Elliptic-curve morphisms#
This class serves as a common parent for various specializations of morphisms between elliptic curves, with the aim of providing a common interface regardless of implementation details.
Current implementations of elliptic-curve morphisms (child classes):
AUTHORS:
See authors of
EllipticCurveIsogeny
. Some of the code in this class was lifted from there.Lorenz Panny (2021): Refactor isogenies and isomorphisms into the common
EllipticCurveHom
interface.
- class sage.schemes.elliptic_curves.hom.EllipticCurveHom#
Bases:
sage.categories.morphism.Morphism
Base class for elliptic-curve morphisms.
- degree()#
Return the degree of this elliptic-curve morphism.
EXAMPLES:
sage: E = EllipticCurve(QQ, [0,0,0,1,0]) sage: phi = EllipticCurveIsogeny(E, E((0,0))) sage: phi.degree() 2 sage: phi = EllipticCurveIsogeny(E, [0,1,0,1]) sage: phi.degree() 4 sage: E = EllipticCurve(GF(31), [1,0,0,1,2]) sage: phi = EllipticCurveIsogeny(E, [17, 1]) sage: phi.degree() 3
Degrees are multiplicative, so the degree of a composite isogeny is the product of the degrees of the individual factors:
sage: from sage.schemes.elliptic_curves.hom_composite import EllipticCurveHom_composite doctest:warning ... sage: E = EllipticCurve(GF(419), [1,0]) sage: P, = E.gens() sage: phi = EllipticCurveHom_composite(E, P+P) sage: phi.degree() 210 sage: phi.degree() == prod(f.degree() for f in phi.factors()) True
Isomorphisms always have degree \(1\) by definition:
sage: E1 = EllipticCurve([1,2,3,4,5]) sage: E2 = EllipticCurve_from_j(E1.j_invariant()) sage: E1.isomorphism_to(E2).degree() 1
- dual()#
Return the dual of this elliptic-curve morphism.
Implemented by child classes. For examples, see:
- formal(prec=20)#
Return the formal isogeny associated to this elliptic-curve morphism as a power series in the variable \(t=-x/y\) on the domain curve.
INPUT:
prec
– (default: 20), the precision with which the computations in the formal group are carried out.
EXAMPLES:
sage: E = EllipticCurve(GF(13),[1,7]) sage: phi = E.isogeny(E(10,4)) sage: phi.formal() t + 12*t^13 + 2*t^17 + 8*t^19 + 2*t^21 + O(t^23)
sage: E = EllipticCurve([0,1]) sage: phi = E.isogeny(E(2,3)) sage: phi.formal(prec=10) t + 54*t^5 + 255*t^7 + 2430*t^9 + 19278*t^11 + O(t^13)
sage: E = EllipticCurve('11a2') sage: R.<x> = QQ[] sage: phi = E.isogeny(x^2 + 101*x + 12751/5) sage: phi.formal(prec=7) t - 2724/5*t^5 + 209046/5*t^7 - 4767/5*t^8 + 29200946/5*t^9 + O(t^10)
- is_injective()#
Determine whether or not this morphism has trivial kernel.
EXAMPLES:
sage: E = EllipticCurve('11a1') sage: R.<x> = QQ[] sage: f = x^2 + x - 29/5 sage: phi = EllipticCurveIsogeny(E, f) sage: phi.is_injective() False sage: phi = EllipticCurveIsogeny(E, R(1)) sage: phi.is_injective() True
sage: F = GF(7) sage: E = EllipticCurve(j=F(0)) sage: phi = EllipticCurveIsogeny(E, [ E((0,-1)), E((0,1))]) sage: phi.is_injective() False sage: phi = EllipticCurveIsogeny(E, E(0)) sage: phi.is_injective() True
- is_normalized()#
Determine whether this morphism is a normalized isogeny.
Note
An isogeny \(\varphi\colon E_1\to E_2\) between two given Weierstrass equations is said to be normalized if the \(\varphi^*(\omega_2) = \omega_1\), where \(\omega_1\) and \(\omega_2\) are the invariant differentials on \(E_1\) and \(E_2\) corresponding to the given equation.
