Canonical heights for elliptic curves over number fields¶

Also, rigorous lower bounds for the canonical height of non-torsion points, implementing the algorithms in [CS2006] (over \(\QQ\)) and [Tho2010], which also refer to [CPS2006].

AUTHORS:

Robert Bradshaw (2010): initial version
John Cremona (2014): added many docstrings and doctests

class sage.schemes.elliptic_curves.height.EllipticCurveCanonicalHeight(E)[source]¶

Bases: object

Class for computing canonical heights of points on elliptic curves defined over number fields, including rigorous lower bounds for the canonical height of non-torsion points.

EXAMPLES:

Sage

sage: from sage.schemes.elliptic_curves.height import EllipticCurveCanonicalHeight
sage: E = EllipticCurve([0,0,0,0,1])
sage: EllipticCurveCanonicalHeight(E)
EllipticCurveCanonicalHeight object associated to
 Elliptic Curve defined by y^2 = x^3 + 1 over Rational Field

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.height import EllipticCurveCanonicalHeight
>>> E = EllipticCurve([Integer(0),Integer(0),Integer(0),Integer(0),Integer(1)])
>>> EllipticCurveCanonicalHeight(E)
EllipticCurveCanonicalHeight object associated to
 Elliptic Curve defined by y^2 = x^3 + 1 over Rational Field

Normally this object would be created like this:

Sage

sage: E.height_function()
EllipticCurveCanonicalHeight object associated to
 Elliptic Curve defined by y^2 = x^3 + 1 over Rational Field

Python

>>> from sage.all import *
>>> E.height_function()
EllipticCurveCanonicalHeight object associated to
 Elliptic Curve defined by y^2 = x^3 + 1 over Rational Field

B(n, mu)[source]¶

Return the value \(B_n(\mu)\).

INPUT:

n – positive integer
mu – positive real number

OUTPUT:

The real value \(B_n(\mu)\) as defined in [Tho2010], section 5.

EXAMPLES:

Example 10.2 from [Tho2010]:

Sage

sage: K.<i> = QuadraticField(-1)                                            # needs sage.rings.number_field
sage: E = EllipticCurve([0, 1-i, i, -i, 0])                                 # needs sage.rings.number_field
sage: H = E.height_function()                                               # needs sage.rings.number_field

Python

>>> from sage.all import *
>>> K = QuadraticField(-Integer(1), names=('i',)); (i,) = K._first_ngens(1)# needs sage.rings.number_field
>>> E = EllipticCurve([Integer(0), Integer(1)-i, i, -i, Integer(0)])                                 # needs sage.rings.number_field
>>> H = E.height_function()                                               # needs sage.rings.number_field

In [Tho2010] the value is given as 0.772:

Sage

sage: RealField(12)( H.B(5, 0.01) )                                         # needs sage.rings.number_field
0.777

Python

>>> from sage.all import *
>>> RealField(Integer(12))( H.B(Integer(5), RealNumber('0.01')) )                                         # needs sage.rings.number_field
0.777

DE(n)[source]¶

Return the value \(D_E(n)\).

INPUT:

n – positive integer

OUTPUT:

The value \(D_E(n)\) as defined in [Tho2010], section 4.

EXAMPLES:

Sage

sage: # needs sage.rings.number_field
sage: K.<i> = QuadraticField(-1)
sage: E = EllipticCurve([0, 0, 0, 1+5*i, 3+i])
sage: H = E.height_function()
sage: [H.DE(n) for n in srange(1,6)]
[0, 2*log(5) + 2*log(2), 0, 2*log(13) + 2*log(5) + 4*log(2), 0]

Python

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> K = QuadraticField(-Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> E = EllipticCurve([Integer(0), Integer(0), Integer(0), Integer(1)+Integer(5)*i, Integer(3)+i])
>>> H = E.height_function()
>>> [H.DE(n) for n in srange(Integer(1),Integer(6))]
[0, 2*log(5) + 2*log(2), 0, 2*log(13) + 2*log(5) + 4*log(2), 0]

ME()[source]¶

Return the norm of the ideal \(M_E\).

OUTPUT:

The norm of the ideal \(M_E\) as defined in [Tho2010], section 3.1. This is \(1\) if \(E\) is a global minimal model, and in general measures the non-minimality of \(E\).

EXAMPLES:

Sage

sage: K.<i> = QuadraticField(-1)                                            # needs sage.rings.number_field
sage: E = EllipticCurve([0, 0, 0, 1+5*i, 3+i])                              # needs sage.rings.number_field
sage: H = E.height_function()                                               # needs sage.rings.number_field
sage: H.ME()                                                                # needs sage.rings.number_field
1
sage: E = EllipticCurve([0,0,0,0,1])
sage: E.height_function().ME()
1
sage: E = EllipticCurve([0,0,0,0,64])
sage: E.height_function().ME()
4096
sage: E.discriminant()/E.minimal_model().discriminant()
4096

Python

>>> from sage.all import *
>>> K = QuadraticField(-Integer(1), names=('i',)); (i,) = K._first_ngens(1)# needs sage.rings.number_field
>>> E = EllipticCurve([Integer(0), Integer(0), Integer(0), Integer(1)+Integer(5)*i, Integer(3)+i])                              # needs sage.rings.number_field
>>> H = E.height_function()                                               # needs sage.rings.number_field
>>> H.ME()                                                                # needs sage.rings.number_field
1
>>> E = EllipticCurve([Integer(0),Integer(0),Integer(0),Integer(0),Integer(1)])
>>> E.height_function().ME()
1
>>> E = EllipticCurve([Integer(0),Integer(0),Integer(0),Integer(0),Integer(64)])
>>> E.height_function().ME()
4096
>>> E.discriminant()/E.minimal_model().discriminant()
4096

S(xi1, xi2, v)[source]¶

Return the union of intervals \(S^{(v)}(\xi_1,\xi_2)\).

INPUT:

xi1, xi2 – real numbers with \(\xi_1\le\xi_2\)
v – a real embedding of the field

OUTPUT:

The union of intervals \(S^{(v)}(\xi_1,\xi_2)\) defined in [Tho2010] section 6.1.

EXAMPLES:

An example over \(\QQ\):

Sage

sage: E = EllipticCurve('389a')
sage: v = QQ.places()[0]
sage: H = E.height_function()
sage: H.S(2, 3, v)
([0.224512677391895, 0.274544821597130] U [0.725455178402870, 0.775487322608105])

Python

>>> from sage.all import *
>>> E = EllipticCurve('389a')
>>> v = QQ.places()[Integer(0)]
>>> H = E.height_function()
>>> H.S(Integer(2), Integer(3), v)
([0.224512677391895, 0.274544821597130] U [0.725455178402870, 0.775487322608105])

An example over a number field:

Sage

sage: # needs sage.rings.number_field
sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^3 - 2)
sage: E = EllipticCurve([0,0,0,0,a])
sage: v = K.real_places()[0]
sage: H = E.height_function()
sage: H.S(9, 10, v)
([0.078119444725347..., 0.082342373201640...] U [0.91765762679836..., 0.92188055527465...])

Python

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> E = EllipticCurve([Integer(0),Integer(0),Integer(0),Integer(0),a])
>>> v = K.real_places()[Integer(0)]
>>> H = E.height_function()
>>> H.S(Integer(9), Integer(10), v)
([0.078119444725347..., 0.082342373201640...] U [0.91765762679836..., 0.92188055527465...])

Sn(xi1, xi2, n, v)[source]¶

Return the union of intervals \(S_n^{(v)}(\xi_1,\xi_2)\).

