# Elliptic curves over a general ring#

Sage defines an elliptic curve over a ring $$R$$ as a Weierstrass Model with five coefficients $$[a_1,a_2,a_3,a_4,a_6]$$ in $$R$$ given by

$y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6.$

Note that the (usual) scheme-theoretic definition of an elliptic curve over $$R$$ would require the discriminant to be a unit in $$R$$; Sage only imposes that the discriminant is non-zero. Also note that in Magma, “Weierstrass Model” refers to a model with $$a_1=a_2=a_3=0$$, which is called Short Weierstrass Model in Sage; these do not always exist in characteristics 2 and 3.

EXAMPLES:

We construct an elliptic curve over an elaborate base ring:

sage: p, a, b = 97, 1, 3
sage: R.<u> = GF(p)[]                                                               # optional - sage.rings.finite_rings
sage: S.<v> = R[]                                                                   # optional - sage.rings.finite_rings
sage: T = S.fraction_field()                                                        # optional - sage.rings.finite_rings
sage: E = EllipticCurve(T, [a, b]); E                                               # optional - sage.rings.finite_rings
Elliptic Curve defined by y^2  = x^3 + x + 3 over Fraction Field of Univariate
Polynomial Ring in v over Univariate Polynomial Ring in u over Finite Field of size 97
sage: latex(E)                                                                      # optional - sage.rings.finite_rings
y^2  = x^{3} + x + 3


AUTHORS:

• William Stein (2005): Initial version

• John Cremona (2008-01): isomorphisms, automorphisms and twists in all characteristics

• Julian Rueth (2014-04-11): improved caching

• Lorenz Panny (2022-04-14): added .montgomery_model()

class sage.schemes.elliptic_curves.ell_generic.EllipticCurve_generic(K, ainvs)#

Elliptic curve over a generic base ring.

EXAMPLES:

sage: E = EllipticCurve([1,2,3/4,7,19]); E
Elliptic Curve defined by y^2 + x*y + 3/4*y = x^3 + 2*x^2 + 7*x + 19 over Rational Field
True
sage: E = EllipticCurve([1,3])
sage: P = E([-1,1,1])
sage: -5*P
(179051/80089 : -91814227/22665187 : 1)

a1()#

Return the $$a_1$$ invariant of this elliptic curve.

EXAMPLES:

sage: E = EllipticCurve([1,2,3,4,6])
sage: E.a1()
1

a2()#

Return the $$a_2$$ invariant of this elliptic curve.

EXAMPLES:

sage: E = EllipticCurve([1,2,3,4,6])
sage: E.a2()
2

a3()#

Return the $$a_3$$ invariant of this elliptic curve.

EXAMPLES:

sage: E = EllipticCurve([1,2,3,4,6])
sage: E.a3()
3

a4()#

Return the $$a_4$$ invariant of this elliptic curve.

EXAMPLES:

sage: E = EllipticCurve([1,2,3,4,6])
sage: E.a4()
4

a6()#

Return the $$a_6$$ invariant of this elliptic curve.

EXAMPLES:

sage: E = EllipticCurve([1,2,3,4,6])
sage: E.a6()
6

a_invariants()#

The $$a$$-invariants of this elliptic curve, as a tuple.

OUTPUT:

(tuple) - a 5-tuple of the $$a$$-invariants of this elliptic curve.

EXAMPLES:

sage: E = EllipticCurve([1,2,3,4,5])
sage: E.a_invariants()
(1, 2, 3, 4, 5)

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

sage: E = EllipticCurve([GF(7)(3),5])                                       # optional - sage.rings.finite_rings
sage: E.a_invariants()                                                      # optional - sage.rings.finite_rings
(0, 0, 0, 3, 5)

ainvs()#

The $$a$$-invariants of this elliptic curve, as a tuple.

OUTPUT:

(tuple) - a 5-tuple of the $$a$$-invariants of this elliptic curve.

EXAMPLES:

sage: E = EllipticCurve([1,2,3,4,5])
sage: E.a_invariants()
(1, 2, 3, 4, 5)

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

sage: E = EllipticCurve([GF(7)(3),5])                                       # optional - sage.rings.finite_rings
sage: E.a_invariants()                                                      # optional - sage.rings.finite_rings
(0, 0, 0, 3, 5)

automorphisms(field=None)#

Return the set of isomorphisms from self to itself (as a list).

The identity and negation morphisms are guaranteed to appear as the first and second entry of the returned list.

INPUT:

• field (default None) – a field into which the coefficients of the curve may be coerced (by default, uses the base field of the curve).

OUTPUT:

(list) A list of WeierstrassIsomorphism objects consisting of all the isomorphisms from the curve self to itself defined over field.

EXAMPLES:

sage: E = EllipticCurve_from_j(QQ(0))  # a curve with j=0 over QQ
sage: E.automorphisms()
[Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3
over Rational Field
Via:  (u,r,s,t) = (1, 0, 0, 0),
Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3
over Rational Field
Via:  (u,r,s,t) = (-1, 0, 0, -1)]


We can also find automorphisms defined over extension fields:

sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^2 + 3)  # adjoin roots of unity                 # optional - sage.rings.number_field
sage: E.automorphisms(K)                                                    # optional - sage.rings.number_field
[Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3
over Number Field in a with defining polynomial x^2 + 3
Via:  (u,r,s,t) = (1, 0, 0, 0),
Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3
over Number Field in a with defining polynomial x^2 + 3
Via:  (u,r,s,t) = (-1, 0, 0, -1),
Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3
over Number Field in a with defining polynomial x^2 + 3
Via:  (u,r,s,t) = (-1/2*a - 1/2, 0, 0, 0),
Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3
over Number Field in a with defining polynomial x^2 + 3
Via:  (u,r,s,t) = (1/2*a + 1/2, 0, 0, -1),
Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3
over Number Field in a with defining polynomial x^2 + 3
Via:  (u,r,s,t) = (1/2*a - 1/2, 0, 0, 0),
Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3
over Number Field in a with defining polynomial x^2 + 3
Via:  (u,r,s,t) = (-1/2*a + 1/2, 0, 0, -1)]

sage: [len(EllipticCurve_from_j(GF(q,'a')(0)).automorphisms())              # optional - sage.rings.finite_rings
....:  for q in [2,4,3,9,5,25,7,49]]
[2, 24, 2, 12, 2, 6, 6, 6]

b2()#

Return the $$b_2$$ invariant of this elliptic curve.

EXAMPLES:

sage: E = EllipticCurve([1,2,3,4,5])
sage: E.b2()
9

b4()#

Return the $$b_4$$ invariant of this elliptic curve.

EXAMPLES:

sage: E = EllipticCurve([1,2,3,4,5])
sage: E.b4()
11

b6()#

Return the $$b_6$$ invariant of this elliptic curve.

EXAMPLES:

sage: E = EllipticCurve([1,2,3,4,5])
sage: E.b6()
29

b8()#

Return the $$b_8$$ invariant of this elliptic curve.

EXAMPLES:

sage: E = EllipticCurve([1,2,3,4,5])
sage: E.b8()
35

b_invariants()#

Return the $$b$$-invariants of this elliptic curve, as a tuple.

OUTPUT:

(tuple) - a 4-tuple of the $$b$$-invariants of this elliptic curve.

This method is cached.

EXAMPLES:

sage: E = EllipticCurve([0, -1, 1, -10, -20])
sage: E.b_invariants()
(-4, -20, -79, -21)

sage: E = EllipticCurve([-4,0])
sage: E.b_invariants()
(0, -8, 0, -16)

sage: E = EllipticCurve([1,2,3,4,5])
sage: E.b_invariants()
(9, 11, 29, 35)
sage: E.b2()
9
sage: E.b4()
11
sage: E.b6()
29
sage: E.b8()
35


ALGORITHM:

These are simple functions of the $$a$$-invariants.

AUTHORS:

• William Stein (2005-04-25)

base_extend(R)#

Return the base extension of self to $$R$$.

INPUT:

• R – either a ring into which the $$a$$-invariants of self may be converted, or a morphism which may be applied to them.

OUTPUT:

An elliptic curve over the new ring whose $$a$$-invariants are the images of the $$a$$-invariants of self.

