all import *
>>> y = EllipticCurve('389a').heegner_point(-Integer(7), Integer(5))
>>> P = y.kolyvagin_point()
>>> P.kolyvagin_cohomology_class(Integer(3))
Kolyvagin cohomology class c(5) in H^1(K,E[3])

>>> y = EllipticCurve('37a').heegner_point(-Integer(7), Integer(5)).kolyvagin_point()
>>> y.kolyvagin_cohomology_class()
Kolyvagin cohomology class c(5) in H^1(K,E[2])

mod(p, prec=53)[source]¶

Return the trace of the reduction \(Q\) modulo a prime over \(p\) of this Kolyvagin point as an element of \(E(\GF{p})\), where \(p\) is any prime that is inert in \(K\) that is coprime to \(NDc\).

The point \(Q\) is only well defined up to an element of \((p+1) E(\GF{p})\), i.e., it gives a well defined element of the abelian group \(E(\GF{p}) / (p+1) E(\GF{p})\).

See [St2011b], Proposition 5.4 for a proof of the above well-definedness assertion.

EXAMPLES:

A Kolyvagin point on a rank 1 curve:

Sage

sage: E = EllipticCurve('37a1'); P = E.kolyvagin_point(-67)
sage: P.mod(2)
(1 : 1 : 1)
sage: P.mod(3)
(1 : 0 : 1)
sage: P.mod(5)
(2 : 2 : 1)
sage: P.mod(7)
(6 : 0 : 1)
sage: P.trace_to_real_numerical()
(1.61355529131986 : -2.18446840788880 : 1.00000000000000)
sage: P._trace_exact_conductor_1()  # the actual point we're reducing
(1357/841 : -53277/24389 : 1)
sage: (P._trace_exact_conductor_1().height() / E.regulator()).sqrt()
12.0000000000000

Python

>>> from sage.all import *
>>> E = EllipticCurve('37a1'); P = E.kolyvagin_point(-Integer(67))
>>> P.mod(Integer(2))
(1 : 1 : 1)
>>> P.mod(Integer(3))
(1 : 0 : 1)
>>> P.mod(Integer(5))
(2 : 2 : 1)
>>> P.mod(Integer(7))
(6 : 0 : 1)
>>> P.trace_to_real_numerical()
(1.61355529131986 : -2.18446840788880 : 1.00000000000000)
>>> P._trace_exact_conductor_1()  # the actual point we're reducing
(1357/841 : -53277/24389 : 1)
>>> (P._trace_exact_conductor_1().height() / E.regulator()).sqrt()
12.0000000000000

Here the Kolyvagin point is a torsion point (since \(E\) has rank 1), and we reduce it modulo several primes.:

Sage

sage: E = EllipticCurve('11a1'); P = E.kolyvagin_point(-7)
sage: P.mod(3,70)  # long time (4s on sage.math, 2013)
(1 : 2 : 1)
sage: P.mod(5,70)
(1 : 4 : 1)
sage: P.mod(7,70)
Traceback (most recent call last):
...
ValueError: p must be coprime to conductors and discriminant
sage: P.mod(11,70)
Traceback (most recent call last):
...
ValueError: p must be coprime to conductors and discriminant
sage: P.mod(13,70)
(3 : 4 : 1)

Python

>>> from sage.all import *
>>> E = EllipticCurve('11a1'); P = E.kolyvagin_point(-Integer(7))
>>> P.mod(Integer(3),Integer(70))  # long time (4s on sage.math, 2013)
(1 : 2 : 1)
>>> P.mod(Integer(5),Integer(70))
(1 : 4 : 1)
>>> P.mod(Integer(7),Integer(70))
Traceback (most recent call last):
...
ValueError: p must be coprime to conductors and discriminant
>>> P.mod(Integer(11),Integer(70))
Traceback (most recent call last):
...
ValueError: p must be coprime to conductors and discriminant
>>> P.mod(Integer(13),Integer(70))
(3 : 4 : 1)

numerical_approx(prec=53)[source]¶

Return a numerical approximation to this Kolyvagin point using prec bits of working precision.

INPUT:

prec – precision in bits (default: 53)

EXAMPLES:

Sage

sage: P = EllipticCurve('37a1').kolyvagin_point(-7); P
Kolyvagin point of discriminant -7 on elliptic curve of conductor 37
sage: P.numerical_approx() # approx. (0 : 0 : 1)
(...e-16 - ...e-16*I : ...e-16 + ...e-16*I : 1.00000000000000)
sage: P.numerical_approx(100)[0].abs() < 2.0^-99
True

sage: P = EllipticCurve('389a1').kolyvagin_point(-7, 5); P
Kolyvagin point of discriminant -7 and conductor 5
 on elliptic curve of conductor 389

Python

>>> from sage.all import *
>>> P = EllipticCurve('37a1').kolyvagin_point(-Integer(7)); P
Kolyvagin point of discriminant -7 on elliptic curve of conductor 37
>>> P.numerical_approx() # approx. (0 : 0 : 1)
(...e-16 - ...e-16*I : ...e-16 + ...e-16*I : 1.00000000000000)
>>> P.numerical_approx(Integer(100))[Integer(0)].abs() < RealNumber('2.0')**-Integer(99)
True

>>> P = EllipticCurve('389a1').kolyvagin_point(-Integer(7), Integer(5)); P
Kolyvagin point of discriminant -7 and conductor 5
 on elliptic curve of conductor 389

Numerical approximation is only implemented for points of conductor 1:

Sage

sage: P.numerical_approx()
Traceback (most recent call last):
...
NotImplementedError

Python

>>> from sage.all import *
>>> P.numerical_approx()
Traceback (most recent call last):
...
NotImplementedError

plot(prec=53, *args, **kwds)[source]¶

Plot a Kolyvagin point \(P_1\) if it is defined over the rational numbers.

EXAMPLES:

Sage

sage: E = EllipticCurve('37a'); P = E.heegner_point(-11).kolyvagin_point()
sage: P.plot(prec=30, pointsize=50, rgbcolor='red') + E.plot()              # needs sage.plot
Graphics object consisting of 3 graphics primitives

Python

>>> from sage.all import *
>>> E = EllipticCurve('37a'); P = E.heegner_point(-Integer(11)).kolyvagin_point()
>>> P.plot(prec=Integer(30), pointsize=Integer(50), rgbcolor='red') + E.plot()              # needs sage.plot
Graphics object consisting of 3 graphics primitives

point_exact(prec=53)[source]¶

INPUT:

prec – precision in bits (default: 53)

EXAMPLES:

A rank 1 curve:

Sage

sage: E = EllipticCurve('37a1'); P = E.kolyvagin_point(-67)
sage: P.point_exact()
(6 : -15 : 1)
sage: P.point_exact(40)
(6 : -15 : 1)
sage: P.point_exact(20)
Traceback (most recent call last):
...
RuntimeError: insufficient precision to find exact point

Python

>>> from sage.all import *
>>> E = EllipticCurve('37a1'); P = E.kolyvagin_point(-Integer(67))
>>> P.point_exact()
(6 : -15 : 1)
>>> P.point_exact(Integer(40))
(6 : -15 : 1)
>>> P.point_exact(Integer(20))
Traceback (most recent call last):
...
RuntimeError: insufficient precision to find exact point

A rank 0 curve:

Sage

sage: E = EllipticCurve('11a1'); P = E.kolyvagin_point(-7)
sage: P.point_exact()
(-1/2*sqrt_minus_7 + 1/2 : -2*sqrt_minus_7 - 2 : 1)

Python

>>> from sage.all import *
>>> E = EllipticCurve('11a1'); P = E.kolyvagin_point(-Integer(7))
>>> P.point_exact()
(-1/2*sqrt_minus_7 + 1/2 : -2*sqrt_minus_7 - 2 : 1)

A rank 2 curve:

Sage

sage: E = EllipticCurve('389a1'); P = E.kolyvagin_point(-7)
sage: P.point_exact()
(0 : 1 : 0)

Python

>>> from sage.all import *
>>> E = EllipticCurve('389a1'); P = E.kolyvagin_point(-Integer(7))
>>> P.point_exact()
(0 : 1 : 0)

satisfies_kolyvagin_hypothesis(n=None)[source]¶

Return True if this Kolyvagin point satisfies the Heegner hypothesis for \(n\), so that it defines a Galois equivariant element of \(E(K_c)/n E(K_c)\).