INPUT:

xi1, xi2 – real numbers with \(\xi_1\le\xi_2\)
n – positive integer
v – a real embedding of the field

OUTPUT:

The union of intervals \(S_n^{(v)}(\xi_1,\xi_2)\) defined in [Tho2010] (Lemma 6.1).

EXAMPLES:

An example over \(\QQ\):

Sage

sage: E = EllipticCurve('389a')
sage: v = QQ.places()[0]
sage: H = E.height_function()
sage: H.S(2, 3, v), H.Sn(2, 3, 1, v)
(([0.224512677391895, 0.274544821597130] U [0.725455178402870, 0.775487322608105]),
([0.224512677391895, 0.274544821597130] U [0.725455178402870, 0.775487322608105]))
sage: H.Sn(2, 3, 6, v)
([0.0374187795653158, 0.0457574702661884] U [0.120909196400478, 0.129247887101351] U [0.204085446231982, 0.212424136932855] U [0.287575863067145, 0.295914553768017] U [0.370752112898649, 0.379090803599522] U [0.454242529733812, 0.462581220434684] U [0.537418779565316, 0.545757470266188] U [0.620909196400478, 0.629247887101351] U [0.704085446231982, 0.712424136932855] U [0.787575863067145, 0.795914553768017] U [0.870752112898649, 0.879090803599522] U [0.954242529733812, 0.962581220434684])

Python

>>> from sage.all import *
>>> E = EllipticCurve('389a')
>>> v = QQ.places()[Integer(0)]
>>> H = E.height_function()
>>> H.S(Integer(2), Integer(3), v), H.Sn(Integer(2), Integer(3), Integer(1), v)
(([0.224512677391895, 0.274544821597130] U [0.725455178402870, 0.775487322608105]),
([0.224512677391895, 0.274544821597130] U [0.725455178402870, 0.775487322608105]))
>>> H.Sn(Integer(2), Integer(3), Integer(6), v)
([0.0374187795653158, 0.0457574702661884] U [0.120909196400478, 0.129247887101351] U [0.204085446231982, 0.212424136932855] U [0.287575863067145, 0.295914553768017] U [0.370752112898649, 0.379090803599522] U [0.454242529733812, 0.462581220434684] U [0.537418779565316, 0.545757470266188] U [0.620909196400478, 0.629247887101351] U [0.704085446231982, 0.712424136932855] U [0.787575863067145, 0.795914553768017] U [0.870752112898649, 0.879090803599522] U [0.954242529733812, 0.962581220434684])

An example over a number field:

Sage

sage: # needs sage.rings.number_field
sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^3 - 2)
sage: E = EllipticCurve([0,0,0,0,a])
sage: v = K.real_places()[0]
sage: H = E.height_function()
sage: H.S(2, 3, v), H.Sn(2, 3, 1, v)
(([0.142172065860075, 0.172845716928584] U [0.827154283071416, 0.857827934139925]),
 ([0.142172065860075, 0.172845716928584] U [0.827154283071416, 0.857827934139925]))
sage: H.Sn(2, 3, 6, v)
([0.0236953443100124, 0.0288076194880974] U [0.137859047178569, 0.142971322356654] U [0.190362010976679, 0.195474286154764] U [0.304525713845236, 0.309637989023321] U [0.357028677643346, 0.362140952821431] U [0.471192380511903, 0.476304655689988] U [0.523695344310012, 0.528807619488097] U [0.637859047178569, 0.642971322356654] U [0.690362010976679, 0.695474286154764] U [0.804525713845236, 0.809637989023321] U [0.857028677643346, 0.862140952821431] U [0.971192380511903, 0.976304655689988])

Python

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> E = EllipticCurve([Integer(0),Integer(0),Integer(0),Integer(0),a])
>>> v = K.real_places()[Integer(0)]
>>> H = E.height_function()
>>> H.S(Integer(2), Integer(3), v), H.Sn(Integer(2), Integer(3), Integer(1), v)
(([0.142172065860075, 0.172845716928584] U [0.827154283071416, 0.857827934139925]),
 ([0.142172065860075, 0.172845716928584] U [0.827154283071416, 0.857827934139925]))
>>> H.Sn(Integer(2), Integer(3), Integer(6), v)
([0.0236953443100124, 0.0288076194880974] U [0.137859047178569, 0.142971322356654] U [0.190362010976679, 0.195474286154764] U [0.304525713845236, 0.309637989023321] U [0.357028677643346, 0.362140952821431] U [0.471192380511903, 0.476304655689988] U [0.523695344310012, 0.528807619488097] U [0.637859047178569, 0.642971322356654] U [0.690362010976679, 0.695474286154764] U [0.804525713845236, 0.809637989023321] U [0.857028677643346, 0.862140952821431] U [0.971192380511903, 0.976304655689988])

alpha(v, tol=0.01)[source]¶

Return the constant \(\alpha_v\) associated to the embedding v.

INPUT:

v – an embedding of the base field into \(\RR\) or \(\CC\)

OUTPUT:

The constant \(\alpha_v\). In the notation of [CPS2006] and [Tho2010] (section 3.2), \(\alpha_v^3=\epsilon_v\). The result is cached since it only depends on the curve.

EXAMPLES:

Example 1 from [CPS2006]:

Sage

sage: # needs sage.rings.number_field
sage: K.<i> = QuadraticField(-1)
sage: E = EllipticCurve([0, 0, 0, 1 + 5*i, 3 + i])
sage: H = E.height_function()
sage: alpha = H.alpha(K.places()[0])
sage: alpha
1.12272013439355

Python

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> K = QuadraticField(-Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> E = EllipticCurve([Integer(0), Integer(0), Integer(0), Integer(1) + Integer(5)*i, Integer(3) + i])
>>> H = E.height_function()
>>> alpha = H.alpha(K.places()[Integer(0)])
>>> alpha
1.12272013439355

Compare with \(\log(\epsilon_v)=0.344562...\) in [CPS2006]:

Sage

sage: 3*alpha.log()                                                         # needs sage.rings.number_field
0.347263296676126

Python

>>> from sage.all import *
>>> Integer(3)*alpha.log()                                                         # needs sage.rings.number_field
0.347263296676126

base_field()[source]¶

Return the base field.

EXAMPLES:

Sage

sage: E = EllipticCurve([0,0,0,0,1])
sage: H = E.height_function()
sage: H.base_field()
Rational Field

Python

>>> from sage.all import *
>>> E = EllipticCurve([Integer(0),Integer(0),Integer(0),Integer(0),Integer(1)])
>>> H = E.height_function()
>>> H.base_field()
Rational Field

complex_intersection_is_empty(Bk, v, verbose=False, use_half=True)[source]¶

Return True iff an intersection of \(T_n^{(v)}\) sets is empty.

INPUT:

Bk – list of reals
v – a complex embedding of the number field
verbose– boolean (default: False); verbosity flag
use_half– boolean (default: False); if True, use only half the fundamental region

OUTPUT:

True or False, according as the intersection of the unions of intervals \(T_n^{(v)}(-b,b)\) for \(b\) in the list Bk (see [Tho2010], section 7) is empty or not. When Bk is the list of \(b=\sqrt{B_n(\mu)}\) for \(n=1,2,3,\dots\) for some \(\mu>0\) this means that all non-torsion points on \(E\) with everywhere good reduction have canonical height strictly greater than \(\mu\), by [Tho2010], Proposition 7.8.