EXAMPLES:

sage: E = EllipticCurve(GF(5), [1,1]); E                                    # optional - sage.rings.finite_rings
Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field of size 5
sage: E1 = E.base_extend(GF(125,'a')); E1                                   # optional - sage.rings.finite_rings
Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field in a of size 5^3

base_ring()#

Return the base ring of the elliptic curve.

EXAMPLES:

sage: E = EllipticCurve(GF(49, 'a'), [3,5])                                 # optional - sage.rings.finite_rings
sage: E.base_ring()                                                         # optional - sage.rings.finite_rings
Finite Field in a of size 7^2

sage: E = EllipticCurve([1,1])
sage: E.base_ring()
Rational Field

sage: E = EllipticCurve(ZZ, [3,5])
sage: E.base_ring()
Integer Ring

c4()#

Return the $$c_4$$ invariant of this elliptic curve.

EXAMPLES:

sage: E = EllipticCurve([0, -1, 1, -10, -20])
sage: E.c4()
496

c6()#

Return the $$c_6$$ invariant of this elliptic curve.

EXAMPLES:

sage: E = EllipticCurve([0, -1, 1, -10, -20])
sage: E.c6()
20008

c_invariants()#

Return the $$c$$-invariants of this elliptic curve, as a tuple.

This method is cached.

OUTPUT:

(tuple) - a 2-tuple of the $$c$$-invariants of the elliptic curve.

EXAMPLES:

sage: E = EllipticCurve([0, -1, 1, -10, -20])
sage: E.c_invariants()
(496, 20008)

sage: E = EllipticCurve([-4,0])
sage: E.c_invariants()
(192, 0)


ALGORITHM:

These are simple functions of the $$a$$-invariants.

AUTHORS:

• William Stein (2005-04-25)

change_ring(R)#

Return the base change of self to $$R$$.

This has the same effect as self.base_extend(R).

EXAMPLES:

sage: F2 = GF(5^2,'a'); a = F2.gen()                                        # optional - sage.rings.finite_rings
sage: F4 = GF(5^4,'b'); b = F4.gen()                                        # optional - sage.rings.finite_rings
sage: roots = a.charpoly().roots(ring=F4, multiplicities=False)             # optional - sage.rings.finite_rings
sage: h = F2.hom([roots], F4)                                            # optional - sage.rings.finite_rings
sage: E = EllipticCurve(F2, [1,a]); E                                       # optional - sage.rings.finite_rings
Elliptic Curve defined by y^2 = x^3 + x + a
over Finite Field in a of size 5^2
sage: E.change_ring(h)                                                      # optional - sage.rings.finite_rings
Elliptic Curve defined by y^2 = x^3 + x + (4*b^3+4*b^2+4*b+3)
over Finite Field in b of size 5^4

change_weierstrass_model(*urst)#

Return a new Weierstrass model of self under the standard transformation $$(u,r,s,t)$$

$(x,y) \mapsto (x',y') = (u^2x + r , u^3y + su^2x + t).$

EXAMPLES:

sage: E = EllipticCurve('15a')
sage: F1 = E.change_weierstrass_model([1/2,0,0,0]); F1
Elliptic Curve defined by y^2 + 2*x*y + 8*y = x^3 + 4*x^2 - 160*x - 640
over Rational Field
sage: F2 = E.change_weierstrass_model([7,2,1/3,5]); F2
Elliptic Curve defined by
y^2 + 5/21*x*y + 13/343*y = x^3 + 59/441*x^2 - 10/7203*x - 58/117649
over Rational Field
sage: F1.is_isomorphic(F2)
True

discriminant()#

Return the discriminant of this elliptic curve.

This method is cached.

EXAMPLES:

sage: E = EllipticCurve([0,0,1,-1,0])
sage: E.discriminant()
37

sage: E = EllipticCurve([0, -1, 1, -10, -20])
sage: E.discriminant()
-161051

sage: E = EllipticCurve([GF(7)(2),1])                                       # optional - sage.rings.finite_rings
sage: E.discriminant()                                                      # optional - sage.rings.finite_rings
1

division_polynomial(m, x=None, two_torsion_multiplicity=2, force_evaluate=None)#

Return the $$m^{th}$$ division polynomial of this elliptic curve evaluated at $$x$$.

The division polynomial is cached if $$x$$ is None.

INPUT:

• m – positive integer.

• x – optional ring element to use as the $$x$$ variable. If $$x$$ is None (omitted), then a new polynomial ring will be constructed over the base ring of the elliptic curve, and its generator will be used as $$x$$. Note that $$x$$ does not need to be a generator of a polynomial ring; any ring element works. This permits fast calculation of the torsion polynomial evaluated on any element of a ring.

• two_torsion_multiplicity – 0, 1, or 2

If 0: For even $$m$$ when $$x$$ is None, a univariate polynomial over the base ring of the curve is returned, which omits factors whose roots are the $$x$$-coordinates of the $$2$$-torsion points. When $$x$$ is not None, the evaluation of such a polynomial at $$x$$ is returned.

If 2: For even $$m$$ when $$x$$ is None, a univariate polynomial over the base ring of the curve is returned, which includes a factor of degree 3 whose roots are the $$x$$-coordinates of the $$2$$-torsion points. Similarly, when $$x$$ is not None, the evaluation of such a polynomial at $$x$$ is returned.

If 1: For even $$m$$ when $$x$$ is None, a bivariate polynomial over the base ring of the curve is returned, which includes a factor $$2y+a_1x+a_3$$ having simple zeros at the $$2$$-torsion points. When $$x$$ is not None, it should be a tuple of length 2, and the evaluation of such a polynomial at $$x$$ is returned.

• force_evaluate (optional) – 0, 1, or 2

By default, this method makes use of previously cached generic division polynomials to compute the value of the polynomial at a given element $$x$$ whenever it appears beneficial to do so. Explicitly setting this flag overrides the default behavior.

Note that the complexity of evaluating a generic division polynomial scales much worse than that of computing the value at a point directly (using the recursive formulas), hence setting this flag can be detrimental to performance.

If 0: Do not use cached generic division polynomials.

If 1: If the generic division polynomial for this $$m$$ has been cached before, evaluate it at $$x$$ to compute the result.

If 2: Compute the value at $$x$$ by evaluating the generic division polynomial. If the generic $$m$$-division polynomial has not yet been cached, compute and cache it first.

EXAMPLES:

sage: E = EllipticCurve([0,0,1,-1,0])
sage: E.division_polynomial(1)
1
sage: E.division_polynomial(2, two_torsion_multiplicity=0)
1
sage: E.division_polynomial(2, two_torsion_multiplicity=1)
2*y + 1
sage: E.division_polynomial(2, two_torsion_multiplicity=2)
4*x^3 - 4*x + 1
sage: E.division_polynomial(2)
4*x^3 - 4*x + 1
sage: [E.division_polynomial(3, two_torsion_multiplicity=i) for i in range(3)]
[3*x^4 - 6*x^2 + 3*x - 1, 3*x^4 - 6*x^2 + 3*x - 1, 3*x^4 - 6*x^2 + 3*x - 1]
sage: [type(E.division_polynomial(3, two_torsion_multiplicity=i)) for i in range(3)]
[<... 'sage.rings.polynomial.polynomial_rational_flint.Polynomial_rational_flint'>,
<... 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular'>,
<... 'sage.rings.polynomial.polynomial_rational_flint.Polynomial_rational_flint'>]

sage: E = EllipticCurve([0, -1, 1, -10, -20])
sage: R.<z> = PolynomialRing(QQ)
sage: E.division_polynomial(4, z, 0)
2*z^6 - 4*z^5 - 100*z^4 - 790*z^3 - 210*z^2 - 1496*z - 5821
sage: E.division_polynomial(4, z)
8*z^9 - 24*z^8 - 464*z^7 - 2758*z^6 + 6636*z^5 + 34356*z^4
+ 53510*z^3 + 99714*z^2 + 351024*z + 459859


This does not work, since when two_torsion_multiplicity is 1, we compute a bivariate polynomial, and must evaluate at a tuple of length 2:

sage: E.division_polynomial(4,z,1)
Traceback (most recent call last):
...
ValueError: x should be a tuple of length 2 (or None)
when two_torsion_multiplicity is 1
sage: R.<z,w> = PolynomialRing(QQ, 2)
sage: E.division_polynomial(4, (z,w), 1).factor()
(2*w + 1) * (2*z^6 - 4*z^5 - 100*z^4 - 790*z^3 - 210*z^2 - 1496*z - 5821)


We can also evaluate this bivariate polynomial at a point:

sage: P = E(5,5)
sage: E.division_polynomial(4,P,two_torsion_multiplicity=1)
-1771561

division_polynomial_0(n, x=None)#

Return the $$n^{th}$$ torsion (division) polynomial, without the 2-torsion factor if $$n$$ is even, as a polynomial in $$x$$.