EXAMPLES:

Sage

sage: y = EllipticCurve('389a').heegner_point(-7,5); P = y.kolyvagin_point()
sage: P.kolyvagin_cohomology_class(3)
Kolyvagin cohomology class c(5) in H^1(K,E[3])
sage: P.satisfies_kolyvagin_hypothesis(3)
True
sage: P.satisfies_kolyvagin_hypothesis(5)
False
sage: P.satisfies_kolyvagin_hypothesis(7)
False
sage: P.satisfies_kolyvagin_hypothesis(11)
False

Python

>>> from sage.all import *
>>> y = EllipticCurve('389a').heegner_point(-Integer(7),Integer(5)); P = y.kolyvagin_point()
>>> P.kolyvagin_cohomology_class(Integer(3))
Kolyvagin cohomology class c(5) in H^1(K,E[3])
>>> P.satisfies_kolyvagin_hypothesis(Integer(3))
True
>>> P.satisfies_kolyvagin_hypothesis(Integer(5))
False
>>> P.satisfies_kolyvagin_hypothesis(Integer(7))
False
>>> P.satisfies_kolyvagin_hypothesis(Integer(11))
False

trace_to_real_numerical(prec=53)[source]¶

Return the trace of this Kolyvagin point down to the real numbers, computed numerically using prec bits of working precision.

EXAMPLES:

Sage

sage: E = EllipticCurve('37a1'); P = E.kolyvagin_point(-67)
sage: PP = P.numerical_approx()
sage: [c.real() for c in PP]
[6.00000000000000, -15.0000000000000, 1.00000000000000]
sage: all(c.imag().abs() < 1e-14 for c in PP)
True
sage: P.trace_to_real_numerical()
(1.61355529131986 : -2.18446840788880 : 1.00000000000000)
sage: P.trace_to_real_numerical(prec=80)  # abs tol 1e-21
(1.6135552913198573127230 : -2.1844684078888023289187 : 1.0000000000000000000000)

Python

>>> from sage.all import *
>>> E = EllipticCurve('37a1'); P = E.kolyvagin_point(-Integer(67))
>>> PP = P.numerical_approx()
>>> [c.real() for c in PP]
[6.00000000000000, -15.0000000000000, 1.00000000000000]
>>> all(c.imag().abs() < RealNumber('1e-14') for c in PP)
True
>>> P.trace_to_real_numerical()
(1.61355529131986 : -2.18446840788880 : 1.00000000000000)
>>> P.trace_to_real_numerical(prec=Integer(80))  # abs tol 1e-21
(1.6135552913198573127230 : -2.1844684078888023289187 : 1.0000000000000000000000)

class sage.schemes.elliptic_curves.heegner.RingClassField(D, c, check=True)[source]¶

Bases: SageObject

A Ring class field of a quadratic imaginary field of given conductor.

Note

This is a ring class field, not a ray class field. In general, the ring class field of given conductor is a subfield of the ray class field of the same conductor.

EXAMPLES:

Sage

sage: heegner_point(37,-7).ring_class_field()
Hilbert class field of QQ[sqrt(-7)]
sage: heegner_point(37,-7,5).ring_class_field()
Ring class field extension of QQ[sqrt(-7)] of conductor 5
sage: heegner_point(37,-7,55).ring_class_field()
Ring class field extension of QQ[sqrt(-7)] of conductor 55

Python

>>> from sage.all import *
>>> heegner_point(Integer(37),-Integer(7)).ring_class_field()
Hilbert class field of QQ[sqrt(-7)]
>>> heegner_point(Integer(37),-Integer(7),Integer(5)).ring_class_field()
Ring class field extension of QQ[sqrt(-7)] of conductor 5
>>> heegner_point(Integer(37),-Integer(7),Integer(55)).ring_class_field()
Ring class field extension of QQ[sqrt(-7)] of conductor 55

absolute_degree()[source]¶

Return the absolute degree of this field over \(\QQ\).

EXAMPLES:

Sage

sage: E = EllipticCurve('389a'); K = E.heegner_point(-7,5).ring_class_field()
sage: K.absolute_degree()
12
sage: K.degree_over_K()
6

Python

>>> from sage.all import *
>>> E = EllipticCurve('389a'); K = E.heegner_point(-Integer(7),Integer(5)).ring_class_field()
>>> K.absolute_degree()
12
>>> K.degree_over_K()
6

conductor()[source]¶

Return the conductor of this ring class field.

EXAMPLES:

Sage

sage: E = EllipticCurve('389a'); K5 = E.heegner_point(-7,5).ring_class_field()
sage: K5.conductor()
5

Python

>>> from sage.all import *
>>> E = EllipticCurve('389a'); K5 = E.heegner_point(-Integer(7),Integer(5)).ring_class_field()
>>> K5.conductor()
5

degree_over_H()[source]¶

Return the degree of this field over the Hilbert class field \(H\) of \(K\).

EXAMPLES:

Sage

sage: E = EllipticCurve('389a')
sage: E.heegner_point(-59).ring_class_field().degree_over_H()
1
sage: E.heegner_point(-59).ring_class_field().degree_over_K()
3
sage: QuadraticField(-59,'a').class_number()
3

Python

>>> from sage.all import *
>>> E = EllipticCurve('389a')
>>> E.heegner_point(-Integer(59)).ring_class_field().degree_over_H()
1
>>> E.heegner_point(-Integer(59)).ring_class_field().degree_over_K()
3
>>> QuadraticField(-Integer(59),'a').class_number()
3

Some examples in which prime dividing c is inert:

Sage

sage: heegner_point(37,-7,3).ring_class_field().degree_over_H()
4
sage: heegner_point(37,-7,3^2).ring_class_field().degree_over_H()
12
sage: heegner_point(37,-7,3^3).ring_class_field().degree_over_H()
36

Python

>>> from sage.all import *
>>> heegner_point(Integer(37),-Integer(7),Integer(3)).ring_class_field().degree_over_H()
4
>>> heegner_point(Integer(37),-Integer(7),Integer(3)**Integer(2)).ring_class_field().degree_over_H()
12
>>> heegner_point(Integer(37),-Integer(7),Integer(3)**Integer(3)).ring_class_field().degree_over_H()
36

The prime dividing c is split. For example, in the first case \(O_K/cO_K\) is isomorphic to a direct sum of two copies of GF(2), so the units are trivial:

Sage

sage: heegner_point(37,-7,2).ring_class_field().degree_over_H()
1
sage: heegner_point(37,-7,4).ring_class_field().degree_over_H()
2
sage: heegner_point(37,-7,8).ring_class_field().degree_over_H()
4

Python

>>> from sage.all import *
>>> heegner_point(Integer(37),-Integer(7),Integer(2)).ring_class_field().degree_over_H()
1
>>> heegner_point(Integer(37),-Integer(7),Integer(4)).ring_class_field().degree_over_H()
2
>>> heegner_point(Integer(37),-Integer(7),Integer(8)).ring_class_field().degree_over_H()
4

Now c is ramified:

Sage

sage: heegner_point(37,-7,7).ring_class_field().degree_over_H()
7
sage: heegner_point(37,-7,7^2).ring_class_field().degree_over_H()
49

Python

>>> from sage.all import *
>>> heegner_point(Integer(37),-Integer(7),Integer(7)).ring_class_field().degree_over_H()
7
>>> heegner_point(Integer(37),-Integer(7),Integer(7)**Integer(2)).ring_class_field().degree_over_H()
49

Check that Issue #15218 is solved:

Sage

sage: E = EllipticCurve("19a");
sage: s = E.heegner_point(-3,2).ring_class_field().galois_group().complex_conjugation()
sage: H = s.domain(); H.absolute_degree()
2

Python

>>> from sage.all import *
>>> E = EllipticCurve("19a");
>>> s = E.heegner_point(-Integer(3),Integer(2)).ring_class_field().galois_group().complex_conjugation()
>>> H = s.domain(); H.absolute_degree()
2

degree_over_K()[source]¶

Return the relative degree of this ring class field over the quadratic imaginary field \(K\).