EXAMPLES:
sage: from sage.schemes.elliptic_curves.weierstrass_morphism import WeierstrassIsomorphism sage: E = EllipticCurve(GF(7), [0,0,0,1,0]) sage: R.<x> = GF(7)[] sage: phi = EllipticCurveIsogeny(E, x) sage: phi.is_normalized() True sage: isom = WeierstrassIsomorphism(phi.codomain(), (3, 0, 0, 0)) sage: phi = isom * phi sage: phi.is_normalized() False sage: isom = WeierstrassIsomorphism(phi.codomain(), (5, 0, 0, 0)) sage: phi = isom * phi sage: phi.is_normalized() True sage: isom = WeierstrassIsomorphism(phi.codomain(), (1, 1, 1, 1)) sage: phi = isom * phi sage: phi.is_normalized() True
sage: F = GF(2^5, 'alpha'); alpha = F.gen() sage: E = EllipticCurve(F, [1,0,1,1,1]) sage: R.<x> = F[] sage: phi = EllipticCurveIsogeny(E, x+1) sage: isom = WeierstrassIsomorphism(phi.codomain(), (alpha, 0, 0, 0)) sage: phi.is_normalized() True sage: phi = isom * phi sage: phi.is_normalized() False sage: isom = WeierstrassIsomorphism(phi.codomain(), (1/alpha, 0, 0, 0)) sage: phi = isom * phi sage: phi.is_normalized() True sage: isom = WeierstrassIsomorphism(phi.codomain(), (1, 1, 1, 1)) sage: phi = isom * phi sage: phi.is_normalized() True
sage: E = EllipticCurve('11a1') sage: R.<x> = QQ[] sage: f = x^3 - x^2 - 10*x - 79/4 sage: phi = EllipticCurveIsogeny(E, f) sage: isom = WeierstrassIsomorphism(phi.codomain(), (2, 0, 0, 0)) sage: phi.is_normalized() True sage: phi = isom * phi sage: phi.is_normalized() False sage: isom = WeierstrassIsomorphism(phi.codomain(), (1/2, 0, 0, 0)) sage: phi = isom * phi sage: phi.is_normalized() True sage: isom = WeierstrassIsomorphism(phi.codomain(), (1, 1, 1, 1)) sage: phi = isom * phi sage: phi.is_normalized() True
ALGORITHM: We check if
scaling_factor()
returns \(1\).
- is_separable()#
Determine whether or not this morphism is separable.
Note
This method currently always returns
True
as Sage does not yet implement inseparable isogenies. This will probably change in the future.EXAMPLES:
sage: E = EllipticCurve(GF(17), [0,0,0,3,0]) sage: phi = EllipticCurveIsogeny(E, E((0,0))) sage: phi.is_separable() True
sage: E = EllipticCurve('11a1') sage: phi = EllipticCurveIsogeny(E, E.torsion_points()) sage: phi.is_separable() True
- is_surjective()#
Determine whether or not this morphism is surjective.
Note
This method currently always returns
True
, since a non-constant map of algebraic curves must be surjective, and Sage does not yet implement the constant zero map. This will probably change in the future.EXAMPLES:
sage: E = EllipticCurve('11a1') sage: R.<x> = QQ[] sage: f = x^2 + x - 29/5 sage: phi = EllipticCurveIsogeny(E, f) sage: phi.is_surjective() True
sage: E = EllipticCurve(GF(7), [0,0,0,1,0]) sage: phi = EllipticCurveIsogeny(E, E((0,0))) sage: phi.is_surjective() True
sage: F = GF(2^5, 'omega') sage: E = EllipticCurve(j=F(0)) sage: R.<x> = F[] sage: phi = EllipticCurveIsogeny(E, x) sage: phi.is_surjective() True
- is_zero()#
Check whether this elliptic-curve morphism is the zero map.
Note
This function currently always returns
True
as Sage does not yet implement the constant zero morphism. This will probably change in the future.EXAMPLES:
sage: E = EllipticCurve(j=GF(7)(0)) sage: phi = EllipticCurveIsogeny(E, [E(0,1), E(0,-1)]) sage: phi.is_zero() False
- kernel_polynomial()#
Return the kernel polynomial of this elliptic-curve morphism.
Implemented by child classes. For examples, see:
- rational_maps()#
Return the pair of explicit rational maps defining this elliptic-curve morphism as fractions of bivariate polynomials in \(x\) and \(y\).
Implemented by child classes. For examples, see:
- scaling_factor()#
Return the Weierstrass scaling factor associated to this elliptic-curve morphism.
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 morphism and \(\omega_i\) are the standard Weierstrass differentials on \(E_i\) defined by \(\mathrm dx/(2y+a_1x+a_3)\).
Implemented by child classes. For examples, see:
- x_rational_map()#
Return the \(x\)-coordinate rational map of this elliptic-curve morphism as a univariate rational expression in \(x\).
Implemented by child classes. For examples, see:
- sage.schemes.elliptic_curves.hom.compare_via_evaluation(left, right)#
Test if two elliptic-curve morphisms are equal by evaluating them at enough points.
INPUT:
left
,right
–EllipticCurveHom
objects
ALGORITHM:
We use the fact that two isogenies of equal degree \(d\) must be the same if and only if they behave identically on more than \(4d\) points. (It suffices to check this on a few points that generate a large enough subgroup.)
If the domain curve does not have sufficiently many rational points, the base field is extended first: Taking an extension of degree \(O(\log(d))\) suffices.
EXAMPLES:
sage: E = EllipticCurve(GF(83), [1,0]) sage: phi = E.isogeny(12*E.0, model='montgomery'); phi Isogeny of degree 7 from Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 83 to Elliptic Curve defined by y^2 = x^3 + 70*x^2 + x over Finite Field of size 83 sage: psi = phi.dual(); psi Isogeny of degree 7 from Elliptic Curve defined by y^2 = x^3 + 70*x^2 + x over Finite Field of size 83 to Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 83 sage: from sage.schemes.elliptic_curves.hom_composite import EllipticCurveHom_composite sage: mu = EllipticCurveHom_composite.from_factors([phi, psi]) sage: from sage.schemes.elliptic_curves.hom import compare_via_evaluation sage: compare_via_evaluation(mu, E.multiplication_by_m_isogeny(7)) True
See also
sage.schemes.elliptic_curves.hom_composite.EllipticCurveHom_composite._richcmp_()