EXAMPLES:

Sage

sage: # needs sage.rings.number_field
sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^3 - 2)
sage: E = EllipticCurve([0,0,0,0,a])
sage: v = K.complex_embeddings()[0]
sage: H = E.height_function()

Python

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> E = EllipticCurve([Integer(0),Integer(0),Integer(0),Integer(0),a])
>>> v = K.complex_embeddings()[Integer(0)]
>>> H = E.height_function()

The following two lines prove that the heights of non-torsion points on \(E\) with everywhere good reduction have canonical height strictly greater than 0.02, but fail to prove the same for 0.03. For the first proof, using only \(n=1,2,3\) is not sufficient:

Sage

sage: H.complex_intersection_is_empty([H.B(n,0.02) for n in [1,2,3]], v)  # long time, needs sage.rings.number_field
False
sage: H.complex_intersection_is_empty([H.B(n,0.02) for n in [1,2,3,4]], v)  # needs sage.rings.number_field
True
sage: H.complex_intersection_is_empty([H.B(n,0.03) for n in [1,2,3,4]], v)  # long time, needs sage.rings.number_field
False

Python

>>> from sage.all import *
>>> H.complex_intersection_is_empty([H.B(n,RealNumber('0.02')) for n in [Integer(1),Integer(2),Integer(3)]], v)  # long time, needs sage.rings.number_field
False
>>> H.complex_intersection_is_empty([H.B(n,RealNumber('0.02')) for n in [Integer(1),Integer(2),Integer(3),Integer(4)]], v)  # needs sage.rings.number_field
True
>>> H.complex_intersection_is_empty([H.B(n,RealNumber('0.03')) for n in [Integer(1),Integer(2),Integer(3),Integer(4)]], v)  # long time, needs sage.rings.number_field
False

Using \(n\le6\) enables us to prove the lower bound 0.03. Note that it takes longer when the result is False than when it is True:

Sage

sage: H.complex_intersection_is_empty([H.B(n,0.03) for n in [1..6]], v)     # needs sage.rings.number_field
True

Python

>>> from sage.all import *
>>> H.complex_intersection_is_empty([H.B(n,RealNumber('0.03')) for n in (ellipsis_range(Integer(1),Ellipsis,Integer(6)))], v)     # needs sage.rings.number_field
True

curve()[source]¶

Return the elliptic curve.

EXAMPLES:

Sage

sage: E = EllipticCurve([0,0,0,0,1])
sage: H = E.height_function()
sage: H.curve()
Elliptic Curve defined by y^2 = x^3 + 1 over Rational Field

Python

>>> from sage.all import *
>>> E = EllipticCurve([Integer(0),Integer(0),Integer(0),Integer(0),Integer(1)])
>>> H = E.height_function()
>>> H.curve()
Elliptic Curve defined by y^2 = x^3 + 1 over Rational Field

e_p(p)[source]¶

Return the exponent of the group over the residue field at p.

INPUT:

p – a prime ideal of \(K\) (or a prime number if \(K=\QQ\))

OUTPUT:

A positive integer \(e_p\), the exponent of the group of nonsingular points on the reduction of the elliptic curve modulo \(p\). The result is cached.

EXAMPLES:

Sage

sage: # needs sage.rings.number_field
sage: K.<i> = QuadraticField(-1)
sage: E = EllipticCurve([0, 0, 0, 1 + 5*i, 3 + i])
sage: H = E.height_function()
sage: H.e_p(K.prime_above(2))
2
sage: H.e_p(K.prime_above(3))
10
sage: H.e_p(K.prime_above(5))
9
sage: E.conductor().norm().factor()
2^10 * 20921
sage: p1, p2 = K.primes_above(20921)
sage: E.local_data(p1)
Local data at Fractional ideal (-40*i + 139):
Reduction type: bad split multiplicative
...
sage: H.e_p(p1)
20920
sage: E.local_data(p2)
Local data at Fractional ideal (40*i + 139):
Reduction type: good
...
sage: H.e_p(p2)
20815

Python

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> K = QuadraticField(-Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> E = EllipticCurve([Integer(0), Integer(0), Integer(0), Integer(1) + Integer(5)*i, Integer(3) + i])
>>> H = E.height_function()
>>> H.e_p(K.prime_above(Integer(2)))
2
>>> H.e_p(K.prime_above(Integer(3)))
10
>>> H.e_p(K.prime_above(Integer(5)))
9
>>> E.conductor().norm().factor()
2^10 * 20921
>>> p1, p2 = K.primes_above(Integer(20921))
>>> E.local_data(p1)
Local data at Fractional ideal (-40*i + 139):
Reduction type: bad split multiplicative
...
>>> H.e_p(p1)
20920
>>> E.local_data(p2)
Local data at Fractional ideal (40*i + 139):
Reduction type: good
...
>>> H.e_p(p2)
20815

fk_intervals(v=None, N=20, domain=Complex Interval Field with 53 bits of precision)[source]¶

Return a function approximating the Weierstrass function, with error.

INPUT:

v – an embedding of the number field. If None (default) use the real embedding if the field is \(\QQ\) and raise an error for other fields.
N – integer; the number of terms to use in the \(q\)-expansion of \(\wp\)
domain – (complex field) the model of \(\CC\) to use, for example CDF of CIF (default)

OUTPUT:

A pair of functions fk, err which can be evaluated at complex numbers \(z\) (in the correct domain) to give an approximation to \(\wp(z)\) and an upper bound on the error, respectively. The Weierstrass function returned is with respect to the normalised lattice \([1,\tau]\) associated to the given embedding.

EXAMPLES:

Sage

sage: E = EllipticCurve('37a')
sage: L = E.period_lattice()
sage: w1, w2 = L.normalised_basis()
sage: z = CDF(0.3, 0.4)

Python

>>> from sage.all import *
>>> E = EllipticCurve('37a')
>>> L = E.period_lattice()
>>> w1, w2 = L.normalised_basis()
>>> z = CDF(RealNumber('0.3'), RealNumber('0.4'))

Compare the value give by the standard elliptic exponential (scaled since fk is with respect to the normalised lattice):

Sage

sage: L.elliptic_exponential(z*w2, to_curve=False)[0] * w2 ** 2
-1.82543539306049 - 2.49336319992847*I

Python

>>> from sage.all import *
>>> L.elliptic_exponential(z*w2, to_curve=False)[Integer(0)] * w2 ** Integer(2)
-1.82543539306049 - 2.49336319992847*I

to the value given by this function, and see the error:

Sage

sage: fk, err = E.height_function().fk_intervals(N=10)
sage: fk(CIF(z))
-1.82543539306049? - 2.49336319992847?*I
sage: err(CIF(z))
2.71750621458744e-31

Python

>>> from sage.all import *
>>> fk, err = E.height_function().fk_intervals(N=Integer(10))
>>> fk(CIF(z))
-1.82543539306049? - 2.49336319992847?*I
>>> err(CIF(z))
2.71750621458744e-31

The same, but in the domain CDF instead of CIF:

Sage

sage: fk, err = E.height_function().fk_intervals(N=10, domain=CDF)
sage: fk(z)
-1.8254353930604... - 2.493363199928...*I

Python

>>> from sage.all import *
>>> fk, err = E.height_function().fk_intervals(N=Integer(10), domain=CDF)
>>> fk(z)
-1.8254353930604... - 2.493363199928...*I

min(tol, n_max, verbose=False)[source]¶

Return a lower bound for all points of infinite order.

INPUT:

tol – tolerance in output (see below)
n_max – how many multiples to use in iteration
verbose– boolean (default: False); verbosity flag

OUTPUT:

A positive real \(\mu\) for which it has been established rigorously that every point of infinite order on the elliptic curve (defined over its ground field) has canonical height greater than \(\mu\), and such that it is not possible (at least without increasing n_max) to prove the same for \(\mu\cdot\text{tol}\).

