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:
sage.schemes.elliptic_curves.hom.EllipticCurveHom
,sage.schemes.elliptic_curves.weierstrass_morphism.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
- 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 \(\varphi^* \omega_2 = u \omega_1\), where \(\varphi: E_1\to E_2\) is this isomorphism and \(\omega_i\) are the standard Weierstrass differentials on \(E_i\) defined by \(\mathrm dx/(2y+a_1x+a_3)\).
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 \(u\not=0\)) which specifies a transformation \((x,y) \mapsto (x',y')\) where
\((x,y) = (u^2x'+r , u^3y' + su^2x' + t).\)
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.isomorphisms(E, F, JustOne=False)#
Return one or all isomorphisms between two elliptic curves.
INPUT:
E
,F
(EllipticCurve) – Two elliptic curves.JustOne
(bool) IfTrue
, returns one isomorphism, orNone
if the curves are not isomorphic. IfFalse
, returns a (possibly empty) list of isomorphisms.
OUTPUT:
Either
None
, or a 4-tuple \((u,r,s,t)\) representing an isomorphism, or a list of these.Note
This function is not intended for users, who should use the interface provided by
ell_generic
.EXAMPLES:
sage: from sage.schemes.elliptic_curves.weierstrass_morphism import * sage: isomorphisms(EllipticCurve_from_j(0),EllipticCurve('27a3')) [(-1, 0, 0, -1), (1, 0, 0, 0)] sage: isomorphisms(EllipticCurve_from_j(0),EllipticCurve('27a3'),JustOne=True) (1, 0, 0, 0) sage: isomorphisms(EllipticCurve_from_j(0),EllipticCurve('27a1')) [] sage: isomorphisms(EllipticCurve_from_j(0),EllipticCurve('27a1'),JustOne=True)