Isomorphisms between Weierstrass models of elliptic curves#

AUTHORS:

Robert Bradshaw (2007): initial version
John Cremona (Jan 2008): isomorphisms, automorphisms and twists in all characteristics
Lorenz Panny (2021): EllipticCurveHom interface

class sage.schemes.elliptic_curves.weierstrass_morphism.WeierstrassIsomorphism(E=None, urst=None, F=None)#

Bases: EllipticCurveHom, baseWI

Class representing a Weierstrass isomorphism between two elliptic curves.

dual()#

Return the dual isogeny of this isomorphism.

For isomorphisms, the dual is just the inverse.

EXAMPLES:

sage: from sage.schemes.elliptic_curves.weierstrass_morphism import WeierstrassIsomorphism
sage: E = EllipticCurve(QuadraticField(-3), [0,1])
sage: w = WeierstrassIsomorphism(E, (CyclotomicField(3).gen(),0,0,0))
sage: (w.dual() * w).rational_maps()
(x, y)

sage: E1 = EllipticCurve([11,22,33,44,55])
sage: E2 = E1.short_weierstrass_model()
sage: iso = E1.isomorphism_to(E2)
sage: iso.dual() == ~iso
True

is_separable()#

Determine whether or not this isogeny is separable.

Since WeierstrassIsomorphism only implements isomorphisms, this method always returns True.

EXAMPLES:

sage: E = EllipticCurve(GF(31337), [0,1])
sage: {f.is_separable() for f in E.automorphisms()}
{True}

kernel_polynomial()#

Return the kernel polynomial of this isomorphism.

Isomorphisms have trivial kernel by definition, hence this method always returns $1$ .

EXAMPLES:

sage: E1 = EllipticCurve([11,22,33,44,55])
sage: E2 = EllipticCurve_from_j(E1.j_invariant())
sage: iso = E1.isomorphism_to(E2)
sage: iso.kernel_polynomial()
1
sage: psi = E1.isogeny(iso.kernel_polynomial(), codomain=E2); psi
Isogeny of degree 1 from Elliptic Curve defined by y^2 + 11*x*y + 33*y = x^3 + 22*x^2 + 44*x + 55 over Rational Field to Elliptic Curve defined by y^2 + x*y = x^3 + x^2 - 684*x + 6681 over Rational Field
sage: psi in {iso, -iso}
True

rational_maps()#

Return the pair of rational maps defining this isomorphism.

EXAMPLES:

sage: E1 = EllipticCurve([11,22,33,44,55])
sage: E2 = EllipticCurve_from_j(E1.j_invariant())
sage: iso = E1.isomorphism_to(E2); iso
Elliptic-curve morphism:
  From: Elliptic Curve defined by y^2 + 11*x*y + 33*y = x^3 + 22*x^2 + 44*x + 55 over Rational Field
  To:   Elliptic Curve defined by y^2 + x*y = x^3 + x^2 - 684*x + 6681 over Rational Field
  Via:  (u,r,s,t) = (1, -17, -5, 77)
sage: iso.rational_maps()
(x + 17, 5*x + y + 8)
sage: f = E2.defining_polynomial()(*iso.rational_maps(), 1)
sage: I = E1.defining_ideal()
sage: x,y,z = I.ring().gens()
sage: f in I + Ideal(z-1)
True

sage: E = EllipticCurve(GF(65537), [1,1,1,1,1])
sage: w = E.isomorphism_to(E.short_weierstrass_model())
sage: f,g = w.rational_maps()
sage: P = E.random_point()
sage: w(P).xy() == (f(P.xy()), g(P.xy()))
True

scaling_factor()#

Return the Weierstrass scaling factor associated to this Weierstrass isomorphism.

The scaling factor is the constant $u$ (in the base field) such that $φ^{*} ω_{2} = u ω_{1}$ , where $φ : E_{1} \to E_{2}$ is this isomorphism and $ω_{i}$ are the standard Weierstrass differentials on $E_{i}$ defined by $d x / (2 y + a_{1} x + a_{3})$ .

EXAMPLES:

sage: E = EllipticCurve(QQbar, [0,1])
sage: all(f.scaling_factor() == f.formal()[1] for f in E.automorphisms())
True

ALGORITHM: The scaling factor equals the $u$ component of the tuple $(u, r, s, t)$ defining the isomorphism.

x_rational_map()#

Return the $x$ -coordinate rational map of this isomorphism.

EXAMPLES:

sage: E1 = EllipticCurve([11,22,33,44,55])
sage: E2 = EllipticCurve_from_j(E1.j_invariant())
sage: iso = E1.isomorphism_to(E2); iso
Elliptic-curve morphism:
  From: Elliptic Curve defined by y^2 + 11*x*y + 33*y = x^3 + 22*x^2 + 44*x + 55 over Rational Field
  To:   Elliptic Curve defined by y^2 + x*y = x^3 + x^2 - 684*x + 6681 over Rational Field
  Via:  (u,r,s,t) = (1, -17, -5, 77)
sage: iso.x_rational_map()
x + 17
sage: iso.x_rational_map() == iso.rational_maps()[0]
True

class sage.schemes.elliptic_curves.weierstrass_morphism.baseWI(u=1, r=0, s=0, t=0)#

Bases: object

This class implements the basic arithmetic of isomorphisms between Weierstrass models of elliptic curves.

These are specified by lists of the form $[u, r, s, t]$ (with $u \neq 0$ ) which specifies a transformation $(x, y) \mapsto (x^{'}, y^{'})$ where

$(x, y) = (u^{2} x^{'} + r, u^{3} y^{'} + s u^{2} x^{'} + t) .$

INPUT:

u,r,s,t (default $(1, 0, 0, 0)$ ) – standard parameters of an isomorphism between Weierstrass models.

EXAMPLES:

sage: from sage.schemes.elliptic_curves.weierstrass_morphism import *
sage: baseWI()
(1, 0, 0, 0)
sage: baseWI(2,3,4,5)
(2, 3, 4, 5)
sage: R.<u,r,s,t> = QQ[]
sage: baseWI(u,r,s,t)
(u, r, s, t)

is_identity()#

Return True if this is the identity isomorphism.

EXAMPLES:

sage: from sage.schemes.elliptic_curves.weierstrass_morphism import *
sage: w = baseWI(); w.is_identity()
True
sage: w = baseWI(2,3,4,5); w.is_identity()
False

tuple()#

Return the parameters $u, r, s, t$ as a tuple.

EXAMPLES:

sage: from sage.schemes.elliptic_curves.weierstrass_morphism import *
sage: w = baseWI(2,3,4,5)
sage: w.tuple()
(2, 3, 4, 5)

sage.schemes.elliptic_curves.weierstrass_morphism.identity_morphism(E)#

Given an elliptic curve $E$ , return the identity morphism on $E$ as a WeierstrassIsomorphism.

EXAMPLES:

sage: from sage.schemes.elliptic_curves.weierstrass_morphism import identity_morphism
sage: E = EllipticCurve([5,6,7,8,9])
sage: id_ = identity_morphism(E)
sage: id_.rational_maps()
(x, y)

sage.schemes.elliptic_curves.weierstrass_morphism.negation_morphism(E)#

Given an elliptic curve $E$ , return the negation endomorphism $[- 1]$ of $E$ as a WeierstrassIsomorphism.

EXAMPLES:

sage: from sage.schemes.elliptic_curves.weierstrass_morphism import negation_morphism
sage: E = EllipticCurve([5,6,7,8,9])
sage: neg = negation_morphism(E)
sage: neg.rational_maps()
(x, -5*x - y - 7)