EXAMPLES:

Sage

sage: E = EllipticCurve('389a'); P = E.heegner_point(-7,5)
sage: K5 = P.ring_class_field(); K5
Ring class field extension of QQ[sqrt(-7)] of conductor 5
sage: K5.degree_over_K()
6
sage: type(K5.degree_over_K())
<... 'sage.rings.integer.Integer'>

sage: E = EllipticCurve('389a'); E.heegner_point(-20).ring_class_field().degree_over_K()
2
sage: E.heegner_point(-20,3).ring_class_field().degree_over_K()
4
sage: kronecker(-20,11)
-1
sage: E.heegner_point(-20,11).ring_class_field().degree_over_K()
24

Python

>>> from sage.all import *
>>> E = EllipticCurve('389a'); P = E.heegner_point(-Integer(7),Integer(5))
>>> K5 = P.ring_class_field(); K5
Ring class field extension of QQ[sqrt(-7)] of conductor 5
>>> K5.degree_over_K()
6
>>> type(K5.degree_over_K())
<... 'sage.rings.integer.Integer'>

>>> E = EllipticCurve('389a'); E.heegner_point(-Integer(20)).ring_class_field().degree_over_K()
2
>>> E.heegner_point(-Integer(20),Integer(3)).ring_class_field().degree_over_K()
4
>>> kronecker(-Integer(20),Integer(11))
-1
>>> E.heegner_point(-Integer(20),Integer(11)).ring_class_field().degree_over_K()
24

degree_over_Q()[source]¶

Return the absolute degree of this field over \(\QQ\).

EXAMPLES:

Sage

sage: E = EllipticCurve('389a'); K = E.heegner_point(-7,5).ring_class_field()
sage: K.absolute_degree()
12
sage: K.degree_over_K()
6

Python

>>> from sage.all import *
>>> E = EllipticCurve('389a'); K = E.heegner_point(-Integer(7),Integer(5)).ring_class_field()
>>> K.absolute_degree()
12
>>> K.degree_over_K()
6

discriminant_of_K()[source]¶

Return the discriminant of the quadratic imaginary field \(K\) contained in self.

EXAMPLES:

Sage

sage: E = EllipticCurve('389a'); K5 = E.heegner_point(-7,5).ring_class_field()
sage: K5.discriminant_of_K()
-7

Python

>>> from sage.all import *
>>> E = EllipticCurve('389a'); K5 = E.heegner_point(-Integer(7),Integer(5)).ring_class_field()
>>> K5.discriminant_of_K()
-7

galois_group(base=Rational Field)[source]¶

Return the Galois group of self over base.

INPUT:

base – (default: \(\QQ\)) a subfield of self or \(\QQ\)

EXAMPLES:

Sage

sage: E = EllipticCurve('389a')
sage: A = E.heegner_point(-7,5).ring_class_field()
sage: A.galois_group()
Galois group of Ring class field extension of QQ[sqrt(-7)] of conductor 5
sage: B = E.heegner_point(-7).ring_class_field()
sage: C = E.heegner_point(-7,15).ring_class_field()
sage: A.galois_group()
Galois group of Ring class field extension of QQ[sqrt(-7)] of conductor 5
sage: A.galois_group(B)
Galois group of Ring class field extension of QQ[sqrt(-7)] of conductor 5
 over Hilbert class field of QQ[sqrt(-7)]
sage: A.galois_group().cardinality()
12
sage: A.galois_group(B).cardinality()
6
sage: C.galois_group(A)
Galois group of Ring class field extension of QQ[sqrt(-7)] of conductor 15
 over Ring class field extension of QQ[sqrt(-7)] of conductor 5
sage: C.galois_group(A).cardinality()
4

Python

>>> from sage.all import *
>>> E = EllipticCurve('389a')
>>> A = E.heegner_point(-Integer(7),Integer(5)).ring_class_field()
>>> A.galois_group()
Galois group of Ring class field extension of QQ[sqrt(-7)] of conductor 5
>>> B = E.heegner_point(-Integer(7)).ring_class_field()
>>> C = E.heegner_point(-Integer(7),Integer(15)).ring_class_field()
>>> A.galois_group()
Galois group of Ring class field extension of QQ[sqrt(-7)] of conductor 5
>>> A.galois_group(B)
Galois group of Ring class field extension of QQ[sqrt(-7)] of conductor 5
 over Hilbert class field of QQ[sqrt(-7)]
>>> A.galois_group().cardinality()
12
>>> A.galois_group(B).cardinality()
6
>>> C.galois_group(A)
Galois group of Ring class field extension of QQ[sqrt(-7)] of conductor 15
 over Ring class field extension of QQ[sqrt(-7)] of conductor 5
>>> C.galois_group(A).cardinality()
4

is_subfield(M)[source]¶

Return True if this ring class field is a subfield of the ring class field \(M\). If \(M\) is not a ring class field, then a TypeError is raised.

EXAMPLES:

Sage

sage: E = EllipticCurve('389a')
sage: A = E.heegner_point(-7,5).ring_class_field()
sage: B = E.heegner_point(-7).ring_class_field()
sage: C = E.heegner_point(-20).ring_class_field()
sage: D = E.heegner_point(-7,15).ring_class_field()
sage: B.is_subfield(A)
True
sage: B.is_subfield(B)
True
sage: B.is_subfield(D)
True
sage: B.is_subfield(C)
False
sage: A.is_subfield(B)
False
sage: A.is_subfield(D)
True

Python

>>> from sage.all import *
>>> E = EllipticCurve('389a')
>>> A = E.heegner_point(-Integer(7),Integer(5)).ring_class_field()
>>> B = E.heegner_point(-Integer(7)).ring_class_field()
>>> C = E.heegner_point(-Integer(20)).ring_class_field()
>>> D = E.heegner_point(-Integer(7),Integer(15)).ring_class_field()
>>> B.is_subfield(A)
True
>>> B.is_subfield(B)
True
>>> B.is_subfield(D)
True
>>> B.is_subfield(C)
False
>>> A.is_subfield(B)
False
>>> A.is_subfield(D)
True

quadratic_field()[source]¶

Return the quadratic imaginary field \(K = \QQ(\sqrt{D})\).

EXAMPLES:

Sage

sage: E = EllipticCurve('389a'); K = E.heegner_point(-7,5).ring_class_field()
sage: K.quadratic_field()
Number Field in sqrt_minus_7 with defining polynomial x^2 + 7
 with sqrt_minus_7 = 2.645751311064591?*I

Python

>>> from sage.all import *
>>> E = EllipticCurve('389a'); K = E.heegner_point(-Integer(7),Integer(5)).ring_class_field()
>>> K.quadratic_field()
Number Field in sqrt_minus_7 with defining polynomial x^2 + 7
 with sqrt_minus_7 = 2.645751311064591?*I

ramified_primes()[source]¶

Return the primes of \(\ZZ\) that ramify in this ring class field.

EXAMPLES:

Sage

sage: E = EllipticCurve('389a'); K55 = E.heegner_point(-7,55).ring_class_field()
sage: K55.ramified_primes()
[5, 7, 11]
sage: E.heegner_point(-7).ring_class_field().ramified_primes()
[7]

Python

>>> from sage.all import *
>>> E = EllipticCurve('389a'); K55 = E.heegner_point(-Integer(7),Integer(55)).ring_class_field()
>>> K55.ramified_primes()
[5, 7, 11]
>>> E.heegner_point(-Integer(7)).ring_class_field().ramified_primes()
[7]

sage.schemes.elliptic_curves.heegner.class_number(D)[source]¶

Return the class number of the quadratic field with fundamental discriminant \(D\).

INPUT:

D – integer

EXAMPLES:

Sage

sage: sage.schemes.elliptic_curves.heegner.class_number(-20)
2
sage: sage.schemes.elliptic_curves.heegner.class_number(-23)
3
sage: sage.schemes.elliptic_curves.heegner.class_number(-163)
1

Python

>>> from sage.all import *
>>> sage.schemes.elliptic_curves.heegner.class_number(-Integer(20))
2
>>> sage.schemes.elliptic_curves.heegner.class_number(-Integer(23))
3
>>> sage.schemes.elliptic_curves.heegner.class_number(-Integer(163))
1

A ValueError is raised when \(D\) is not a fundamental discriminant:

Sage

sage: sage.schemes.elliptic_curves.heegner.class_number(-5)
Traceback (most recent call last):
...
ValueError: D (=-5) must be a fundamental discriminant

Python

>>> from sage.all import *
>>> sage.schemes.elliptic_curves.heegner.class_number(-Integer(5))
Traceback (most recent call last):
...
ValueError: D (=-5) must be a fundamental discriminant

sage.schemes.elliptic_curves.heegner.ell_heegner_discriminants(self, bound)[source]¶

Return the list of self’s Heegner discriminants between -1 and -bound.