These are the polynomials $$g_n$$ defined in [MT1991], but with the sign flipped for even $$n$$, so that the leading coefficient is always positive.

Note

This function is intended for internal use; users should use division_polynomial().

INPUT:

• n – positive integer, or the special values -1 and -2 which mean $$B_6 = (2y + a_1 x + a_3)^2$$ and $$B_6^2$$ respectively (in the notation of [MT1991]); or a list of integers.

• x – a ring element to use as the “x” variable or None (default: None). If None, then a new polynomial ring will be constructed over the base ring of the elliptic curve, and its generator will be used as x. Note that x does not need to be a generator of a polynomial ring; any ring element is ok. This permits fast calculation of the torsion polynomial evaluated on any element of a ring.

ALGORITHM:

Recursion described in [MT1991]. The recursive formulae are evaluated $$O(\log^2 n)$$ times.

AUTHORS:

• David Harvey (2006-09-24): initial version

• John Cremona (2008-08-26): unified division polynomial code

EXAMPLES:

sage: E = EllipticCurve("37a")
sage: E.division_polynomial_0(1)
1
sage: E.division_polynomial_0(2)
1
sage: E.division_polynomial_0(3)
3*x^4 - 6*x^2 + 3*x - 1
sage: E.division_polynomial_0(4)
2*x^6 - 10*x^4 + 10*x^3 - 10*x^2 + 2*x + 1
sage: E.division_polynomial_0(5)
5*x^12 - 62*x^10 + 95*x^9 - 105*x^8 - 60*x^7 + 285*x^6 - 174*x^5 - 5*x^4 - 5*x^3 + 35*x^2 - 15*x + 2
sage: E.division_polynomial_0(6)
3*x^16 - 72*x^14 + 168*x^13 - 364*x^12 + 1120*x^10 - 1144*x^9 + 300*x^8 - 540*x^7 + 1120*x^6 - 588*x^5 - 133*x^4 + 252*x^3 - 114*x^2 + 22*x - 1
sage: E.division_polynomial_0(7)
7*x^24 - 308*x^22 + 986*x^21 - 2954*x^20 + 28*x^19 + 17171*x^18 - 23142*x^17 + 511*x^16 - 5012*x^15 + 43804*x^14 - 7140*x^13 - 96950*x^12 + 111356*x^11 - 19516*x^10 - 49707*x^9 + 40054*x^8 - 124*x^7 - 18382*x^6 + 13342*x^5 - 4816*x^4 + 1099*x^3 - 210*x^2 + 35*x - 3
sage: E.division_polynomial_0(8)
4*x^30 - 292*x^28 + 1252*x^27 - 5436*x^26 + 2340*x^25 + 39834*x^24 - 79560*x^23 + 51432*x^22 - 142896*x^21 + 451596*x^20 - 212040*x^19 - 1005316*x^18 + 1726416*x^17 - 671160*x^16 - 954924*x^15 + 1119552*x^14 + 313308*x^13 - 1502818*x^12 + 1189908*x^11 - 160152*x^10 - 399176*x^9 + 386142*x^8 - 220128*x^7 + 99558*x^6 - 33528*x^5 + 6042*x^4 + 310*x^3 - 406*x^2 + 78*x - 5

sage: E.division_polynomial_0(18) % E.division_polynomial_0(6) == 0
True


An example to illustrate the relationship with torsion points:

sage: F = GF(11)                                                            # optional - sage.rings.finite_rings
sage: E = EllipticCurve(F, [0, 2]); E                                       # optional - sage.rings.finite_rings
Elliptic Curve defined by y^2  = x^3 + 2 over Finite Field of size 11
sage: f = E.division_polynomial_0(5); f                                     # optional - sage.rings.finite_rings
5*x^12 + x^9 + 8*x^6 + 4*x^3 + 7
sage: f.factor()                                                            # optional - sage.rings.finite_rings
(5) * (x^2 + 5) * (x^2 + 2*x + 5) * (x^2 + 5*x + 7)
* (x^2 + 7*x + 7) * (x^2 + 9*x + 5) * (x^2 + 10*x + 7)


This indicates that the $$x$$-coordinates of all the 5-torsion points of $$E$$ are in $$\GF{11^2}$$, and therefore the $$y$$-coordinates are in $$\GF{11^4}$$:

sage: K = GF(11^4, 'a')                                                     # optional - sage.rings.finite_rings
sage: X = E.change_ring(K)                                                  # optional - sage.rings.finite_rings
sage: f = X.division_polynomial_0(5)                                        # optional - sage.rings.finite_rings
sage: x_coords = f.roots(multiplicities=False); x_coords                    # optional - sage.rings.finite_rings
[10*a^3 + 4*a^2 + 5*a + 6,
9*a^3 + 8*a^2 + 10*a + 8,
8*a^3 + a^2 + 4*a + 10,
8*a^3 + a^2 + 4*a + 8,
8*a^3 + a^2 + 4*a + 4,
6*a^3 + 9*a^2 + 3*a + 4,
5*a^3 + 2*a^2 + 8*a + 7,
3*a^3 + 10*a^2 + 7*a + 8,
3*a^3 + 10*a^2 + 7*a + 3,
3*a^3 + 10*a^2 + 7*a + 1,
2*a^3 + 3*a^2 + a + 7,
a^3 + 7*a^2 + 6*a]


Now we check that these are exactly the $$x$$-coordinates of the 5-torsion points of $$E$$:

sage: for x in x_coords:                                                    # optional - sage.rings.finite_rings
....:     assert X.lift_x(x).order() == 5


The roots of the polynomial are the $$x$$-coordinates of the points $$P$$ such that $$mP=0$$ but $$2P\not=0$$:

sage: E = EllipticCurve('14a1')
sage: T = E.torsion_subgroup()
sage: [n*T.0 for n in range(6)]
[(0 : 1 : 0),
(9 : 23 : 1),
(2 : 2 : 1),
(1 : -1 : 1),
(2 : -5 : 1),
(9 : -33 : 1)]
sage: pol = E.division_polynomial_0(6)
sage: xlist = pol.roots(multiplicities=False); xlist
[9, 2, -1/3, -5]
sage: [E.lift_x(x, all=True) for x in xlist]
[[(9 : -33 : 1), (9 : 23 : 1)], [(2 : -5 : 1), (2 : 2 : 1)], [], []]


Note

The point of order 2 and the identity do not appear. The points with $$x=-1/3$$ and $$x=-5$$ are not rational.

formal()#

Return the formal group associated to this elliptic curve.

This method is cached.

EXAMPLES:

sage: E = EllipticCurve("37a")
sage: E.formal_group()
Formal Group associated to the Elliptic Curve
defined by y^2 + y = x^3 - x over Rational Field

formal_group()#

Return the formal group associated to this elliptic curve.

This method is cached.

EXAMPLES:

sage: E = EllipticCurve("37a")
sage: E.formal_group()
Formal Group associated to the Elliptic Curve
defined by y^2 + y = x^3 - x over Rational Field

frobenius_isogeny(n=1)#

Return the $$n$$-power Frobenius isogeny from this curve to its Galois conjugate.

The Frobenius endomorphism is the special case where $$n$$ is divisible by the degree of the base ring of the curve.