EXAMPLES:

Example 1 from [CS2006] (where the same lower bound of 0.1126 was given):

Sage

sage: E = EllipticCurve([1, 0, 1, 421152067, 105484554028056]) # 60490d1
sage: E.height_function().min(.0001, 5)
0.0011263287309893311

Python

>>> from sage.all import *
>>> E = EllipticCurve([Integer(1), Integer(0), Integer(1), Integer(421152067), Integer(105484554028056)]) # 60490d1
>>> E.height_function().min(RealNumber('.0001'), Integer(5))
0.0011263287309893311

Example 10.1 from [Tho2010] (where a lower bound of 0.18 was given):

Sage

sage: # needs sage.rings.number_field
sage: K.<i> = QuadraticField(-1)
sage: E = EllipticCurve([0, 0, 0, 91 - 26*i, -144 - 323*i])
sage: H = E.height_function()
sage: H.min(0.1, 4)                 # long time
0.1621049443313762

Python

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> K = QuadraticField(-Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> E = EllipticCurve([Integer(0), Integer(0), Integer(0), Integer(91) - Integer(26)*i, -Integer(144) - Integer(323)*i])
>>> H = E.height_function()
>>> H.min(RealNumber('0.1'), Integer(4))                 # long time
0.1621049443313762

Example 10.2 from [Tho2010]:

Sage

sage: # needs sage.rings.number_field
sage: K.<i> = QuadraticField(-1)
sage: E = EllipticCurve([0, 1 - i, i, -i, 0])
sage: H = E.height_function()
sage: H.min(0.01, 5)                # long time
0.020153685521979152

Python

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> K = QuadraticField(-Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> E = EllipticCurve([Integer(0), Integer(1) - i, i, -i, Integer(0)])
>>> H = E.height_function()
>>> H.min(RealNumber('0.01'), Integer(5))                # long time
0.020153685521979152

In this example the point \(P=(0,0)\) has height 0.023 so our lower bound is quite good:

Sage

sage: P = E((0,0))                                                          # needs sage.rings.number_field
sage: P.height()                                                            # needs sage.rings.number_field
0.0230242154471211

Python

>>> from sage.all import *
>>> P = E((Integer(0),Integer(0)))                                                          # needs sage.rings.number_field
>>> P.height()                                                            # needs sage.rings.number_field
0.0230242154471211

Example 10.3 from [Tho2010] (where the same bound of 0.0625 is given):

Sage

sage: # needs sage.rings.number_field
sage: K.<a> = NumberField(x^3 - 2)
sage: E = EllipticCurve([0, 0, 0, -3*a - a^2, a^2])
sage: H = E.height_function()
sage: H.min(0.1, 5)                 # long time
0.0625

Python

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> E = EllipticCurve([Integer(0), Integer(0), Integer(0), -Integer(3)*a - a**Integer(2), a**Integer(2)])
>>> H = E.height_function()
>>> H.min(RealNumber('0.1'), Integer(5))                 # long time
0.0625

More examples over \(\QQ\):

Sage

sage: E = EllipticCurve('37a')
sage: h = E.height_function()
sage: h.min(.01, 5)
0.03987318057488725
sage: E.gen(0).height()
0.0511114082399688

Python

>>> from sage.all import *
>>> E = EllipticCurve('37a')
>>> h = E.height_function()
>>> h.min(RealNumber('.01'), Integer(5))
0.03987318057488725
>>> E.gen(Integer(0)).height()
0.0511114082399688

After base change the lower bound can decrease:

Sage

sage: K.<a> = QuadraticField(-5)                                            # needs sage.rings.number_field
sage: E.change_ring(K).height_function().min(0.5, 10)       # long time, needs sage.rings.number_field
0.04419417382415922

sage: E = EllipticCurve('389a')
sage: h = E.height_function()
sage: h.min(0.1, 5)
0.05731275270029196
sage: [P.height() for P in E.gens()]
[0.686667083305587, 0.327000773651605]

Python

>>> from sage.all import *
>>> K = QuadraticField(-Integer(5), names=('a',)); (a,) = K._first_ngens(1)# needs sage.rings.number_field
>>> E.change_ring(K).height_function().min(RealNumber('0.5'), Integer(10))       # long time, needs sage.rings.number_field
0.04419417382415922

>>> E = EllipticCurve('389a')
>>> h = E.height_function()
>>> h.min(RealNumber('0.1'), Integer(5))
0.05731275270029196
>>> [P.height() for P in E.gens()]
[0.686667083305587, 0.327000773651605]

min_gr(tol, n_max, verbose=False)[source]¶

Return a lower bound for points of infinite order with good reduction.

INPUT:

tol – tolerance in output (see below)
n_max – how many multiples to use in iteration
verbose– boolean (default: False); verbosity flag

OUTPUT:

A positive real \(\mu\) for which it has been established rigorously that every point of infinite order on the elliptic curve (defined over its ground field), which has good reduction at all primes, has canonical height greater than \(\mu\), and such that it is not possible (at least without increasing n_max) to prove the same for \(\mu\cdot\text{tol}\).

EXAMPLES:

Example 1 from [CS2006] (where a lower bound of 1.9865 was given):

Sage

sage: E = EllipticCurve([1, 0, 1, 421152067, 105484554028056]) # 60490d1
sage: E.height_function().min_gr(.0001, 5)
1.98684388146518

Python

>>> from sage.all import *
>>> E = EllipticCurve([Integer(1), Integer(0), Integer(1), Integer(421152067), Integer(105484554028056)]) # 60490d1
>>> E.height_function().min_gr(RealNumber('.0001'), Integer(5))
1.98684388146518

Example 10.1 from [Tho2010] (where a lower bound of 0.18 was given):

Sage

sage: # needs sage.rings.number_field
sage: K.<i> = QuadraticField(-1)
sage: E = EllipticCurve([0, 0, 0, 91 - 26*i, -144 - 323*i])
sage: H = E.height_function()
sage: H.min_gr(0.1, 4)              # long time
0.1621049443313762

Python

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> K = QuadraticField(-Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> E = EllipticCurve([Integer(0), Integer(0), Integer(0), Integer(91) - Integer(26)*i, -Integer(144) - Integer(323)*i])
>>> H = E.height_function()
>>> H.min_gr(RealNumber('0.1'), Integer(4))              # long time
0.1621049443313762

Example 10.2 from [Tho2010]:

Sage

sage: # needs sage.rings.number_field
sage: K.<i> = QuadraticField(-1)
sage: E = EllipticCurve([0, 1 - i, i, -i, 0])
sage: H = E.height_function()
sage: H.min_gr(0.01, 5)             # long time
0.020153685521979152

Python

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> K = QuadraticField(-Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> E = EllipticCurve([Integer(0), Integer(1) - i, i, -i, Integer(0)])
>>> H = E.height_function()
>>> H.min_gr(RealNumber('0.01'), Integer(5))             # long time
0.020153685521979152

In this example the point \(P=(0,0)\) has height 0.023 so our lower bound is quite good:

Sage

sage: P = E((0,0))                                                          # needs sage.rings.number_field
sage: P.has_good_reduction()                                                # needs sage.rings.number_field
True
sage: P.height()                                                            # needs sage.rings.number_field
0.0230242154471211

Python

>>> from sage.all import *
>>> P = E((Integer(0),Integer(0)))                                                          # needs sage.rings.number_field
>>> P.has_good_reduction()                                                # needs sage.rings.number_field
True
>>> P.height()                                                            # needs sage.rings.number_field
0.0230242154471211

Example 10.3 from [Tho2010] (where the same bound of 0.25 is given):

Sage

sage: # needs sage.rings.number_field
sage: K.<a> = NumberField(x^3 - 2)
sage: E = EllipticCurve([0, 0, 0, -3*a - a^2, a^2])
sage: H = E.height_function()
sage: H.min_gr(0.1, 5)              # long time
0.25

Python

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> E = EllipticCurve([Integer(0), Integer(0), Integer(0), -Integer(3)*a - a**Integer(2), a**Integer(2)])
>>> H = E.height_function()
>>> H.min_gr(RealNumber('0.1'), Integer(5))              # long time
0.25

psi(xi, v)[source]¶

Return the normalised elliptic log of a point with this x-coordinate.