INPUT:

bound – integer; upper bound for -discriminant

OUTPUT: the list of Heegner discriminants between -1 and -bound for the given elliptic curve

EXAMPLES:

Sage

sage: E=EllipticCurve('11a')
sage: E.heegner_discriminants(30)                     # indirect doctest
[-7, -8, -19, -24]

Python

>>> from sage.all import *
>>> E=EllipticCurve('11a')
>>> E.heegner_discriminants(Integer(30))                     # indirect doctest
[-7, -8, -19, -24]

sage.schemes.elliptic_curves.heegner.ell_heegner_discriminants_list(self, n)[source]¶

Return the list of self’s first \(n\) Heegner discriminants smaller than -5.

INPUT:

n – integer; the number of discriminants to compute

OUTPUT: the list of the first \(n\) Heegner discriminants smaller than \(-5\) for the given elliptic curve

EXAMPLES:

Sage

sage: E=EllipticCurve('11a')
sage: E.heegner_discriminants_list(4)                     # indirect doctest
[-7, -8, -19, -24]

Python

>>> from sage.all import *
>>> E=EllipticCurve('11a')
>>> E.heegner_discriminants_list(Integer(4))                     # indirect doctest
[-7, -8, -19, -24]

sage.schemes.elliptic_curves.heegner.ell_heegner_point(self, D, c=1, f=None, check=True)[source]¶

Return the Heegner point on this curve associated to the quadratic imaginary field \(K=\QQ(\sqrt{D})\).

If the optional parameter \(c\) is given, returns the higher Heegner point associated to the order of conductor \(c\).

INPUT:

D – a Heegner discriminant
c – (default: 1) conductor, must be coprime to \(DN\)
f – binary quadratic form or 3-tuple \((A,B,C)\) of coefficients of \(AX^2 + BXY + CY^2\)
check – boolean (default: True)

OUTPUT: the Heegner point \(y_c\)

EXAMPLES:

Sage

sage: E = EllipticCurve('37a')
sage: E.heegner_discriminants_list(10)
[-7, -11, -40, -47, -67, -71, -83, -84, -95, -104]
sage: P = E.heegner_point(-7); P                          # indirect doctest
Heegner point of discriminant -7 on elliptic curve of conductor 37
sage: z = P.point_exact(); z == E(0, 0, 1)  or -z == E(0, 0, 1)
True
sage: P.curve()
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
sage: P = E.heegner_point(-40).point_exact(); P
(a : -a + 1 : 1)
sage: P = E.heegner_point(-47).point_exact(); P
(a : a^4 + a - 1 : 1)
sage: P[0].parent()
Number Field in a with defining polynomial x^5 - x^4 + x^3 + x^2 - 2*x + 1

Python

>>> from sage.all import *
>>> E = EllipticCurve('37a')
>>> E.heegner_discriminants_list(Integer(10))
[-7, -11, -40, -47, -67, -71, -83, -84, -95, -104]
>>> P = E.heegner_point(-Integer(7)); P                          # indirect doctest
Heegner point of discriminant -7 on elliptic curve of conductor 37
>>> z = P.point_exact(); z == E(Integer(0), Integer(0), Integer(1))  or -z == E(Integer(0), Integer(0), Integer(1))
True
>>> P.curve()
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
>>> P = E.heegner_point(-Integer(40)).point_exact(); P
(a : -a + 1 : 1)
>>> P = E.heegner_point(-Integer(47)).point_exact(); P
(a : a^4 + a - 1 : 1)
>>> P[Integer(0)].parent()
Number Field in a with defining polynomial x^5 - x^4 + x^3 + x^2 - 2*x + 1

Working out the details manually:

Sage

sage: P = E.heegner_point(-47).numerical_approx(prec=200)
sage: f = algdep(P[0], 5); f
x^5 - x^4 + x^3 + x^2 - 2*x + 1
sage: f.discriminant().factor()
47^2

Python

>>> from sage.all import *
>>> P = E.heegner_point(-Integer(47)).numerical_approx(prec=Integer(200))
>>> f = algdep(P[Integer(0)], Integer(5)); f
x^5 - x^4 + x^3 + x^2 - 2*x + 1
>>> f.discriminant().factor()
47^2

The Heegner hypothesis is checked:

Sage

sage: E = EllipticCurve('389a'); P = E.heegner_point(-5,7);
Traceback (most recent call last):
...
ValueError: N (=389) and D (=-5) must satisfy the Heegner hypothesis

Python

>>> from sage.all import *
>>> E = EllipticCurve('389a'); P = E.heegner_point(-Integer(5),Integer(7));
Traceback (most recent call last):
...
ValueError: N (=389) and D (=-5) must satisfy the Heegner hypothesis

We can specify the quadratic form:

Sage

sage: P = EllipticCurve('389a').heegner_point(-7, 5, (778,925,275)); P
Heegner point of discriminant -7 and conductor 5
 on elliptic curve of conductor 389
sage: P.quadratic_form()
778*x^2 + 925*x*y + 275*y^2

Python

>>> from sage.all import *
>>> P = EllipticCurve('389a').heegner_point(-Integer(7), Integer(5), (Integer(778),Integer(925),Integer(275))); P
Heegner point of discriminant -7 and conductor 5
 on elliptic curve of conductor 389
>>> P.quadratic_form()
778*x^2 + 925*x*y + 275*y^2

sage.schemes.elliptic_curves.heegner.heegner_index(self, D, min_p=2, prec=5, descent_second_limit=12, verbose_mwrank=False, check_rank=True)[source]¶

Return an interval that contains the index of the Heegner point \(y_K\) in the group of \(K\)-rational points modulo torsion on this elliptic curve, computed using the Gross-Zagier formula and/or a point search, or possibly half the index if the rank is greater than one.

If the curve has rank > 1, then the returned index is infinity.

Note

If min_p is bigger than 2 then the index can be off by any prime less than min_p. This function returns the index divided by \(2\) exactly when the rank of \(E(K)\) is greater than 1 and \(E(\QQ)_{/tor} \oplus E^D(\QQ)_{/tor}\) has index \(2\) in \(E(K)_{/tor}\), where the second factor undergoes a twist.

INPUT:

D – integer; Heegner discriminant
min_p – integer (default: 2); only rule out primes = min_p dividing the index
verbose_mwrank – boolean (default: False); print lots of mwrank search status information when computing regulator
prec – integer (default: 5);, use prec*sqrt(N) + 20 terms of \(L\)-series in computations, where N is the conductor
descent_second_limit – (default: 12)- used in 2-descent when computing regulator of the twist
check_rank – whether to check if the rank is at least 2 by computing the Mordell-Weil rank directly

OUTPUT: an interval that contains the index, or half the index

EXAMPLES:

Sage

sage: E = EllipticCurve('11a')
sage: E.heegner_discriminants(50)
[-7, -8, -19, -24, -35, -39, -40, -43]
sage: E.heegner_index(-7)
1.00000?

Python

>>> from sage.all import *
>>> E = EllipticCurve('11a')
>>> E.heegner_discriminants(Integer(50))
[-7, -8, -19, -24, -35, -39, -40, -43]
>>> E.heegner_index(-Integer(7))
1.00000?

Sage

sage: E = EllipticCurve('37b')
sage: E.heegner_discriminants(100)
[-3, -4, -7, -11, -40, -47, -67, -71, -83, -84, -95]
sage: E.heegner_index(-95)          # long time (1 second)
2.00000?

Python

>>> from sage.all import *
>>> E = EllipticCurve('37b')
>>> E.heegner_discriminants(Integer(100))
[-3, -4, -7, -11, -40, -47, -67, -71, -83, -84, -95]
>>> E.heegner_index(-Integer(95))          # long time (1 second)
2.00000?

This tests doing direct computation of the Mordell-Weil group.

Sage

sage: EllipticCurve('675b').heegner_index(-11)
3.0000?

Python

>>> from sage.all import *
>>> EllipticCurve('675b').heegner_index(-Integer(11))
3.0000?