EXAMPLES:

sage: z3, = GF(13^3).gens()                                                 # optional - sage.rings.finite_rings
sage: E = EllipticCurve([z3, z3^2])                                         # optional - sage.rings.finite_rings
sage: E.frobenius_isogeny()                                                 # optional - sage.rings.finite_rings
Frobenius isogeny of degree 13:
From: Elliptic Curve defined by y^2 = x^3 + z3*x + z3^2
over Finite Field in z3 of size 13^3
To:   Elliptic Curve defined by y^2 = x^3 + (5*z3^2+7*z3+11)*x + (5*z3^2+12*z3+1)
over Finite Field in z3 of size 13^3
sage: E.frobenius_isogeny(3)                                                # optional - sage.rings.finite_rings
Frobenius endomorphism of degree 2197 = 13^3:
From: Elliptic Curve defined by y^2 = x^3 + z3*x + z3^2
over Finite Field in z3 of size 13^3
To:   Elliptic Curve defined by y^2 = x^3 + z3*x + z3^2
over Finite Field in z3 of size 13^3

gen(i)#

Function returning the i’th generator of this elliptic curve.

Note

Relies on gens() being implemented.

EXAMPLES:

sage: R.<a1,a2,a3,a4,a6> = QQ[]
sage: E = EllipticCurve([a1,a2,a3,a4,a6])
sage: E.gen(0)
Traceback (most recent call last):
...
NotImplementedError: not implemented.

gens()#

Placeholder function to return generators of an elliptic curve.

Note

This functionality is implemented in certain derived classes, such as EllipticCurve_rational_field.

EXAMPLES:

sage: R.<a1,a2,a3,a4,a6> = QQ[]
sage: E = EllipticCurve([a1,a2,a3,a4,a6])
sage: E.gens()
Traceback (most recent call last):
...
NotImplementedError: not implemented.
sage: E = EllipticCurve(QQ, [1,1])
sage: E.gens()
[(0 : 1 : 1)]

hyperelliptic_polynomials()#

Return a pair of polynomials $$g(x)$$, $$h(x)$$ such that this elliptic curve can be defined by the standard hyperelliptic equation

$y^2 + h(x)y = g(x).$

EXAMPLES:

sage: R.<a1,a2,a3,a4,a6>=QQ[]
sage: E = EllipticCurve([a1,a2,a3,a4,a6])
sage: E.hyperelliptic_polynomials()
(x^3 + a2*x^2 + a4*x + a6, a1*x + a3)

is_isomorphic(other, field=None)#

Return whether or not self is isomorphic to other.

INPUT:

• other – another elliptic curve.

• field (default None) – a field into which the coefficients of the curves may be coerced (by default, uses the base field of the curves).

OUTPUT:

(bool) True if there is an isomorphism from curve self to curve other defined over field.

EXAMPLES:

sage: E = EllipticCurve('389a')
sage: F = E.change_weierstrass_model([2,3,4,5]); F
Elliptic Curve defined by y^2 + 4*x*y + 11/8*y = x^3 - 3/2*x^2 - 13/16*x
over Rational Field
sage: E.is_isomorphic(F)
True
sage: E.is_isomorphic(F.change_ring(CC))
False

is_on_curve(x, y)#

Return True if $$(x,y)$$ is an affine point on this curve.

INPUT:

• x, y – elements of the base ring of the curve.

EXAMPLES:

sage: E = EllipticCurve(QQ,[1,1])
sage: E.is_on_curve(0,1)
True
sage: E.is_on_curve(1,1)
False

is_x_coord(x)#

Return True if x is the $$x$$-coordinate of a point on this curve.

Note

See also lift_x() to find the point(s) with a given $$x$$-coordinate. This function may be useful in cases where testing an element of the base field for being a square is faster than finding its square root.

EXAMPLES:

sage: E = EllipticCurve('37a'); E
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
sage: E.is_x_coord(1)
True
sage: E.is_x_coord(2)
True


There are no rational points with x-coordinate 3:

sage: E.is_x_coord(3)
False


However, there are such points in $$E(\RR)$$:

sage: E.change_ring(RR).is_x_coord(3)
True


And of course it always works in $$E(\CC)$$:

sage: E.change_ring(RR).is_x_coord(-3)
False
sage: E.change_ring(CC).is_x_coord(-3)
True


AUTHORS:

isomorphism_to(other)#

Given another weierstrass model other of self, return an isomorphism from self to other.

INPUT:

• other – an elliptic curve isomorphic to self.

OUTPUT:

(Weierstrassmorphism) An isomorphism from self to other.

Note

If the curves in question are not isomorphic, a ValueError is raised.

EXAMPLES:

sage: E = EllipticCurve('37a')
sage: F = E.short_weierstrass_model()
sage: w = E.isomorphism_to(F); w
Elliptic-curve morphism:
From: Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
To:   Elliptic Curve defined by y^2  = x^3 - 16*x + 16 over Rational Field
Via:  (u,r,s,t) = (1/2, 0, 0, -1/2)
sage: P = E(0,-1,1)
sage: w(P)
(0 : -4 : 1)
sage: w(5*P)
(1 : 1 : 1)
sage: 5*w(P)
(1 : 1 : 1)
sage: 120*w(P) == w(120*P)
True


We can also handle injections to different base rings:

sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^3 - 7)                                          # optional - sage.rings.number_field
sage: E.isomorphism_to(E.change_ring(K))                                    # optional - sage.rings.number_field
Elliptic-curve morphism:
From: Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
To:   Elliptic Curve defined by y^2 + y = x^3 + (-1)*x
over Number Field in a with defining polynomial x^3 - 7
Via:  (u,r,s,t) = (1, 0, 0, 0)

isomorphisms(other, field=None)#

Return the set of isomorphisms from self to other (as a list).

INPUT:

• other – another elliptic curve.

• field (default None) – a field into which the coefficients of the curves may be coerced (by default, uses the base field of the curves).

OUTPUT:

(list) A list of WeierstrassIsomorphism objects consisting of all the isomorphisms from the curve self to the curve other defined over field.

EXAMPLES:

sage: E = EllipticCurve_from_j(QQ(0)) # a curve with j=0 over QQ
sage: F = EllipticCurve('27a3') # should be the same one
sage: E.isomorphisms(F)
[Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3
over Rational Field
Via:  (u,r,s,t) = (1, 0, 0, 0),
Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3
over Rational Field
Via:  (u,r,s,t) = (-1, 0, 0, -1)]


We can also find isomorphisms defined over extension fields:

sage: E = EllipticCurve(GF(7), [0,0,0,1,1])                                 # optional - sage.rings.finite_rings
sage: F = EllipticCurve(GF(7), [0,0,0,1,-1])                                # optional - sage.rings.finite_rings
sage: E.isomorphisms(F)                                                     # optional - sage.rings.finite_rings
[]
sage: E.isomorphisms(F, GF(49,'a'))                                         # optional - sage.rings.finite_rings
[Elliptic-curve morphism:
From: Elliptic Curve defined by y^2 = x^3 + x + 1
over Finite Field in a of size 7^2
To:   Elliptic Curve defined by y^2 = x^3 + x + 6
over Finite Field in a of size 7^2
Via:  (u,r,s,t) = (a + 3, 0, 0, 0),
Elliptic-curve morphism:
From: Elliptic Curve defined by y^2 = x^3 + x + 1
over Finite Field in a of size 7^2
To:   Elliptic Curve defined by y^2 = x^3 + x + 6
over Finite Field in a of size 7^2
Via:  (u,r,s,t) = (6*a + 4, 0, 0, 0)]

j_invariant()#

Return the $$j$$-invariant of this elliptic curve.

This method is cached.

EXAMPLES:

sage: E = EllipticCurve([0,0,1,-1,0])
sage: E.j_invariant()
110592/37

sage: E = EllipticCurve([0, -1, 1, -10, -20])
sage: E.j_invariant()
-122023936/161051

sage: E = EllipticCurve([-4,0])
sage: E.j_invariant()
1728

sage: E = EllipticCurve([GF(7)(2),1])                                       # optional - sage.rings.finite_rings
sage: E.j_invariant()                                                       # optional - sage.rings.finite_rings
1

lift_x(x, all=False, extend=False)#

Return one or all points with given $$x$$-coordinate.

This method is deterministic: It returns the same data each time when called again with the same $$x$$.

INPUT:

• x – an element of the base ring of the curve, or of an extension.