INPUT:

xi – real; the real x-coordinate of a point on the curve in the connected component with respect to a real embedding
v – a real embedding of the number field

OUTPUT:

A real number in the interval [0.5,1] giving the elliptic logarithm of a point on \(E\) with \(x\)-coordinate xi, on the connected component with respect to the embedding \(v\), scaled by the real period.

EXAMPLES:

An example over \(\QQ\):

Sage

sage: E = EllipticCurve('389a')
sage: v = QQ.places()[0]
sage: L = E.period_lattice(v)
sage: P = E.lift_x(10/9)
sage: L(P)
0.958696500380439
sage: L(P) / L.real_period()
0.384985810227885
sage: H = E.height_function()
sage: H.psi(10/9, v)
0.615014189772115

Python

>>> from sage.all import *
>>> E = EllipticCurve('389a')
>>> v = QQ.places()[Integer(0)]
>>> L = E.period_lattice(v)
>>> P = E.lift_x(Integer(10)/Integer(9))
>>> L(P)
0.958696500380439
>>> L(P) / L.real_period()
0.384985810227885
>>> H = E.height_function()
>>> H.psi(Integer(10)/Integer(9), v)
0.615014189772115

An example over a number field:

Sage

sage: # needs sage.rings.number_field
sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^3 - 2)
sage: E = EllipticCurve([0,0,0,0,a])
sage: P = E.lift_x(1/3*a^2 + a + 5/3)
sage: v = K.real_places()[0]
sage: L = E.period_lattice(v)
sage: L(P)
3.51086196882538
sage: L(P) / L.real_period()
0.867385122699931
sage: xP = v(P.x())
sage: H = E.height_function()
sage: H.psi(xP, v)
0.867385122699931
sage: H.psi(1.23, v)
0.785854718241495

Python

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> E = EllipticCurve([Integer(0),Integer(0),Integer(0),Integer(0),a])
>>> P = E.lift_x(Integer(1)/Integer(3)*a**Integer(2) + a + Integer(5)/Integer(3))
>>> v = K.real_places()[Integer(0)]
>>> L = E.period_lattice(v)
>>> L(P)
3.51086196882538
>>> L(P) / L.real_period()
0.867385122699931
>>> xP = v(P.x())
>>> H = E.height_function()
>>> H.psi(xP, v)
0.867385122699931
>>> H.psi(RealNumber('1.23'), v)
0.785854718241495

real_intersection_is_empty(Bk, v)[source]¶

Return True iff an intersection of \(S_n^{(v)}\) sets is empty.

INPUT:

Bk – list of reals
v – a real embedding of the number field

OUTPUT:

True or False, according as the intersection of the unions of intervals \(S_n^{(v)}(-b,b)\) for \(b\) in the list Bk is empty or not. When Bk is the list of \(b=B_n(\mu)\) for \(n=1,2,3,\dots\) for some \(\mu>0\) this means that all non-torsion points on \(E\) with everywhere good reduction have canonical height strictly greater than \(\mu\), by [Tho2010], Proposition 6.2.

EXAMPLES:

An example over \(\QQ\):

Sage

sage: E = EllipticCurve('389a')
sage: v = QQ.places()[0]
sage: H = E.height_function()

Python

>>> from sage.all import *
>>> E = EllipticCurve('389a')
>>> v = QQ.places()[Integer(0)]
>>> H = E.height_function()

The following two lines prove that the heights of non-torsion points on \(E\) with everywhere good reduction have canonical height strictly greater than 0.2, but fail to prove the same for 0.3:

Sage

sage: H.real_intersection_is_empty([H.B(n,0.2) for n in srange(1,10)], v)
True
sage: H.real_intersection_is_empty([H.B(n,0.3) for n in srange(1,10)], v)
False

Python

>>> from sage.all import *
>>> H.real_intersection_is_empty([H.B(n,RealNumber('0.2')) for n in srange(Integer(1),Integer(10))], v)
True
>>> H.real_intersection_is_empty([H.B(n,RealNumber('0.3')) for n in srange(Integer(1),Integer(10))], v)
False

An example over a number field:

Sage

sage: # needs sage.rings.number_field
sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^3 - 2)
sage: E = EllipticCurve([0,0,0,0,a])
sage: v = K.real_places()[0]
sage: H = E.height_function()

Python

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> E = EllipticCurve([Integer(0),Integer(0),Integer(0),Integer(0),a])
>>> v = K.real_places()[Integer(0)]
>>> H = E.height_function()

The following two lines prove that the heights of non-torsion points on \(E\) with everywhere good reduction have canonical height strictly greater than 0.07, but fail to prove the same for 0.08:

Sage

sage: H.real_intersection_is_empty([H.B(n,0.07) for n in srange(1,5)], v)  # long time, needs sage.rings.number_field
True
sage: H.real_intersection_is_empty([H.B(n,0.08) for n in srange(1,5)], v)   # needs sage.rings.number_field
False

Python

>>> from sage.all import *
>>> H.real_intersection_is_empty([H.B(n,RealNumber('0.07')) for n in srange(Integer(1),Integer(5))], v)  # long time, needs sage.rings.number_field
True
>>> H.real_intersection_is_empty([H.B(n,RealNumber('0.08')) for n in srange(Integer(1),Integer(5))], v)   # needs sage.rings.number_field
False

tau(v)[source]¶

Return the normalised upper half-plane parameter \(\tau\) for the period lattice with respect to the embedding \(v\).

INPUT:

v – a real or complex embedding of the number field

OUTPUT:

(Complex) \(\tau = \omega_1/\omega_2\) in the fundamental region of the upper half-plane.

EXAMPLES:

Sage

sage: E = EllipticCurve('37a')
sage: H = E.height_function()
sage: H.tau(QQ.places()[0])
1.22112736076463*I

Python

>>> from sage.all import *
>>> E = EllipticCurve('37a')
>>> H = E.height_function()
>>> H.tau(QQ.places()[Integer(0)])
1.22112736076463*I

test_mu(mu, N, verbose=True)[source]¶

Return True if we can prove that \(\mu\) is a lower bound.

INPUT:

mu – positive real number
N – integer; upper bound on the multiples to be used
verbose– boolean (default: True); verbosity flag

OUTPUT:

True or False, according to whether we succeed in proving that \(\mu\) is a lower bound for the canonical heights of points of infinite order with everywhere good reduction.

Note

A True result is rigorous; False only means that the attempt failed: trying again with larger \(N\) may yield True.