Currently discriminants -3 and -4 are not supported:

Sage

sage: E.heegner_index(-3)
Traceback (most recent call last):
...
ArithmeticError: Discriminant (=-3) must not be -3 or -4.

Python

>>> from sage.all import *
>>> E.heegner_index(-Integer(3))
Traceback (most recent call last):
...
ArithmeticError: Discriminant (=-3) must not be -3 or -4.

The curve 681b returns the true index, which is \(3\):

Sage

sage: E = EllipticCurve('681b')
sage: I = E.heegner_index(-8); I
3.0000?

Python

>>> from sage.all import *
>>> E = EllipticCurve('681b')
>>> I = E.heegner_index(-Integer(8)); I
3.0000?

In fact, whenever the returned index has a denominator of \(2\), the true index is got by multiplying the returned index by \(2\). Unfortunately, this is not an if and only if condition, i.e., sometimes the index must be multiplied by \(2\) even though the denominator is not \(2\).

This example demonstrates the descent_second_limit option, which can be used to fine tune the 2-descent used to compute the regulator of the twist:

Sage

sage: E = EllipticCurve([1,-1,0,-1228,-16267])
sage: E.heegner_index(-8)
Traceback (most recent call last):
...
RuntimeError: ...

Python

>>> from sage.all import *
>>> E = EllipticCurve([Integer(1),-Integer(1),Integer(0),-Integer(1228),-Integer(16267)])
>>> E.heegner_index(-Integer(8))
Traceback (most recent call last):
...
RuntimeError: ...

However when we search higher, we find the points we need:

Sage

sage: E.heegner_index(-8, descent_second_limit=16, check_rank=False)  # long time
2.00000?

Python

>>> from sage.all import *
>>> E.heegner_index(-Integer(8), descent_second_limit=Integer(16), check_rank=False)  # long time
2.00000?

Two higher rank examples (of ranks 2 and 3):

Sage

sage: E = EllipticCurve('389a')
sage: E.heegner_index(-7)
+Infinity
sage: E = EllipticCurve('5077a')
sage: E.heegner_index(-7)
+Infinity
sage: E.heegner_index(-7, check_rank=False)
0.001?
sage: E.heegner_index(-7, check_rank=False).lower() == 0
True

Python

>>> from sage.all import *
>>> E = EllipticCurve('389a')
>>> E.heegner_index(-Integer(7))
+Infinity
>>> E = EllipticCurve('5077a')
>>> E.heegner_index(-Integer(7))
+Infinity
>>> E.heegner_index(-Integer(7), check_rank=False)
0.001?
>>> E.heegner_index(-Integer(7), check_rank=False).lower() == Integer(0)
True

sage.schemes.elliptic_curves.heegner.heegner_index_bound(self, D=0, prec=5, max_height=None)[source]¶

Assume self has rank 0.

Return a list \(v\) of primes such that if an odd prime \(p\) divides the index of the Heegner point in the group of rational points modulo torsion, then \(p\) is in \(v\).

If 0 is in the interval of the height of the Heegner point computed to the given prec, then this function returns \(v = 0\). This does not mean that the Heegner point is torsion, just that it is very likely torsion.

If we obtain no information from a search up to max_height, e.g., if the Siksek et al. bound is bigger than max_height, then we return \(v = -1\).

INPUT:

D – integer (default: 0); Heegner discriminant; if 0, use the first discriminant -4 that satisfies the Heegner hypothesis
verbose – boolean (default: True)
prec – integer (default: 5); use \(prec \cdot \sqrt(N) + 20\) terms of \(L\)-series in computations, where \(N\) is the conductor
max_height (float) – should be = 21; bound on logarithmic naive height used in point searches. Make smaller to make this function faster, at the expense of possibly obtaining a worse answer. A good range is between 13 and 21.

OUTPUT:

v – list or int (bad primes or 0 or -1)
D – the discriminant that was used (this is useful if \(D\) was automatically selected)
exact – either False, or the exact Heegner index (up to factors of 2)

EXAMPLES:

Sage

sage: E = EllipticCurve('11a1')
sage: E.heegner_index_bound()
([2], -7, 2)

Python

>>> from sage.all import *
>>> E = EllipticCurve('11a1')
>>> E.heegner_index_bound()
([2], -7, 2)

sage.schemes.elliptic_curves.heegner.heegner_point(N, D=None, c=1)[source]¶

Return a specific Heegner point of level \(N\) with given discriminant and conductor. If \(D\) is not specified, then the first valid Heegner discriminant is used. If \(c\) is not given, then \(c=1\) is used.

INPUT:

N – level (positive integer)
D – discriminant (optional: default first valid \(D\))
c – conductor (positive integer, default: 1)

EXAMPLES:

Sage

sage: heegner_point(389)
Heegner point 1/778*sqrt(-7) - 185/778 of discriminant -7 on X_0(389)
sage: heegner_point(389,-7)
Heegner point 1/778*sqrt(-7) - 185/778 of discriminant -7 on X_0(389)
sage: heegner_point(389,-7,5)
Heegner point 5/778*sqrt(-7) - 147/778 of discriminant -7 and conductor 5 on X_0(389)
sage: heegner_point(389,-20)
Heegner point 1/778*sqrt(-20) - 165/389 of discriminant -20 on X_0(389)

Python

>>> from sage.all import *
>>> heegner_point(Integer(389))
Heegner point 1/778*sqrt(-7) - 185/778 of discriminant -7 on X_0(389)
>>> heegner_point(Integer(389),-Integer(7))
Heegner point 1/778*sqrt(-7) - 185/778 of discriminant -7 on X_0(389)
>>> heegner_point(Integer(389),-Integer(7),Integer(5))
Heegner point 5/778*sqrt(-7) - 147/778 of discriminant -7 and conductor 5 on X_0(389)
>>> heegner_point(Integer(389),-Integer(20))
Heegner point 1/778*sqrt(-20) - 165/389 of discriminant -20 on X_0(389)

sage.schemes.elliptic_curves.heegner.heegner_point_height(self, D, prec=2, check_rank=True)[source]¶

Use the Gross-Zagier formula to compute the Neron-Tate canonical height over \(K\) of the Heegner point corresponding to \(D\), as an interval (it is computed to some precision using \(L\)-functions).

If the curve has rank at least 2, then the returned height is the exact Sage integer 0.

INPUT:

D – integer; fundamental discriminant (=/= -3, -4)
prec – integer (default: 2); use \(prec \cdot \sqrt(N) + 20\) terms of \(L\)-series in computations, where \(N\) is the conductor
check_rank – whether to check if the rank is at least 2 by computing the Mordell-Weil rank directly

OUTPUT: interval that contains the height of the Heegner point

EXAMPLES:

Sage

sage: E = EllipticCurve('11a')
sage: E.heegner_point_height(-7)
0.22227?

Python

>>> from sage.all import *
>>> E = EllipticCurve('11a')
>>> E.heegner_point_height(-Integer(7))
0.22227?

Some higher rank examples:

Sage

sage: E = EllipticCurve('389a')
sage: E.heegner_point_height(-7)
0
sage: E = EllipticCurve('5077a')
sage: E.heegner_point_height(-7)
0
sage: E.heegner_point_height(-7, check_rank=False)
0.0000?

Python

>>> from sage.all import *
>>> E = EllipticCurve('389a')
>>> E.heegner_point_height(-Integer(7))
0
>>> E = EllipticCurve('5077a')
>>> E.heegner_point_height(-Integer(7))
0
>>> E.heegner_point_height(-Integer(7), check_rank=False)
0.0000?

sage.schemes.elliptic_curves.heegner.heegner_points(N, D=None, c=None)[source]¶

Return all Heegner points of given level \(N\). Can also restrict to Heegner points with specified discriminant \(D\) and optionally conductor \(c\).