• all (bool, default False) – if True, return a (possibly empty) list of all points; if False, return just one point, or raise a ValueError if there are none.

• extend (bool, default False) –

• if False, extend the base if necessary and possible to include $$x$$, and only return point(s) defined over this ring, or raise an error when there are none with this $$x$$-coordinate;

• If True, the base ring will be extended if necessary to contain the $$y$$-coordinates of the point(s) with this $$x$$-coordinate, in addition to a possible base change to include $$x$$.

OUTPUT:

A point or list of up to 2 points on this curve, or a base-change of this curve to a larger ring.

EXAMPLES:

sage: E = EllipticCurve('37a'); E
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
sage: E.lift_x(1)
(1 : -1 : 1)
sage: E.lift_x(2)
(2 : -3 : 1)
sage: E.lift_x(1/4, all=True)
[(1/4 : -5/8 : 1), (1/4 : -3/8 : 1)]


There are no rational points with $$x$$-coordinate 3:

sage: E.lift_x(3)
Traceback (most recent call last):
...
ValueError: No point with x-coordinate 3
on Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field


We can use the extend parameter to make the necessary quadratic extension. Note that in such cases the returned point is a point on a new curve object, the result of changing the base ring to the parent of $$x$$:

sage: P = E.lift_x(3, extend=True); P                                       # optional - sage.rings.number_field
(3 : -y - 1 : 1)
sage: P.curve()                                                             # optional - sage.rings.number_field
Elliptic Curve defined by y^2 + y = x^3 + (-1)*x
over Number Field in y with defining polynomial y^2 + y - 24


Or we can extend scalars. There are two such points in $$E(\RR)$$:

sage: E.change_ring(RR).lift_x(3, all=True)
[(3.00000000000000 : -5.42442890089805 : 1.00000000000000),
(3.00000000000000 : 4.42442890089805 : 1.00000000000000)]


And of course it always works in $$E(\CC)$$:

sage: E.change_ring(RR).lift_x(.5, all=True)
[]
sage: E.change_ring(CC).lift_x(.5)
(0.500000000000000 : -0.500000000000000 - 0.353553390593274*I : 1.00000000000000)


In this example we start with a curve defined over $$\QQ$$ which has no rational points with $$x=0$$, but using extend = True we can construct such a point over a quadratic field:

sage: E = EllipticCurve([0,0,0,0,2]); E
Elliptic Curve defined by y^2 = x^3 + 2 over Rational Field
sage: P = E.lift_x(0, extend=True); P                                       # optional - sage.rings.number_field
(0 : -y : 1)
sage: P.curve()                                                             # optional - sage.rings.number_field
Elliptic Curve defined by y^2 = x^3 + 2
over Number Field in y with defining polynomial y^2 - 2


We can perform these operations over finite fields too:

sage: E = EllipticCurve('37a').change_ring(GF(17)); E                       # optional - sage.rings.finite_rings
Elliptic Curve defined by y^2 + y = x^3 + 16*x over Finite Field of size 17
sage: E.lift_x(7)                                                           # optional - sage.rings.finite_rings
(7 : 5 : 1)
sage: E.lift_x(3)                                                           # optional - sage.rings.finite_rings
Traceback (most recent call last):
...
ValueError: No point with x-coordinate 3 on
Elliptic Curve defined by y^2 + y = x^3 + 16*x over Finite Field of size 17


Note that there is only one lift with $$x$$-coordinate 10 in $$E(\GF{17})$$:

sage: E.lift_x(10, all=True)                                                # optional - sage.rings.finite_rings
[(10 : 8 : 1)]


We can lift over more exotic rings too. If the supplied x value is in an extension of the base, note that the point returned is on the base-extended curve:

sage: E = EllipticCurve('37a')
(6 + O(17^5) : 14 + O(17^5) : 1 + O(17^5))
sage: P.curve()                                                             # optional - sage.rings.padics
Elliptic Curve defined by
y^2 + (1+O(17^5))*y = x^3 + (16+16*17+16*17^2+16*17^3+16*17^4+O(17^5))*x
over 17-adic Field with capped relative precision 5
sage: K.<t> = PowerSeriesRing(QQ, 't', 5)
sage: P = E.lift_x(1 + t); P
(1 + t : -1 - 2*t + t^2 - 5*t^3 + 21*t^4 + O(t^5) : 1)
sage: K.<a> = GF(16)                                                        # optional - sage.rings.finite_rings
sage: P = E.change_ring(K).lift_x(a^3); P                                   # optional - sage.rings.finite_rings
(a^3 : a^3 + a : 1)
sage: P.curve()                                                             # optional - sage.rings.finite_rings
Elliptic Curve defined by y^2 + y = x^3 + x over Finite Field in a of size 2^4


We can extend the base field to include the associated $$y$$ value(s):

sage: E = EllipticCurve([0,0,0,0,2]); E
Elliptic Curve defined by y^2 = x^3 + 2 over Rational Field
sage: x = polygen(QQ)
sage: P = E.lift_x(x, extend=True); P
(x : -y : 1)


This point is a generic point on E:

sage: P.curve()
Elliptic Curve defined by y^2 = x^3 + 2
over Univariate Quotient Polynomial Ring in y
over Fraction Field of Univariate Polynomial Ring in x over Rational Field
with modulus y^2 - x^3 - 2
sage: -P
(x : y : 1)
sage: 2*P
((1/4*x^4 - 4*x)/(x^3 + 2) : ((-1/8*x^6 - 5*x^3 + 4)/(x^6 + 4*x^3 + 4))*y : 1)


Check that github issue #30297 is fixed:

sage: K = Qp(5)                                                             # optional - sage.rings.padics
sage: E = EllipticCurve([K(0), K(1)])                                       # optional - sage.rings.padics
sage: E.lift_x(1, extend=True)                                              # optional - sage.rings.padics
(1 + O(5^20) : y + O(5^20) : 1 + O(5^20))


AUTHORS:

• John Cremona (2017-11-10)

montgomery_model(twisted=False, morphism=False)#

Return a (twisted or untwisted) Montgomery model for this elliptic curve, if possible.

A Montgomery curve is a smooth projective curve of the form

$BY^2 = X^3 + AX^2 + X.$

The Montgomery curve is called untwisted if $$B=1$$.

INPUT:

• twisted – boolean (default: False); allow $$B \neq 1$$

• morphism – boolean (default: False); also return an isomorphism from this curve to the computed Montgomery model

OUTPUT:

If twisted is False (the default), an EllipticCurve_generic object encapsulating an untwisted Montgomery curve. Otherwise, a ProjectivePlaneCurve object encapsulating a (potentially twisted) Montgomery curve.

If morphism is True, this method returns a tuple consisting of such a curve together with an isomorphism of suitable type (either WeierstrassIsomorphism or WeierstrassTransformationWithInverse) from this curve to the Montgomery model.

EXAMPLES:

sage: E = EllipticCurve(QQbar, '11a1')                                      # optional - sage.rings.number_field
sage: E.montgomery_model()                                                  # optional - sage.rings.number_field
Elliptic Curve defined by y^2 = x^3 + (-1.953522420987248?)*x^2 + x
over Algebraic Field

sage: E = EllipticCurve(GF(431^2), [7,7])                                   # optional - sage.rings.finite_rings
sage: E.montgomery_model()                                                  # optional - sage.rings.finite_rings
Elliptic Curve defined by y^2 = x^3 + (51*z2+190)*x^2 + x
over Finite Field in z2 of size 431^2


An isomorphism between the Montgomery and Weierstrass form can be obtained using the morphism parameter:

sage: E.montgomery_model(morphism=True)                                     # optional - sage.rings.finite_rings
(Elliptic Curve defined by y^2 = x^3 + (51*z2+190)*x^2 + x
over Finite Field in z2 of size 431^2,
Elliptic-curve morphism:
From: Elliptic Curve defined by y^2 = x^3 + 7*x + 7
over Finite Field in z2 of size 431^2
To:   Elliptic Curve defined by y^2 = x^3 + (51*z2+190)*x^2 + x
over Finite Field in z2 of size 431^2
Via:  (u,r,s,t) = (64*z2 + 407, 159, 0, 0))