EXAMPLES:

Sage

sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^3 - 2)                                          # needs sage.rings.number_field
sage: E = EllipticCurve([0,0,0,0,a])                                        # needs sage.rings.number_field
sage: H = E.height_function()                                               # needs sage.rings.number_field

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)# needs sage.rings.number_field
>>> E = EllipticCurve([Integer(0),Integer(0),Integer(0),Integer(0),a])                                        # needs sage.rings.number_field
>>> H = E.height_function()                                               # needs sage.rings.number_field

This curve does have a point of good reduction whose canonical point is approximately 1.68:

Sage

sage: P = E.gens(lim3=5)[0]; P                                              # needs sage.rings.number_field
(1/3*a^2 + a + 5/3 : -2*a^2 - 4/3*a - 5/3 : 1)
sage: P.height()                                                            # needs sage.rings.number_field
1.68038085233673
sage: P.has_good_reduction()                                                # needs sage.rings.number_field
True

Python

>>> from sage.all import *
>>> P = E.gens(lim3=Integer(5))[Integer(0)]; P                                              # needs sage.rings.number_field
(1/3*a^2 + a + 5/3 : -2*a^2 - 4/3*a - 5/3 : 1)
>>> P.height()                                                            # needs sage.rings.number_field
1.68038085233673
>>> P.has_good_reduction()                                                # needs sage.rings.number_field
True

Using \(N=5\) we can prove that 0.1 is a lower bound (in fact we only need \(N=2\)), but not that 0.2 is:

Sage

sage: H.test_mu(0.1, 5)                                                     # needs sage.rings.number_field
B_1(0.100000000000000) = 1.51580969677387
B_2(0.100000000000000) = 0.932072561526720
True
sage: H.test_mu(0.2, 5)                                                     # needs sage.rings.number_field
B_1(0.200000000000000) = 2.04612906979932
B_2(0.200000000000000) = 3.09458988474327
B_3(0.200000000000000) = 27.6251108409484
B_4(0.200000000000000) = 1036.24722370223
B_5(0.200000000000000) = 3.67090854562318e6
False

Python

>>> from sage.all import *
>>> H.test_mu(RealNumber('0.1'), Integer(5))                                                     # needs sage.rings.number_field
B_1(0.100000000000000) = 1.51580969677387
B_2(0.100000000000000) = 0.932072561526720
True
>>> H.test_mu(RealNumber('0.2'), Integer(5))                                                     # needs sage.rings.number_field
B_1(0.200000000000000) = 2.04612906979932
B_2(0.200000000000000) = 3.09458988474327
B_3(0.200000000000000) = 27.6251108409484
B_4(0.200000000000000) = 1036.24722370223
B_5(0.200000000000000) = 3.67090854562318e6
False

Since 0.1 is a lower bound we can deduce that the point \(P\) is either primitive or divisible by either 2 or 3. In fact it is primitive:

Sage

sage: (P.height()/0.1).sqrt()                                               # needs sage.rings.number_field
4.09924487233530
sage: P.division_points(2)                                                  # needs sage.rings.number_field
[]
sage: P.division_points(3)                                                  # needs sage.rings.number_field
[]

Python

>>> from sage.all import *
>>> (P.height()/RealNumber('0.1')).sqrt()                                               # needs sage.rings.number_field
4.09924487233530
>>> P.division_points(Integer(2))                                                  # needs sage.rings.number_field
[]
>>> P.division_points(Integer(3))                                                  # needs sage.rings.number_field
[]

wp_c(v)[source]¶

Return a bound for the Weierstrass \(\wp\)-function.

INPUT:

v – a real or complex embedding of the number field

OUTPUT:

(Real) \(c>0\) such that

\[|\wp(z) - z^-2| \le \frac{c^2|z|^2}{1-c|z|^2}\]

whenever \(c|z|^2<1\). Given the recurrence relations for the Laurent series expansion of \(\wp\), it is easy to see that there is such a constant \(c\). [Reference?]

EXAMPLES:

Sage

sage: E = EllipticCurve('37a')
sage: H = E.height_function()
sage: H.wp_c(QQ.places()[0])
2.68744508779950

sage: # needs sage.rings.number_field
sage: K.<i> = QuadraticField(-1)
sage: E = EllipticCurve([0, 0, 0, 1 + 5*i, 3 + i])
sage: H = E.height_function()
sage: H.wp_c(K.places()[0])
2.66213425640096

Python

>>> from sage.all import *
>>> E = EllipticCurve('37a')
>>> H = E.height_function()
>>> H.wp_c(QQ.places()[Integer(0)])
2.68744508779950

>>> # needs sage.rings.number_field
>>> K = QuadraticField(-Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> E = EllipticCurve([Integer(0), Integer(0), Integer(0), Integer(1) + Integer(5)*i, Integer(3) + i])
>>> H = E.height_function()
>>> H.wp_c(K.places()[Integer(0)])
2.66213425640096

wp_intervals(v=None, N=20, abs_only=False)[source]¶

Return a function approximating the Weierstrass function.

INPUT:

v – an embedding of the number field; if None (default) use the real embedding if the field is \(\QQ\) and raise an error for other fields
N – integer (default: 20); The number of terms to use in the \(q\)-expansion of \(\wp\)
abs_only– boolean (default: False); flag to determine whether (if True) the error adjustment should use the absolute value or (if False) the real and imaginary parts

OUTPUT:

A function wp which can be evaluated at complex numbers \(z\) to give an approximation to \(\wp(z)\). The Weierstrass function returned is with respect to the normalised lattice \([1,\tau]\) associated to the given embedding. For \(z\) which are not near a lattice point the function fk is used, otherwise a better approximation is used.

EXAMPLES:

Sage

sage: E = EllipticCurve('37a')
sage: wp = E.height_function().wp_intervals()
sage: z = CDF(0.3, 0.4)
sage: wp(CIF(z))
-1.82543539306049? - 2.4933631999285?*I

sage: L = E.period_lattice()
sage: w1, w2 = L.normalised_basis()
sage: L.elliptic_exponential(z*w2, to_curve=False)[0] * w2^2
-1.82543539306049 - 2.49336319992847*I

sage: z = CDF(0.3, 0.1)
sage: wp(CIF(z))
8.5918243572165? - 5.4751982004351?*I
sage: L.elliptic_exponential(z*w2, to_curve=False)[0] * w2^2
8.59182435721650 - 5.47519820043503*I

Python

>>> from sage.all import *
>>> E = EllipticCurve('37a')
>>> wp = E.height_function().wp_intervals()
>>> z = CDF(RealNumber('0.3'), RealNumber('0.4'))
>>> wp(CIF(z))
-1.82543539306049? - 2.4933631999285?*I

>>> L = E.period_lattice()
>>> w1, w2 = L.normalised_basis()
>>> L.elliptic_exponential(z*w2, to_curve=False)[Integer(0)] * w2**Integer(2)
-1.82543539306049 - 2.49336319992847*I

>>> z = CDF(RealNumber('0.3'), RealNumber('0.1'))
>>> wp(CIF(z))
8.5918243572165? - 5.4751982004351?*I
>>> L.elliptic_exponential(z*w2, to_curve=False)[Integer(0)] * w2**Integer(2)
8.59182435721650 - 5.47519820043503*I

wp_on_grid(v, N, half=False)[source]¶

Return an array of the values of \(\wp\) on an \(N\times N\) grid.

INPUT:

v – an embedding of the number field
N – integer; the number of terms to use in the \(q\)-expansion of \(\wp\)
half– boolean (default: False); if True, use an array of size \(N\times N/2\) instead of \(N\times N\)

OUTPUT:

An array of size either \(N\times N/2\) or \(N\times N\) whose \((i,j)\) entry is the value of the Weierstrass \(\wp\)-function at \((i+.5)/N + (j+.5)*\tau/N\), a grid of points in the fundamental region for the lattice \([1,\tau]\).