INPUT:

N – level (positive integer)
D – discriminant (negative integer)
c – conductor (positive integer)

EXAMPLES:

Sage

sage: heegner_points(389, -7)
Set of all Heegner points on X_0(389) associated to QQ[sqrt(-7)]
sage: heegner_points(389, -7, 1)
All Heegner points of conductor 1 on X_0(389) associated to QQ[sqrt(-7)]
sage: heegner_points(389, -7, 5)
All Heegner points of conductor 5 on X_0(389) associated to QQ[sqrt(-7)]

Python

>>> from sage.all import *
>>> heegner_points(Integer(389), -Integer(7))
Set of all Heegner points on X_0(389) associated to QQ[sqrt(-7)]
>>> heegner_points(Integer(389), -Integer(7), Integer(1))
All Heegner points of conductor 1 on X_0(389) associated to QQ[sqrt(-7)]
>>> heegner_points(Integer(389), -Integer(7), Integer(5))
All Heegner points of conductor 5 on X_0(389) associated to QQ[sqrt(-7)]

sage.schemes.elliptic_curves.heegner.heegner_sha_an(self, D, prec=53)[source]¶

Return the conjectural (analytic) order of Sha for E over the field \(K=\QQ(\sqrt{D})\).

INPUT:

D – negative integer; the Heegner discriminant
prec – integer (default: 53); bits of precision to compute analytic order of Sha

OUTPUT:

(floating point number) an approximation to the conjectural order of Sha.

Note

Often you’ll want to do proof.elliptic_curve(False) when using this function, since often the twisted elliptic curves that come up have enormous conductor, and Sha is nontrivial, which makes provably finding the Mordell-Weil group using 2-descent difficult.

EXAMPLES:

An example where E has conductor 11:

Sage

sage: E = EllipticCurve('11a')
sage: E.heegner_sha_an(-7)                                  # long time
1.00000000000000

Python

>>> from sage.all import *
>>> E = EllipticCurve('11a')
>>> E.heegner_sha_an(-Integer(7))                                  # long time
1.00000000000000

The cache works:

Sage

sage: E.heegner_sha_an(-7) is E.heegner_sha_an(-7)          # long time
True

Python

>>> from sage.all import *
>>> E.heegner_sha_an(-Integer(7)) is E.heegner_sha_an(-Integer(7))          # long time
True

Lower precision:

Sage

sage: E.heegner_sha_an(-7,10)                               # long time
1.0

Python

>>> from sage.all import *
>>> E.heegner_sha_an(-Integer(7),Integer(10))                               # long time
1.0

Checking that the cache works for any precision:

Sage

sage: E.heegner_sha_an(-7,10) is E.heegner_sha_an(-7,10)    # long time
True

Python

>>> from sage.all import *
>>> E.heegner_sha_an(-Integer(7),Integer(10)) is E.heegner_sha_an(-Integer(7),Integer(10))    # long time
True

Next we consider a rank 1 curve with nontrivial Sha over the quadratic imaginary field \(K\); however, there is no Sha for \(E\) over \(\QQ\) or for the quadratic twist of \(E\):

Sage

sage: E = EllipticCurve('37a')
sage: E.heegner_sha_an(-40)                                 # long time
4.00000000000000
sage: E.quadratic_twist(-40).sha().an()                     # long time
1
sage: E.sha().an()                                          # long time
1

Python

>>> from sage.all import *
>>> E = EllipticCurve('37a')
>>> E.heegner_sha_an(-Integer(40))                                 # long time
4.00000000000000
>>> E.quadratic_twist(-Integer(40)).sha().an()                     # long time
1
>>> E.sha().an()                                          # long time
1

A rank 2 curve:

Sage

sage: E = EllipticCurve('389a')                             # long time
sage: E.heegner_sha_an(-7)                                  # long time
1.00000000000000

Python

>>> from sage.all import *
>>> E = EllipticCurve('389a')                             # long time
>>> E.heegner_sha_an(-Integer(7))                                  # long time
1.00000000000000

If we remove the hypothesis that \(E(K)\) has rank 1 in Conjecture 2.3 in [GZ1986] page 311, then that conjecture is false, as the following example shows:

Sage

sage: # long time
sage: E = EllipticCurve('65a')
sage: E.heegner_sha_an(-56)
1.00000000000000
sage: E.torsion_order()
2
sage: E.tamagawa_product()
1
sage: E.quadratic_twist(-56).rank()
2

Python

>>> from sage.all import *
>>> # long time
>>> E = EllipticCurve('65a')
>>> E.heegner_sha_an(-Integer(56))
1.00000000000000
>>> E.torsion_order()
2
>>> E.tamagawa_product()
1
>>> E.quadratic_twist(-Integer(56)).rank()
2

sage.schemes.elliptic_curves.heegner.is_inert(D, p)[source]¶

Return True if p is an inert prime in the field \(\QQ(\sqrt{D})\).

INPUT:

D – fundamental discriminant
p – prime integer

EXAMPLES:

Sage

sage: sage.schemes.elliptic_curves.heegner.is_inert(-7,3)
True
sage: sage.schemes.elliptic_curves.heegner.is_inert(-7,7)
False
sage: sage.schemes.elliptic_curves.heegner.is_inert(-7,11)
False

Python

>>> from sage.all import *
>>> sage.schemes.elliptic_curves.heegner.is_inert(-Integer(7),Integer(3))
True
>>> sage.schemes.elliptic_curves.heegner.is_inert(-Integer(7),Integer(7))
False
>>> sage.schemes.elliptic_curves.heegner.is_inert(-Integer(7),Integer(11))
False

sage.schemes.elliptic_curves.heegner.is_kolyvagin_conductor(N, E, D, r, n, c)[source]¶

Return True if \(c\) is a Kolyvagin conductor for level \(N\), discriminant \(D\), mod \(n\), etc., i.e., \(c\) is divisible by exactly \(r\) prime factors, is coprime to \(ND\), each prime dividing \(c\) is inert, and if \(E\) is not None then \(n | \gcd(p+1, a_p(E))\) for each prime \(p\) dividing \(c\).

INPUT:

N – level (positive integer)
E – elliptic curve or None
D – negative fundamental discriminant
r – number of prime factors (nonnegative integer) or None
n – torsion order (i.e., do we get class in \((E(K_c)/n E(K_c))^{Gal(K_c/K)}\)?)
c – conductor (positive integer)

EXAMPLES:

Sage

sage: from sage.schemes.elliptic_curves.heegner import is_kolyvagin_conductor
sage: is_kolyvagin_conductor(389, None, -7, 1, None, 5)
True
sage: is_kolyvagin_conductor(389, None, -7, 1, None, 7)
False
sage: is_kolyvagin_conductor(389, None, -7, 1, None, 11)
False
sage: is_kolyvagin_conductor(389, EllipticCurve('389a'), -7, 1, 3, 5)
True
sage: is_kolyvagin_conductor(389, EllipticCurve('389a'), -7, 1, 11, 5)
False

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.heegner import is_kolyvagin_conductor
>>> is_kolyvagin_conductor(Integer(389), None, -Integer(7), Integer(1), None, Integer(5))
True
>>> is_kolyvagin_conductor(Integer(389), None, -Integer(7), Integer(1), None, Integer(7))
False
>>> is_kolyvagin_conductor(Integer(389), None, -Integer(7), Integer(1), None, Integer(11))
False
>>> is_kolyvagin_conductor(Integer(389), EllipticCurve('389a'), -Integer(7), Integer(1), Integer(3), Integer(5))
True
>>> is_kolyvagin_conductor(Integer(389), EllipticCurve('389a'), -Integer(7), Integer(1), Integer(11), Integer(5))
False

sage.schemes.elliptic_curves.heegner.is_ramified(D, p)[source]¶

Return True if p is a ramified prime in the field \(\QQ(\sqrt{D})\).

INPUT:

D – fundamental discriminant
p – prime integer

EXAMPLES:

Sage

sage: sage.schemes.elliptic_curves.heegner.is_ramified(-7,2)
False
sage: sage.schemes.elliptic_curves.heegner.is_ramified(-7,7)
True
sage: sage.schemes.elliptic_curves.heegner.is_ramified(-1,2)
True

Python

>>> from sage.all import *
>>> sage.schemes.elliptic_curves.heegner.is_ramified(-Integer(7),Integer(2))
False
>>> sage.schemes.elliptic_curves.heegner.is_ramified(-Integer(7),Integer(7))
True
>>> sage.schemes.elliptic_curves.heegner.is_ramified(-Integer(1),Integer(2))
True

sage.schemes.elliptic_curves.heegner.is_split(D, p)[source]¶

Return True if p is a split prime in the field \(\QQ(\sqrt{D})\).