Not all elliptic curves have a Montgomery model over their field of definition:

sage: E = EllipticCurve(GF(257), [1,1])                                     # optional - sage.rings.finite_rings
sage: E.montgomery_model()                                                  # optional - sage.rings.finite_rings
Traceback (most recent call last):
...
ValueError: Elliptic Curve defined by y^2 = x^3 + x + 1
over Finite Field of size 257 has no Montgomery model

sage: E = EllipticCurve(GF(257), [10,10])                                   # optional - sage.rings.finite_rings
sage: E.montgomery_model()                                                  # optional - sage.rings.finite_rings
Traceback (most recent call last):
...
ValueError: Elliptic Curve defined by y^2 = x^3 + 10*x + 10
over Finite Field of size 257 has no untwisted Montgomery model


However, as hinted by the error message, the latter curve does admit a twisted Montgomery model, which can be computed by passing twisted=True:

sage: E.montgomery_model(twisted=True)                                      # optional - sage.rings.finite_rings
Projective Plane Curve over Finite Field of size 257
defined by -x^3 + 8*x^2*z - 127*y^2*z - x*z^2


Since Sage internally represents elliptic curves as (long) Weierstrass curves, which do not feature the Montgomery $$B$$ coefficient, the returned curve in this case is merely a ProjectivePlaneCurve rather than the usual EllipticCurve_generic.

Arithmetic on curves of this type is not implemented natively, but can easily be emulated by mapping back and forth to the corresponding Weierstrass curve:

sage: C, f = E.montgomery_model(twisted=True, morphism=True)                # optional - sage.rings.finite_rings
sage: f                                                                     # optional - sage.rings.finite_rings
Scheme morphism:
From: Elliptic Curve defined by y^2 = x^3 + 10*x + 10
over Finite Field of size 257
To:   Projective Plane Curve over Finite Field of size 257
defined by -x^3 + 8*x^2*z - 127*y^2*z - x*z^2
Defn: Defined on coordinates by sending (x : y : z) to
(x + 116*z : -y : -85*z)
sage: g = f.inverse(); g                                                    # optional - sage.rings.finite_rings
Scheme morphism:
From: Projective Plane Curve over Finite Field of size 257
defined by -x^3 + 8*x^2*z - 127*y^2*z - x*z^2
To:   Elliptic Curve defined by y^2 = x^3 + 10*x + 10
over Finite Field of size 257
Defn: Defined on coordinates by sending (x : y : z) to
(-85*x - 116*z : 85*y : z)
sage: P = C(70, 8)                                                          # optional - sage.rings.finite_rings
sage: Q = C(17, 17)                                                         # optional - sage.rings.finite_rings
sage: P + Q             # this doesn't work...                              # optional - sage.rings.finite_rings
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for +: ...
sage: f(g(P) + g(Q))    # ...but this does                                  # optional - sage.rings.finite_rings
(107 : 168 : 1)


Using the fact that the Weil pairing satisfies $$e(\psi(P),\psi(Q)) = e(P,Q)^{\deg\psi}$$, even pairings can be emulated in this way (note that isomorphisms have degree $$1$$):

sage: F.<z2> = GF(257^2)                                                    # optional - sage.rings.finite_rings
sage: C_ = C.change_ring(F)                                                 # optional - sage.rings.finite_rings
sage: g_ = g.change_ring(F)                                                 # optional - sage.rings.finite_rings
sage: g_(P).order()                                                         # optional - sage.rings.finite_rings
12
sage: T = C_(-7 * z2 - 57, 31 * z2 - 52, 1)                                 # optional - sage.rings.finite_rings
sage: g_(T).order()                                                         # optional - sage.rings.finite_rings
12
sage: g_(P).weil_pairing(g_(T), 12)                                         # optional - sage.rings.finite_rings
15*z2 + 204


Another alternative is to simply extend the base field enough for the curve to have an untwisted Montgomery model:

sage: C_ = E.change_ring(F).montgomery_model(); C_                          # optional - sage.rings.finite_rings
Elliptic Curve defined by y^2 = x^3 + 249*x^2 + x
over Finite Field in z2 of size 257^2
sage: h = C.defining_polynomial().change_ring(F); h                         # optional - sage.rings.finite_rings
-x^3 + 8*x^2*z - 127*y^2*z - x*z^2
sage: C_.is_isomorphic(EllipticCurve_from_cubic(h).codomain())              # optional - sage.rings.finite_rings
True


The inverse conversion — computing a Weierstrass model for a given Montgomery curve — can be performed using EllipticCurve_from_cubic().

ALGORITHM: [CS2018], §2.4

REFERENCES:

multiplication_by_m(m, x_only=False)#

Return the multiplication-by-$$m$$ map from self to self

The result is a pair of rational functions in two variables $$x$$, $$y$$ (or a rational function in one variable $$x$$ if x_only is True).

INPUT:

• m – a nonzero integer

• x_only – boolean (default: False) if True, return only the $$x$$-coordinate of the map (as a rational function in one variable).

OUTPUT:

• a pair $$(f(x), g(x,y))$$, where $$f$$ and $$g$$ are rational functions with the degree of $$y$$ in $$g(x,y)$$ exactly 1,

• or just $$f(x)$$ if x_only is True

Note

• The result is not cached.

• m is allowed to be negative (but not 0).

EXAMPLES:

sage: E = EllipticCurve([-1,3])


We verify that multiplication by 1 is just the identity:

sage: E.multiplication_by_m(1)
(x, y)


Multiplication by 2 is more complicated:

sage: f = E.multiplication_by_m(2)
sage: f
((x^4 + 2*x^2 - 24*x + 1)/(4*x^3 - 4*x + 12),
(8*x^6*y - 40*x^4*y + 480*x^3*y - 40*x^2*y + 96*x*y - 568*y)/(64*x^6 - 128*x^4 + 384*x^3 + 64*x^2 - 384*x + 576))


Grab only the x-coordinate (less work):

sage: mx = E.multiplication_by_m(2, x_only=True); mx
(1/4*x^4 + 1/2*x^2 - 6*x + 1/4)/(x^3 - x + 3)
sage: mx.parent()
Fraction Field of Univariate Polynomial Ring in x over Rational Field


We check that it works on a point:

sage: P = E([2,3])
sage: eval = lambda f,P: [fi(P,P) for fi in f]
sage: assert E(eval(f,P)) == 2*P


We do the same but with multiplication by 3:

sage: f = E.multiplication_by_m(3)
sage: assert E(eval(f,P)) == 3*P


And the same with multiplication by 4:

sage: f = E.multiplication_by_m(4)
sage: assert E(eval(f,P)) == 4*P


And the same with multiplication by -1,-2,-3,-4:

sage: for m in [-1,-2,-3,-4]:
....:     f = E.multiplication_by_m(m)
....:     assert E(eval(f,P)) == m*P

multiplication_by_m_isogeny(m)#

Return the EllipticCurveIsogeny object associated to the multiplication-by-$$m$$ map on this elliptic curve.

The resulting isogeny will have the associated rational maps (i.e., those returned by multiplication_by_m()) already computed.

NOTE: This function is currently much slower than the result of self.multiplication_by_m(), because constructing an isogeny precomputes a significant amount of information. See github issue #7368 and github issue #8014 for the status of improving this situation.

INPUT:

• m – a nonzero integer

OUTPUT:

• An EllipticCurveIsogeny object associated to the multiplication-by-$$m$$ map on this elliptic curve.

EXAMPLES:

sage: E = EllipticCurve('11a1')
sage: E.multiplication_by_m_isogeny(7)
doctest:warning ... DeprecationWarning: ...
Isogeny of degree 49
from Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20
over Rational Field
to   Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20
over Rational Field

pari_curve()#

Return the PARI curve corresponding to this elliptic curve.

The result is cached.