EXAMPLES:

Sage

sage: E = EllipticCurve('37a')
sage: H = E.height_function()
sage: v = QQ.places()[0]

Python

>>> from sage.all import *
>>> E = EllipticCurve('37a')
>>> H = E.height_function()
>>> v = QQ.places()[Integer(0)]

The array of values on the grid shows symmetry, since \(\wp\) is even:

Sage

sage: H.wp_on_grid(v,4)
array([[25.43920182,  5.28760943,  5.28760943, 25.43920182],
       [ 6.05099485,  1.83757786,  1.83757786,  6.05099485],
       [ 6.05099485,  1.83757786,  1.83757786,  6.05099485],
       [25.43920182,  5.28760943,  5.28760943, 25.43920182]])

Python

>>> from sage.all import *
>>> H.wp_on_grid(v,Integer(4))
array([[25.43920182,  5.28760943,  5.28760943, 25.43920182],
       [ 6.05099485,  1.83757786,  1.83757786,  6.05099485],
       [ 6.05099485,  1.83757786,  1.83757786,  6.05099485],
       [25.43920182,  5.28760943,  5.28760943, 25.43920182]])

The array of values on the half-grid:

Sage

sage: H.wp_on_grid(v,4,True)
array([[25.43920182,  5.28760943],
       [ 6.05099485,  1.83757786],
       [ 6.05099485,  1.83757786],
       [25.43920182,  5.28760943]])

Python

>>> from sage.all import *
>>> H.wp_on_grid(v,Integer(4),True)
array([[25.43920182,  5.28760943],
       [ 6.05099485,  1.83757786],
       [ 6.05099485,  1.83757786],
       [25.43920182,  5.28760943]])

class sage.schemes.elliptic_curves.height.UnionOfIntervals(endpoints)[source]¶

Bases: object

A class representing a finite union of closed intervals in \(\RR\) which can be scaled, shifted, intersected, etc.

The intervals are represented as an ordered list of their endpoints, which may include \(-\infty\) and \(+\infty\).

EXAMPLES:

Sage

sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
sage: R = UnionOfIntervals([1, 2, 3, infinity]); R
([1, 2] U [3, +Infinity])
sage: R + 5
([6, 7] U [8, +Infinity])
sage: ~R
([-Infinity, 1] U [2, 3])
sage: ~R | (10*R + 100)
([-Infinity, 1] U [2, 3] U [110, 120] U [130, +Infinity])

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.height import UnionOfIntervals
>>> R = UnionOfIntervals([Integer(1), Integer(2), Integer(3), infinity]); R
([1, 2] U [3, +Infinity])
>>> R + Integer(5)
([6, 7] U [8, +Infinity])
>>> ~R
([-Infinity, 1] U [2, 3])
>>> ~R | (Integer(10)*R + Integer(100))
([-Infinity, 1] U [2, 3] U [110, 120] U [130, +Infinity])

Todo

Unify UnionOfIntervals with the class RealSet introduced by Issue #13125; see Issue #16063.

finite_endpoints()[source]¶

Return the finite endpoints of this union of intervals.

EXAMPLES:

Sage

sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
sage: UnionOfIntervals([0, 1]).finite_endpoints()
[0, 1]
sage: UnionOfIntervals([-infinity, 0, 1, infinity]).finite_endpoints()
[0, 1]

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.height import UnionOfIntervals
>>> UnionOfIntervals([Integer(0), Integer(1)]).finite_endpoints()
[0, 1]
>>> UnionOfIntervals([-infinity, Integer(0), Integer(1), infinity]).finite_endpoints()
[0, 1]

classmethod intersection(L)[source]¶

Return the intersection of a list of UnionOfIntervals.

INPUT:

L – list of UnionOfIntervals instances

OUTPUT:

A new UnionOfIntervals instance representing the intersection of the UnionOfIntervals in the list.

Note

This is a class method.

EXAMPLES:

Sage

sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
sage: A = UnionOfIntervals([1,3,5,7]); A
([1, 3] U [5, 7])
sage: B = A + 1; B
([2, 4] U [6, 8])
sage: A.intersection([A,B])
([2, 3] U [6, 7])

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.height import UnionOfIntervals
>>> A = UnionOfIntervals([Integer(1),Integer(3),Integer(5),Integer(7)]); A
([1, 3] U [5, 7])
>>> B = A + Integer(1); B
([2, 4] U [6, 8])
>>> A.intersection([A,B])
([2, 3] U [6, 7])

intervals()[source]¶

Return the intervals in self, as a list of 2-tuples.

EXAMPLES:

Sage

sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
sage: UnionOfIntervals(list(range(10))).intervals()
[(0, 1), (2, 3), (4, 5), (6, 7), (8, 9)]
sage: UnionOfIntervals([-infinity, pi, 17, infinity]).intervals()           # needs sage.symbolic
[(-Infinity, pi), (17, +Infinity)]

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.height import UnionOfIntervals
>>> UnionOfIntervals(list(range(Integer(10)))).intervals()
[(0, 1), (2, 3), (4, 5), (6, 7), (8, 9)]
>>> UnionOfIntervals([-infinity, pi, Integer(17), infinity]).intervals()           # needs sage.symbolic
[(-Infinity, pi), (17, +Infinity)]

is_empty()[source]¶

Return whether self is empty.

EXAMPLES:

Sage

sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
sage: UnionOfIntervals([3, 4]).is_empty()
False
sage: all = UnionOfIntervals([-infinity, infinity])
sage: all.is_empty()
False
sage: (~all).is_empty()
True
sage: A = UnionOfIntervals([0, 1]) & UnionOfIntervals([2, 3])
sage: A.is_empty()
True

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.height import UnionOfIntervals
>>> UnionOfIntervals([Integer(3), Integer(4)]).is_empty()
False
>>> all = UnionOfIntervals([-infinity, infinity])
>>> all.is_empty()
False
>>> (~all).is_empty()
True
>>> A = UnionOfIntervals([Integer(0), Integer(1)]) & UnionOfIntervals([Integer(2), Integer(3)])
>>> A.is_empty()
True

static join(L, condition)[source]¶

Utility function to form the union or intersection of a list of UnionOfIntervals.

INPUT:

L – list of UnionOfIntervals instances
condition – function; either any or all, or some other boolean function of a list of boolean values

OUTPUT:

A new UnionOfIntervals instance representing the subset of ‘RR’ equal to those reals in any/all/condition of the UnionOfIntervals in the list.

Note

This is a static method for the class.

EXAMPLES:

Sage

sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
sage: A = UnionOfIntervals([1,3,5,7]); A
([1, 3] U [5, 7])
sage: B = A + 1; B
([2, 4] U [6, 8])
sage: A.join([A,B], any) # union
([1, 4] U [5, 8])
sage: A.join([A,B], all) # intersection
([2, 3] U [6, 7])
sage: A.join([A,B], sum) # symmetric difference
([1, 2] U [3, 4] U [5, 6] U [7, 8])

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.height import UnionOfIntervals
>>> A = UnionOfIntervals([Integer(1),Integer(3),Integer(5),Integer(7)]); A
([1, 3] U [5, 7])
>>> B = A + Integer(1); B
([2, 4] U [6, 8])
>>> A.join([A,B], any) # union
([1, 4] U [5, 8])
>>> A.join([A,B], all) # intersection
([2, 3] U [6, 7])
>>> A.join([A,B], sum) # symmetric difference
([1, 2] U [3, 4] U [5, 6] U [7, 8])

classmethod union(L)[source]¶

Return the union of a list of UnionOfIntervals.

INPUT:

L – list of UnionOfIntervals instances

OUTPUT:

A new UnionOfIntervals instance representing the union of the UnionOfIntervals in the list.

Note

This is a class method.