INPUT:

D – fundamental discriminant
p – prime integer

EXAMPLES:

Sage

sage: sage.schemes.elliptic_curves.heegner.is_split(-7,3)
False
sage: sage.schemes.elliptic_curves.heegner.is_split(-7,7)
False
sage: sage.schemes.elliptic_curves.heegner.is_split(-7,11)
True

Python

>>> from sage.all import *
>>> sage.schemes.elliptic_curves.heegner.is_split(-Integer(7),Integer(3))
False
>>> sage.schemes.elliptic_curves.heegner.is_split(-Integer(7),Integer(7))
False
>>> sage.schemes.elliptic_curves.heegner.is_split(-Integer(7),Integer(11))
True

sage.schemes.elliptic_curves.heegner.kolyvagin_point(self, D, c=1, check=True)[source]¶

Return the Kolyvagin point on this curve associated to the quadratic imaginary field \(K=\QQ(\sqrt{D})\) and conductor \(c\).

INPUT:

D – a Heegner discriminant
c – (default: 1) conductor, must be coprime to \(DN\)
check – boolean (default: True)

OUTPUT: the Kolyvagin point \(P\) of conductor \(c\)

EXAMPLES:

Sage

sage: E = EllipticCurve('37a1')
sage: P = E.kolyvagin_point(-67); P
Kolyvagin point of discriminant -67 on elliptic curve of conductor 37
sage: P.numerical_approx()  # abs tol 1e-14
(6.00000000000000 : -15.0000000000000 : 1.00000000000000)
sage: P.index()
6
sage: g = E((0,-1,1)) # a generator
sage: E.regulator() == E.regulator_of_points([g])
True
sage: 6*g
(6 : -15 : 1)

Python

>>> from sage.all import *
>>> E = EllipticCurve('37a1')
>>> P = E.kolyvagin_point(-Integer(67)); P
Kolyvagin point of discriminant -67 on elliptic curve of conductor 37
>>> P.numerical_approx()  # abs tol 1e-14
(6.00000000000000 : -15.0000000000000 : 1.00000000000000)
>>> P.index()
6
>>> g = E((Integer(0),-Integer(1),Integer(1))) # a generator
>>> E.regulator() == E.regulator_of_points([g])
True
>>> Integer(6)*g
(6 : -15 : 1)

sage.schemes.elliptic_curves.heegner.kolyvagin_reduction_data(E, q, first_only=True)[source]¶

Given an elliptic curve of positive rank and a prime \(q\), this function returns data about how to use Kolyvagin’s \(q\)-torsion Heegner point Euler system to do computations with this curve. See the precise description of the output below.

INPUT:

E – elliptic curve over \(\QQ\) of rank 1 or 2
q – an odd prime that does not divide the order of the rational torsion subgroup of \(E\)
first_only – boolean (default: True); whether two only return the first prime that one can work modulo to get data about the Euler system

OUTPUT in the rank 1 case or when the default flag first_only=True:

\(\ell\) – first good odd prime satisfying the Kolyvagin condition that \(q\) divides gcd(a_{ell},ell+1)` and the reduction map is surjective to \(E(\GF{\ell}) / q E(\GF{\ell})\)
D – discriminant of the first quadratic imaginary field
\(K\) that satisfies the Heegner hypothesis for \(E\) such that both \(\ell\) is inert in \(K\), and the twist \(E^D\) has analytic rank \(\leq 1\)
h_D – the class number of \(K\)
the dimension of the Brandt module \(B(\ell,N)\), where \(N\) is the conductor of \(E\)

OUTPUT in the rank 2 case:

\(\ell_1\) – first prime (as above in the rank 1 case) where reduction map is surjective
\(\ell_2\) – second prime (as above) where reduction map is surjective
D – discriminant of the first quadratic imaginary field
\(K\) that satisfies the Heegner hypothesis for \(E\) such that both \(\ell_1\) and \(\ell_2\) are simultaneously inert in \(K\), and the twist \(E^D\) has analytic rank \(\leq 1\)
h_D – the class number of \(K\)
the dimension of the Brandt module \(B(\ell_1,N)\), where \(N\) is the conductor of \(E\)
the dimension of the Brandt module \(B(\ell_2,N)\)

EXAMPLES:

Import this function:

Sage

sage: from sage.schemes.elliptic_curves.heegner import kolyvagin_reduction_data

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.heegner import kolyvagin_reduction_data

A rank 1 example:

Sage

sage: kolyvagin_reduction_data(EllipticCurve('37a1'), 3)
(17, -7, 1, 52)

Python

>>> from sage.all import *
>>> kolyvagin_reduction_data(EllipticCurve('37a1'), Integer(3))
(17, -7, 1, 52)

A rank 3 example:

Sage

sage: kolyvagin_reduction_data(EllipticCurve('5077a1'), 3)
(11, -47, 5, 4234)
sage: H = heegner_points(5077, -47)
sage: [c for c in H.kolyvagin_conductors(2, 10, EllipticCurve('5077a1'), 3)
....:    if c % 11]
[667, 943, 1189, 2461]
sage: factor(667)
23 * 29

Python

>>> from sage.all import *
>>> kolyvagin_reduction_data(EllipticCurve('5077a1'), Integer(3))
(11, -47, 5, 4234)
>>> H = heegner_points(Integer(5077), -Integer(47))
>>> [c for c in H.kolyvagin_conductors(Integer(2), Integer(10), EllipticCurve('5077a1'), Integer(3))
...    if c % Integer(11)]
[667, 943, 1189, 2461]
>>> factor(Integer(667))
23 * 29

A rank 4 example (the first Kolyvagin class that we could try to compute would be \(P_{23\cdot 29\cdot 41}\), and would require working in a space of dimension 293060 (so prohibitive at present):

Sage

sage: E = elliptic_curves.rank(4)[0]
sage: kolyvagin_reduction_data(E,3)              # long time
(11, -71, 7, 293060)
sage: H = heegner_points(293060, -71)
sage: H.kolyvagin_conductors(1,4,E,3)
[11, 17, 23, 41]

Python

>>> from sage.all import *
>>> E = elliptic_curves.rank(Integer(4))[Integer(0)]
>>> kolyvagin_reduction_data(E,Integer(3))              # long time
(11, -71, 7, 293060)
>>> H = heegner_points(Integer(293060), -Integer(71))
>>> H.kolyvagin_conductors(Integer(1),Integer(4),E,Integer(3))
[11, 17, 23, 41]

The first rank 2 example:

Sage

sage: kolyvagin_reduction_data(EllipticCurve('389a'), 3)
(5, -7, 1, 130)
sage: kolyvagin_reduction_data(EllipticCurve('389a'), 3, first_only=False)
(5, 17, -7, 1, 130, 520)

Python

>>> from sage.all import *
>>> kolyvagin_reduction_data(EllipticCurve('389a'), Integer(3))
(5, -7, 1, 130)
>>> kolyvagin_reduction_data(EllipticCurve('389a'), Integer(3), first_only=False)
(5, 17, -7, 1, 130, 520)

A large \(q = 7\):

Sage

sage: kolyvagin_reduction_data(EllipticCurve('1143c1'), 7, first_only=False)
(13, 83, -59, 3, 1536, 10496)

Python

>>> from sage.all import *
>>> kolyvagin_reduction_data(EllipticCurve('1143c1'), Integer(7), first_only=False)
(13, 83, -59, 3, 1536, 10496)

Additive reduction:

Sage

sage: kolyvagin_reduction_data(EllipticCurve('2350g1'), 5, first_only=False)
(19, 239, -311, 19, 6480, 85680)

Python

>>> from sage.all import *
>>> kolyvagin_reduction_data(EllipticCurve('2350g1'), Integer(5), first_only=False)
(19, 239, -311, 19, 6480, 85680)

sage.schemes.elliptic_curves.heegner.make_monic(f)[source]¶

Return a monic integral polynomial \(g\) and an integer \(d\) such that if \(\alpha\) is a root of \(g\), then \(\alpha/d\) is a root of \(f\). In other words, \(c f(x) = g(d x)\) for some scalar \(c\).