EXAMPLES:

sage: E = EllipticCurve([RR(0), RR(0), RR(1), RR(-1), RR(0)])
sage: e = E.pari_curve()                                                    # optional - sage.libs.pari
sage: type(e)                                                               # optional - sage.libs.pari
<... 'cypari2.gen.Gen'>
sage: e.type()                                                              # optional - sage.libs.pari
't_VEC'
sage: e.disc()                                                              # optional - sage.libs.pari
37.0000000000000


Over a finite field:

sage: EllipticCurve(GF(41), [2,5]).pari_curve()                             # optional - sage.libs.pari sage.rings.finite_rings
[Mod(0, 41), Mod(0, 41), Mod(0, 41), Mod(2, 41), Mod(5, 41),
Mod(0, 41), Mod(4, 41), Mod(20, 41), Mod(37, 41), Mod(27, 41),
Mod(26, 41), Mod(4, 41), Mod(11, 41),
Vecsmall(),
[41, [9, 31, [6, 0, 0, 0]]], [0, 0, 0, 0]]


Over a $$p$$-adic field:

sage: Qp = pAdicField(5, prec=3)                                            # optional - sage.libs.pari sage.rings.padics
sage: E = EllipticCurve(Qp, [3, 4])                                         # optional - sage.libs.pari sage.rings.padics
sage: E.pari_curve()                                                        # optional - sage.libs.pari sage.rings.padics
[0, 0, 0, 3, 4, 0, 6, 16, -9, -144, -3456, -8640, 1728/5,
Vecsmall(), [O(5^3)], [0, 0]]
sage: E.j_invariant()                                                       # optional - sage.libs.pari sage.rings.padics
3*5^-1 + O(5)


Over a number field:

sage: K.<a> = QuadraticField(2)                                             # optional - sage.libs.pari sage.rings.number_field
sage: E = EllipticCurve([1,a])                                              # optional - sage.libs.pari sage.rings.number_field
sage: E.pari_curve()                                                        # optional - sage.libs.pari sage.rings.number_field
[0, 0, 0, Mod(1, y^2 - 2),
Mod(y, y^2 - 2), 0, Mod(2, y^2 - 2), Mod(4*y, y^2 - 2),
Mod(-1, y^2 - 2), Mod(-48, y^2 - 2), Mod(-864*y, y^2 - 2),
Mod(-928, y^2 - 2), Mod(3456/29, y^2 - 2),
Vecsmall(),
[[y^2 - 2, [2, 0], 8, 1, [[1, -1.41421356237310; 1, 1.41421356237310],
[1, -1.41421356237310; 1, 1.41421356237310],
[16, -23; 16, 23], [2, 0; 0, 4], [4, 0; 0, 2], [2, 0; 0, 1],
[2, [0, 2; 1, 0]], ], [-1.41421356237310, 1.41421356237310],
[1, y], [1, 0; 0, 1], [1, 0, 0, 2; 0, 1, 1, 0]]], [0, 0, 0, 0, 0]]


PARI no longer requires that the $$j$$-invariant has negative $$p$$-adic valuation:

sage: E = EllipticCurve(Qp,[1, 1])                                          # optional - sage.libs.pari sage.rings.padics
sage: E.j_invariant() # the j-invariant is a p-adic integer                 # optional - sage.libs.pari sage.rings.padics
2 + 4*5^2 + O(5^3)
sage: E.pari_curve()                                                        # optional - sage.libs.pari sage.rings.padics
[0, 0, 0, 1, 1, 0, 2, 4, -1, -48, -864, -496, 6912/31,
Vecsmall(), [O(5^3)], [0, 0]]

plot(xmin=None, xmax=None, components='both', **args)#

Draw a graph of this elliptic curve.

The plot method is only implemented when there is a natural coercion from the base ring of self to RR. In this case, self is plotted as if it was defined over RR.

INPUT:

• xmin, xmax – (optional) points will be computed at least within this range, but possibly farther.

• components – a string, one of the following:

• both – (default), scale so that both bounded and unbounded components appear

• bounded – scale the plot to show the bounded component. Raises an error if there is only one real component.

• unbounded – scale the plot to show the unbounded component, including the two flex points.

• plot_points – passed to sage.plot.generate_plot_points()

• adaptive_tolerance – passed to sage.plot.generate_plot_points()

• adaptive_recursion – passed to sage.plot.generate_plot_points()

• randomize – passed to sage.plot.generate_plot_points()

• **args – all other options are passed to sage.plot.line.Line

EXAMPLES:

sage: E = EllipticCurve([0, -1])
sage: plot(E, rgbcolor=hue(0.7))                                            # optional - sage.plot
Graphics object consisting of 1 graphics primitive
sage: E = EllipticCurve('37a')
sage: plot(E)                                                               # optional - sage.plot
Graphics object consisting of 2 graphics primitives
sage: plot(E, xmin=25, xmax=26)                                             # optional - sage.plot
Graphics object consisting of 2 graphics primitives


With github issue #12766 we added the components keyword:

sage: E.real_components()
2
sage: E.plot(components='bounded')                                          # optional - sage.plot
Graphics object consisting of 1 graphics primitive
sage: E.plot(components='unbounded')                                        # optional - sage.plot
Graphics object consisting of 1 graphics primitive


If there is only one component then specifying components=’bounded’ raises a ValueError:

sage: E = EllipticCurve('9990be2')
sage: E.plot(components='bounded')                                          # optional - sage.plot
Traceback (most recent call last):
...
ValueError: no bounded component for this curve


An elliptic curve defined over the Complex Field can not be plotted:

sage: E = EllipticCurve(CC, [0,0,1,-1,0])
sage: E.plot()                                                              # optional - sage.plot
Traceback (most recent call last):
...
NotImplementedError: plotting of curves over Complex Field
with 53 bits of precision is not implemented yet

rst_transform(r, s, t)#

Return the transform of the curve by $$(r,s,t)$$ (with $$u=1$$).

INPUT:

• r, s, t – three elements of the base ring.

OUTPUT:

The elliptic curve obtained from self by the standard Weierstrass transformation $$(u,r,s,t)$$ with $$u=1$$.

Note

This is just a special case of change_weierstrass_model(), with $$u=1$$.

EXAMPLES:

sage: R.<r,s,t> = QQ[]
sage: E = EllipticCurve([1,2,3,4,5])
sage: E.rst_transform(r, s, t)
Elliptic Curve defined by y^2 + (2*s+1)*x*y + (r+2*t+3)*y
= x^3 + (-s^2+3*r-s+2)*x^2 + (3*r^2-r*s-2*s*t+4*r-3*s-t+4)*x
+ (r^3+2*r^2-r*t-t^2+4*r-3*t+5)
over Multivariate Polynomial Ring in r, s, t over Rational Field

scalar_multiplication(m)#

Return the scalar-multiplication map $$[m]$$ on this elliptic curve as a sage.schemes.elliptic_curves.hom_scalar.EllipticCurveHom_scalar object.

EXAMPLES:

sage: E = EllipticCurve('77a1')
sage: m = E.scalar_multiplication(-7); m
Scalar-multiplication endomorphism [-7]
of Elliptic Curve defined by y^2 + y = x^3 + 2*x over Rational Field
sage: m.degree()
49
sage: P = E(2,3)
sage: m(P)
(-26/225 : -2132/3375 : 1)
sage: m.rational_maps() == E.multiplication_by_m(-7)
True

scale_curve(u)#

Return the transform of the curve by scale factor $$u$$.

INPUT:

• u – an invertible element of the base ring.

OUTPUT:

The elliptic curve obtained from self by the standard Weierstrass transformation $$(u,r,s,t)$$ with $$r=s=t=0$$.

Note

This is just a special case of change_weierstrass_model(), with $$r=s=t=0$$.

EXAMPLES:

sage: K = Frac(PolynomialRing(QQ, 'u'))
sage: u = K.gen()
sage: E = EllipticCurve([1,2,3,4,5])
sage: E.scale_curve(u)
Elliptic Curve defined by
y^2 + u*x*y + 3*u^3*y = x^3 + 2*u^2*x^2 + 4*u^4*x + 5*u^6
over Fraction Field of Univariate Polynomial Ring in u over Rational Field

short_weierstrass_model(complete_cube=True)#

Return a short Weierstrass model for self.

INPUT:

• complete_cube – boolean (default: True); for meaning, see below.

OUTPUT:

An elliptic curve.

If complete_cube=True: Return a model of the form $$y^2 = x^3 + a*x + b$$ for this curve. The characteristic must not be 2; in characteristic 3, it is only possible if $$b_2=0$$.