EXAMPLES:

Sage

sage: from sage.schemes.elliptic_curves.height import UnionOfIntervals
sage: A = UnionOfIntervals([1,3,5,7]); A
([1, 3] U [5, 7])
sage: B = A + 1; B
([2, 4] U [6, 8])
sage: A.union([A,B])
([1, 4] U [5, 8])

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.height import UnionOfIntervals
>>> A = UnionOfIntervals([Integer(1),Integer(3),Integer(5),Integer(7)]); A
([1, 3] U [5, 7])
>>> B = A + Integer(1); B
([2, 4] U [6, 8])
>>> A.union([A,B])
([1, 4] U [5, 8])

sage.schemes.elliptic_curves.height.eps(err, is_real)[source]¶

Return a Real or Complex interval centered on 0 with radius err.

INPUT:

err – a positive real number; the radius of the interval
is_real – boolean; if True, returns a real interval in RIF, else a complex interval in CIF

OUTPUT:

An element of RIF or CIF (as specified), centered on 0, with given radius.

EXAMPLES:

Sage

sage: from sage.schemes.elliptic_curves.height import eps
sage: eps(0.01, True)
0.0?
sage: eps(0.01, False)
0.0? + 0.0?*I

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.height import eps
>>> eps(RealNumber('0.01'), True)
0.0?
>>> eps(RealNumber('0.01'), False)
0.0? + 0.0?*I

sage.schemes.elliptic_curves.height.inf_max_abs(f, g, D)[source]¶

Return \(\inf_D(\max(|f|, |g|))\).

INPUT:

f, g – real univariate polynomials
D – UnionOfIntervals; a subset of \(\RR\)

OUTPUT:

A real number approximating the value of \(\inf_D(\max(|f|, |g|))\).

ALGORITHM:

The extreme values must occur at an endpoint of a subinterval of \(D\) or at a point where one of \(f\), \(f'\), \(g\), \(g'\), \(f\pm g\) is zero.

EXAMPLES:

Sage

sage: from sage.schemes.elliptic_curves.height import inf_max_abs, UnionOfIntervals
sage: x = polygen(RR)
sage: f = (x-10)^4 + 1
sage: g = 2*x^3 + 100
sage: inf_max_abs(f, g, UnionOfIntervals([1,2,3,4,5,6]))
425.638201706391
sage: r0 = (f - g).roots()[0][0]
sage: r0
5.46053402234697
sage: max(abs(f(r0)), abs(g(r0)))
425.638201706391

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.height import inf_max_abs, UnionOfIntervals
>>> x = polygen(RR)
>>> f = (x-Integer(10))**Integer(4) + Integer(1)
>>> g = Integer(2)*x**Integer(3) + Integer(100)
>>> inf_max_abs(f, g, UnionOfIntervals([Integer(1),Integer(2),Integer(3),Integer(4),Integer(5),Integer(6)]))
425.638201706391
>>> r0 = (f - g).roots()[Integer(0)][Integer(0)]
>>> r0
5.46053402234697
>>> max(abs(f(r0)), abs(g(r0)))
425.638201706391

sage.schemes.elliptic_curves.height.min_on_disk(f, tol, max_iter=10000)[source]¶

Return the minimum of a real-valued complex function on a square.

INPUT:

f – a function from CIF to RIF
tol – a positive real number
max_iter – integer (default: 10000); a positive integer bounding the number of iterations to be used

OUTPUT:

A 2-tuple \((s,t)\), where \(t=f(s)\) and \(s\) is a CIF element contained in the disk \(|z|\le1\), at which \(f\) takes its minimum value.

EXAMPLES:

Sage

sage: from sage.schemes.elliptic_curves.height import min_on_disk
sage: f = lambda x: (x^2 + 100).abs()
sage: s, t = min_on_disk(f, 0.0001)
sage: s, f(s), t
(0.01? + 1.00?*I, 99.01?, 99.0000000000000)

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.height import min_on_disk
>>> f = lambda x: (x**Integer(2) + Integer(100)).abs()
>>> s, t = min_on_disk(f, RealNumber('0.0001'))
>>> s, f(s), t
(0.01? + 1.00?*I, 99.01?, 99.0000000000000)

sage.schemes.elliptic_curves.height.nonneg_region(f)[source]¶

Return the UnionOfIntervals representing the region where f is nonnegative.

INPUT:

f – a univariate polynomial over \(\RR\)

OUTPUT:

A UnionOfIntervals representing the set \(\{x \in\RR mid f(x) \ge 0\}\).

EXAMPLES:

Sage

sage: from sage.schemes.elliptic_curves.height import nonneg_region
sage: x = polygen(RR)
sage: nonneg_region(x^2 - 1)
([-Infinity, -1.00000000000000] U [1.00000000000000, +Infinity])
sage: nonneg_region(1 - x^2)
([-1.00000000000000, 1.00000000000000])
sage: nonneg_region(1 - x^3)
([-Infinity, 1.00000000000000])
sage: nonneg_region(x^3 - 1)
([1.00000000000000, +Infinity])
sage: nonneg_region((x-1)*(x-2))
([-Infinity, 1.00000000000000] U [2.00000000000000, +Infinity])
sage: nonneg_region(-(x-1)*(x-2))
([1.00000000000000, 2.00000000000000])
sage: nonneg_region((x-1)*(x-2)*(x-3))
([1.00000000000000, 2.00000000000000] U [3.00000000000000, +Infinity])
sage: nonneg_region(-(x-1)*(x-2)*(x-3))
([-Infinity, 1.00000000000000] U [2.00000000000000, 3.00000000000000])
sage: nonneg_region(x^4 + 1)
([-Infinity, +Infinity])
sage: nonneg_region(-x^4 - 1)
()

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.height import nonneg_region
>>> x = polygen(RR)
>>> nonneg_region(x**Integer(2) - Integer(1))
([-Infinity, -1.00000000000000] U [1.00000000000000, +Infinity])
>>> nonneg_region(Integer(1) - x**Integer(2))
([-1.00000000000000, 1.00000000000000])
>>> nonneg_region(Integer(1) - x**Integer(3))
([-Infinity, 1.00000000000000])
>>> nonneg_region(x**Integer(3) - Integer(1))
([1.00000000000000, +Infinity])
>>> nonneg_region((x-Integer(1))*(x-Integer(2)))
([-Infinity, 1.00000000000000] U [2.00000000000000, +Infinity])
>>> nonneg_region(-(x-Integer(1))*(x-Integer(2)))
([1.00000000000000, 2.00000000000000])
>>> nonneg_region((x-Integer(1))*(x-Integer(2))*(x-Integer(3)))
([1.00000000000000, 2.00000000000000] U [3.00000000000000, +Infinity])
>>> nonneg_region(-(x-Integer(1))*(x-Integer(2))*(x-Integer(3)))
([-Infinity, 1.00000000000000] U [2.00000000000000, 3.00000000000000])
>>> nonneg_region(x**Integer(4) + Integer(1))
([-Infinity, +Infinity])
>>> nonneg_region(-x**Integer(4) - Integer(1))
()

sage.schemes.elliptic_curves.height.rat_term_CIF(z, try_strict=True)[source]¶

Compute the value of \(u/(1-u)^2\) in CIF, where \(u=\exp(2\pi i z)\).

INPUT:

z – complex; a CIF element
try_strict – boolean; flag

EXAMPLES:

Sage

sage: from sage.schemes.elliptic_curves.height import rat_term_CIF
sage: z = CIF(0.5,0.2)
sage: rat_term_CIF(z)
-0.172467461182437? + 0.?e-16*I
sage: rat_term_CIF(z, False)
-0.172467461182437? + 0.?e-16*I

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.height import rat_term_CIF
>>> z = CIF(RealNumber('0.5'),RealNumber('0.2'))
>>> rat_term_CIF(z)
-0.172467461182437? + 0.?e-16*I
>>> rat_term_CIF(z, False)
-0.172467461182437? + 0.?e-16*I