INPUT:

f – polynomial over the rational numbers

OUTPUT: a monic integral polynomial and an integer

EXAMPLES:

Sage

sage: from sage.schemes.elliptic_curves.heegner import make_monic
sage: R.<x> = QQ[]
sage: make_monic(3*x^3 + 14*x^2 - 7*x + 5)
(x^3 + 14*x^2 - 21*x + 45, 3)

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.heegner import make_monic
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> make_monic(Integer(3)*x**Integer(3) + Integer(14)*x**Integer(2) - Integer(7)*x + Integer(5))
(x^3 + 14*x^2 - 21*x + 45, 3)

In this example we verify that make_monic does what we claim it does:

Sage

sage: K.<a> = NumberField(x^3 + 17*x - 3)
sage: f = (a/7+2/3).minpoly(); f
x^3 - 2*x^2 + 247/147*x - 4967/9261
sage: g, d = make_monic(f); (g, d)
(x^3 - 42*x^2 + 741*x - 4967, 21)
sage: K.<b> = NumberField(g)
sage: (b/d).minpoly()
x^3 - 2*x^2 + 247/147*x - 4967/9261

Python

>>> from sage.all import *
>>> K = NumberField(x**Integer(3) + Integer(17)*x - Integer(3), names=('a',)); (a,) = K._first_ngens(1)
>>> f = (a/Integer(7)+Integer(2)/Integer(3)).minpoly(); f
x^3 - 2*x^2 + 247/147*x - 4967/9261
>>> g, d = make_monic(f); (g, d)
(x^3 - 42*x^2 + 741*x - 4967, 21)
>>> K = NumberField(g, names=('b',)); (b,) = K._first_ngens(1)
>>> (b/d).minpoly()
x^3 - 2*x^2 + 247/147*x - 4967/9261

sage.schemes.elliptic_curves.heegner.nearby_rational_poly(f, **kwds)[source]¶

Return a polynomial whose coefficients are rational numbers close to the coefficients of \(f\).

INPUT:

f – polynomial with real floating point entries
**kwds – passed on to nearby_rational method

EXAMPLES:

Sage

sage: from sage.schemes.elliptic_curves.heegner import nearby_rational_poly
sage: R.<x> = RR[]
sage: nearby_rational_poly(2.1*x^2 + 3.5*x - 1.2, max_error=10e-16)
21/10*X^2 + 7/2*X - 6/5
sage: nearby_rational_poly(2.1*x^2 + 3.5*x - 1.2, max_error=10e-17)
4728779608739021/2251799813685248*X^2 + 7/2*X - 5404319552844595/4503599627370496
sage: RR(4728779608739021/2251799813685248  - 21/10)
8.88178419700125e-17

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.heegner import nearby_rational_poly
>>> R = RR['x']; (x,) = R._first_ngens(1)
>>> nearby_rational_poly(RealNumber('2.1')*x**Integer(2) + RealNumber('3.5')*x - RealNumber('1.2'), max_error=RealNumber('10e-16'))
21/10*X^2 + 7/2*X - 6/5
>>> nearby_rational_poly(RealNumber('2.1')*x**Integer(2) + RealNumber('3.5')*x - RealNumber('1.2'), max_error=RealNumber('10e-17'))
4728779608739021/2251799813685248*X^2 + 7/2*X - 5404319552844595/4503599627370496
>>> RR(Integer(4728779608739021)/Integer(2251799813685248)  - Integer(21)/Integer(10))
8.88178419700125e-17

sage.schemes.elliptic_curves.heegner.quadratic_order(D, c, names='a')[source]¶

Return order of conductor \(c\) in quadratic field with fundamental discriminant \(D\).

INPUT:

D – fundamental discriminant
c – conductor
names – string (default: 'a')

OUTPUT:

order \(R\) of conductor \(c\) in an imaginary quadratic field
the element \(c\sqrt{D}\) as an element of \(R\)

The generator for the field is named ‘a’ by default.

EXAMPLES:

Sage

sage: sage.schemes.elliptic_curves.heegner.quadratic_order(-7,3)
(Order of conductor 6 generated by 3*a in Number Field in a
 with defining polynomial x^2 + 7 with a = 2.645751311064591?*I,
 3*a)
sage: sage.schemes.elliptic_curves.heegner.quadratic_order(-7,3,'alpha')
(Order of conductor 6 generated by 3*alpha in Number Field in alpha
 with defining polynomial x^2 + 7 with alpha = 2.645751311064591?*I,
 3*alpha)

Python

>>> from sage.all import *
>>> sage.schemes.elliptic_curves.heegner.quadratic_order(-Integer(7),Integer(3))
(Order of conductor 6 generated by 3*a in Number Field in a
 with defining polynomial x^2 + 7 with a = 2.645751311064591?*I,
 3*a)
>>> sage.schemes.elliptic_curves.heegner.quadratic_order(-Integer(7),Integer(3),'alpha')
(Order of conductor 6 generated by 3*alpha in Number Field in alpha
 with defining polynomial x^2 + 7 with alpha = 2.645751311064591?*I,
 3*alpha)

sage.schemes.elliptic_curves.heegner.satisfies_heegner_hypothesis(self, D)[source]¶

Return True precisely when \(D\) is a fundamental discriminant that satisfies the Heegner hypothesis for this elliptic curve.

EXAMPLES:

Sage

sage: E = EllipticCurve('11a1')
sage: E.satisfies_heegner_hypothesis(-7)
True
sage: E.satisfies_heegner_hypothesis(-11)
False

Python

>>> from sage.all import *
>>> E = EllipticCurve('11a1')
>>> E.satisfies_heegner_hypothesis(-Integer(7))
True
>>> E.satisfies_heegner_hypothesis(-Integer(11))
False

sage.schemes.elliptic_curves.heegner.satisfies_weak_heegner_hypothesis(N, D)[source]¶

Check that \(D\) satisfies the weak Heegner hypothesis relative to \(N\). This is all that is needed to define Heegner points.

The condition is that \(D<0\) is a fundamental discriminant and that each unramified prime dividing \(N\) splits in \(K=\QQ(\sqrt{D})\) and each ramified prime exactly divides \(N\). We also do not require that \(D<-4\).

INPUT:

N – positive integer
D – negative integer

EXAMPLES:

Sage

sage: s = sage.schemes.elliptic_curves.heegner.satisfies_weak_heegner_hypothesis
sage: s(37,-7)
True
sage: s(37,-37)
False
sage: s(37,-37*4)
True
sage: s(100,-4)
False
sage: [D for D in [-1,-2,..,-40] if s(37,D)]
[-3, -4, -7, -11, -40]
sage: [D for D in [-1,-2,..,-100] if s(37,D)]
[-3, -4, -7, -11, -40, -47, -67, -71, -83, -84, -95]
sage: EllipticCurve('37a').heegner_discriminants_list(10)
[-7, -11, -40, -47, -67, -71, -83, -84, -95, -104]

Python

>>> from sage.all import *
>>> s = sage.schemes.elliptic_curves.heegner.satisfies_weak_heegner_hypothesis
>>> s(Integer(37),-Integer(7))
True
>>> s(Integer(37),-Integer(37))
False
>>> s(Integer(37),-Integer(37)*Integer(4))
True
>>> s(Integer(100),-Integer(4))
False
>>> [D for D in (ellipsis_range(-Integer(1),-Integer(2),Ellipsis,-Integer(40))) if s(Integer(37),D)]
[-3, -4, -7, -11, -40]
>>> [D for D in (ellipsis_range(-Integer(1),-Integer(2),Ellipsis,-Integer(100))) if s(Integer(37),D)]
[-3, -4, -7, -11, -40, -47, -67, -71, -83, -84, -95]
>>> EllipticCurve('37a').heegner_discriminants_list(Integer(10))
[-7, -11, -40, -47, -67, -71, -83, -84, -95, -104]

sage.schemes.elliptic_curves.heegner.simplest_rational_poly(f, prec)[source]¶

Return a polynomial whose coefficients are as simple as possible rationals that are also close to the coefficients of f.

INPUT:

f – polynomial with real floating point entries
prec – positive integer

EXAMPLES:

Sage

sage: from sage.schemes.elliptic_curves.heegner import simplest_rational_poly
sage: R.<x> = RR[]
sage: simplest_rational_poly(2.1*x^2 + 3.5*x - 1.2, 53)
21/10*X^2 + 7/2*X - 6/5

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.heegner import simplest_rational_poly
>>> R = RR['x']; (x,) = R._first_ngens(1)
>>> simplest_rational_poly(RealNumber('2.1')*x**Integer(2) + RealNumber('3.5')*x - RealNumber('1.2'), Integer(53))
21/10*X^2 + 7/2*X - 6/5