If complete_cube=False: Return a model of the form $$y^2 = x^3 + ax^2 + bx + c$$ for this curve. The characteristic must not be 2.

EXAMPLES:

sage: E = EllipticCurve([1,2,3,4,5])
sage: E
Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Rational Field
sage: F = E.short_weierstrass_model()
sage: F
Elliptic Curve defined by y^2  = x^3 + 4941*x + 185166 over Rational Field
sage: E.is_isomorphic(F)
True
sage: F = E.short_weierstrass_model(complete_cube=False)
sage: F
Elliptic Curve defined by y^2  = x^3 + 9*x^2 + 88*x + 464 over Rational Field
sage: E.is_isomorphic(F)
True

sage: E = EllipticCurve(GF(3), [1,2,3,4,5])                                 # optional - sage.rings.finite_rings
sage: E.short_weierstrass_model(complete_cube=False)                        # optional - sage.rings.finite_rings
Elliptic Curve defined by y^2 = x^3 + x + 2 over Finite Field of size 3


This used to be different see github issue #3973:

sage: E.short_weierstrass_model()                                           # optional - sage.rings.finite_rings
Elliptic Curve defined by y^2 = x^3 + x + 2 over Finite Field of size 3


More tests in characteristic 3:

sage: E = EllipticCurve(GF(3), [0,2,1,2,1])                                 # optional - sage.rings.finite_rings
sage: E.short_weierstrass_model()                                           # optional - sage.rings.finite_rings
Traceback (most recent call last):
...
ValueError: short_weierstrass_model(): no short model for Elliptic Curve
defined by y^2 + y = x^3 + 2*x^2 + 2*x + 1 over Finite Field of size 3
(characteristic is 3)
sage: E.short_weierstrass_model(complete_cube=False)                        # optional - sage.rings.finite_rings
Elliptic Curve defined by y^2 = x^3 + 2*x^2 + 2*x + 2
over Finite Field of size 3
sage: E.short_weierstrass_model(complete_cube=False).is_isomorphic(E)       # optional - sage.rings.finite_rings
True

torsion_polynomial(m, x=None, two_torsion_multiplicity=2, force_evaluate=None)#

Return the $$m^{th}$$ division polynomial of this elliptic curve evaluated at $$x$$.

The division polynomial is cached if $$x$$ is None.

INPUT:

• m – positive integer.

• x – optional ring element to use as the $$x$$ variable. If $$x$$ is None (omitted), then a new polynomial ring will be constructed over the base ring of the elliptic curve, and its generator will be used as $$x$$. Note that $$x$$ does not need to be a generator of a polynomial ring; any ring element works. This permits fast calculation of the torsion polynomial evaluated on any element of a ring.

• two_torsion_multiplicity – 0, 1, or 2

If 0: For even $$m$$ when $$x$$ is None, a univariate polynomial over the base ring of the curve is returned, which omits factors whose roots are the $$x$$-coordinates of the $$2$$-torsion points. When $$x$$ is not None, the evaluation of such a polynomial at $$x$$ is returned.

If 2: For even $$m$$ when $$x$$ is None, a univariate polynomial over the base ring of the curve is returned, which includes a factor of degree 3 whose roots are the $$x$$-coordinates of the $$2$$-torsion points. Similarly, when $$x$$ is not None, the evaluation of such a polynomial at $$x$$ is returned.

If 1: For even $$m$$ when $$x$$ is None, a bivariate polynomial over the base ring of the curve is returned, which includes a factor $$2y+a_1x+a_3$$ having simple zeros at the $$2$$-torsion points. When $$x$$ is not None, it should be a tuple of length 2, and the evaluation of such a polynomial at $$x$$ is returned.

• force_evaluate (optional) – 0, 1, or 2

By default, this method makes use of previously cached generic division polynomials to compute the value of the polynomial at a given element $$x$$ whenever it appears beneficial to do so. Explicitly setting this flag overrides the default behavior.

Note that the complexity of evaluating a generic division polynomial scales much worse than that of computing the value at a point directly (using the recursive formulas), hence setting this flag can be detrimental to performance.

If 0: Do not use cached generic division polynomials.

If 1: If the generic division polynomial for this $$m$$ has been cached before, evaluate it at $$x$$ to compute the result.

If 2: Compute the value at $$x$$ by evaluating the generic division polynomial. If the generic $$m$$-division polynomial has not yet been cached, compute and cache it first.

EXAMPLES:

sage: E = EllipticCurve([0,0,1,-1,0])
sage: E.division_polynomial(1)
1
sage: E.division_polynomial(2, two_torsion_multiplicity=0)
1
sage: E.division_polynomial(2, two_torsion_multiplicity=1)
2*y + 1
sage: E.division_polynomial(2, two_torsion_multiplicity=2)
4*x^3 - 4*x + 1
sage: E.division_polynomial(2)
4*x^3 - 4*x + 1
sage: [E.division_polynomial(3, two_torsion_multiplicity=i) for i in range(3)]
[3*x^4 - 6*x^2 + 3*x - 1, 3*x^4 - 6*x^2 + 3*x - 1, 3*x^4 - 6*x^2 + 3*x - 1]
sage: [type(E.division_polynomial(3, two_torsion_multiplicity=i)) for i in range(3)]
[<... 'sage.rings.polynomial.polynomial_rational_flint.Polynomial_rational_flint'>,
<... 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular'>,
<... 'sage.rings.polynomial.polynomial_rational_flint.Polynomial_rational_flint'>]

sage: E = EllipticCurve([0, -1, 1, -10, -20])
sage: R.<z> = PolynomialRing(QQ)
sage: E.division_polynomial(4, z, 0)
2*z^6 - 4*z^5 - 100*z^4 - 790*z^3 - 210*z^2 - 1496*z - 5821
sage: E.division_polynomial(4, z)
8*z^9 - 24*z^8 - 464*z^7 - 2758*z^6 + 6636*z^5 + 34356*z^4
+ 53510*z^3 + 99714*z^2 + 351024*z + 459859


This does not work, since when two_torsion_multiplicity is 1, we compute a bivariate polynomial, and must evaluate at a tuple of length 2:

sage: E.division_polynomial(4,z,1)
Traceback (most recent call last):
...
ValueError: x should be a tuple of length 2 (or None)
when two_torsion_multiplicity is 1
sage: R.<z,w> = PolynomialRing(QQ, 2)
sage: E.division_polynomial(4, (z,w), 1).factor()
(2*w + 1) * (2*z^6 - 4*z^5 - 100*z^4 - 790*z^3 - 210*z^2 - 1496*z - 5821)


We can also evaluate this bivariate polynomial at a point:

sage: P = E(5,5)
sage: E.division_polynomial(4,P,two_torsion_multiplicity=1)
-1771561

two_division_polynomial(x=None)#

Return the 2-division polynomial of this elliptic curve evaluated at x.

INPUT:

• x – optional ring element to use as the $$x$$ variable. If x is None, then a new polynomial ring will be constructed over the base ring of the elliptic curve, and its generator will be used as x. Note that x does not need to be a generator of a polynomial ring; any ring element is acceptable. This permits fast calculation of the torsion polynomial evaluated on any element of a ring.

EXAMPLES:

sage: E = EllipticCurve('5077a1')
sage: E.two_division_polynomial()
4*x^3 - 28*x + 25
sage: E = EllipticCurve(GF(3^2,'a'), [1,1,1,1,1])                           # optional - sage.rings.finite_rings
sage: E.two_division_polynomial()                                           # optional - sage.rings.finite_rings
x^3 + 2*x^2 + 2
sage: E.two_division_polynomial().roots()                                   # optional - sage.rings.finite_rings
[(2, 1), (2*a, 1), (a + 2, 1)]

sage.schemes.elliptic_curves.ell_generic.is_EllipticCurve(x)#

Utility function to test if x is an instance of an Elliptic Curve class.

EXAMPLES:

sage: from sage.schemes.elliptic_curves.ell_generic import is_EllipticCurve
sage: E = EllipticCurve([1,2,3/4,7,19])
sage: is_EllipticCurve(E)
True
sage: is_EllipticCurve(0)
False