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):

• EllipticCurveIsogeny

• WeierstrassIsomorphism

• EllipticCurveHom_composite

• EllipticCurveHom_sum

• EllipticCurveHom_scalar

• EllipticCurveHom_frobenius

• EllipticCurveHom_velusqrt

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.

• Lorenz Panny (2022): matrix_on_subgroup()

• Lorenz Panny (2023): trace(), characteristic_polynomial()

class sage.schemes.elliptic_curves.hom.EllipticCurveHom(*args, **kwds)

Bases: Morphism

Base class for elliptic-curve morphisms.

as_morphism()

Return self as a morphism of projective schemes.

EXAMPLES:

Sage

sage: k = GF(11)
sage: E = EllipticCurve(k, [1,1])
sage: Q = E(6,5)
sage: phi = E.isogeny(Q)
sage: mor = phi.as_morphism()
sage: mor.domain() == E
True
sage: mor.codomain() == phi.codomain()
True
sage: mor(Q) == phi(Q)
True

Python

>>> from sage.all import *
>>> k = GF(Integer(11))
>>> E = EllipticCurve(k, [Integer(1),Integer(1)])
>>> Q = E(Integer(6),Integer(5))
>>> phi = E.isogeny(Q)
>>> mor = phi.as_morphism()
>>> mor.domain() == E
True
>>> mor.codomain() == phi.codomain()
True
>>> mor(Q) == phi(Q)
True

Sage Live

k = GF(11)
E = EllipticCurve(k, [1,1])
Q = E(6,5)
phi = E.isogeny(Q)
mor = phi.as_morphism()
mor.domain() == E
mor.codomain() == phi.codomain()
mor(Q) == phi(Q)

characteristic_polynomial()

Return the characteristic polynomial of this elliptic-curve morphism, which must be an endomorphism.

See also

• degree()

• trace()

EXAMPLES:

Sage

sage: E = EllipticCurve(QQ, [42, 42])
sage: m5 = E.scalar_multiplication(5)
sage: m5.characteristic_polynomial()
x^2 - 10*x + 25

Python

>>> from sage.all import *
>>> E = EllipticCurve(QQ, [Integer(42), Integer(42)])
>>> m5 = E.scalar_multiplication(Integer(5))
>>> m5.characteristic_polynomial()
x^2 - 10*x + 25

Sage Live

E = EllipticCurve(QQ, [42, 42])
m5 = E.scalar_multiplication(5)
m5.characteristic_polynomial()

Sage

sage: E = EllipticCurve(GF(71), [42, 42])
sage: pi = E.frobenius_endomorphism()
sage: pi.characteristic_polynomial()
x^2 - 8*x + 71
sage: E.frobenius().charpoly()
x^2 - 8*x + 71

Python

>>> from sage.all import *
>>> E = EllipticCurve(GF(Integer(71)), [Integer(42), Integer(42)])
>>> pi = E.frobenius_endomorphism()
>>> pi.characteristic_polynomial()
x^2 - 8*x + 71
>>> E.frobenius().charpoly()
x^2 - 8*x + 71

Sage Live

E = EllipticCurve(GF(71), [42, 42])
pi = E.frobenius_endomorphism()
pi.characteristic_polynomial()
E.frobenius().charpoly()

degree()

Return the degree of this elliptic-curve morphism.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> E = EllipticCurve(QQ, [Integer(0),Integer(0),Integer(0),Integer(1),Integer(0)])
>>> phi = EllipticCurveIsogeny(E, E((Integer(0),Integer(0))))
>>> phi.degree()
2
>>> phi = EllipticCurveIsogeny(E, [Integer(0),Integer(1),Integer(0),Integer(1)])
>>> phi.degree()
4

>>> E = EllipticCurve(GF(Integer(31)), [Integer(1),Integer(0),Integer(0),Integer(1),Integer(2)])
>>> phi = EllipticCurveIsogeny(E, [Integer(17), Integer(1)])
>>> phi.degree()
3

Sage Live

E = EllipticCurve(QQ, [0,0,0,1,0])
phi = EllipticCurveIsogeny(E, E((0,0)))
phi.degree()
phi = EllipticCurveIsogeny(E, [0,1,0,1])
phi.degree()
E = EllipticCurve(GF(31), [1,0,0,1,2])
phi = EllipticCurveIsogeny(E, [17, 1])
phi.degree()

Degrees are multiplicative, so the degree of a composite isogeny is the product of the degrees of the individual factors:

Sage

sage: from sage.schemes.elliptic_curves.hom_composite import EllipticCurveHom_composite
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

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.hom_composite import EllipticCurveHom_composite
>>> E = EllipticCurve(GF(Integer(419)), [Integer(1),Integer(0)])
>>> P, = E.gens()
>>> phi = EllipticCurveHom_composite(E, P+P)
>>> phi.degree()
210
>>> phi.degree() == prod(f.degree() for f in phi.factors())
True

Sage Live

from sage.schemes.elliptic_curves.hom_composite import EllipticCurveHom_composite
E = EllipticCurve(GF(419), [1,0])
P, = E.gens()
phi = EllipticCurveHom_composite(E, P+P)
phi.degree()
phi.degree() == prod(f.degree() for f in phi.factors())

Isomorphisms always have degree $1$ by definition:

Sage

sage: E1 = EllipticCurve([1,2,3,4,5])
sage: E2 = EllipticCurve_from_j(E1.j_invariant())
sage: E1.isomorphism_to(E2).degree()
1

Python

>>> from sage.all import *
>>> E1 = EllipticCurve([Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)])
>>> E2 = EllipticCurve_from_j(E1.j_invariant())
>>> E1.isomorphism_to(E2).degree()
1

Sage Live

E1 = EllipticCurve([1,2,3,4,5])
E2 = EllipticCurve_from_j(E1.j_invariant())
E1.isomorphism_to(E2).degree()

dual()

Return the dual of this elliptic-curve morphism.

Implemented by child classes. For examples, see:

• EllipticCurveIsogeny.dual()

• sage.schemes.elliptic_curves.weierstrass_morphism.WeierstrassIsomorphism.dual()

• sage.schemes.elliptic_curves.hom_composite.EllipticCurveHom_composite.dual()

• sage.schemes.elliptic_curves.hom_sum.EllipticCurveHom_sum.dual()

• sage.schemes.elliptic_curves.hom_scalar.EllipticCurveHom_scalar.dual()

• sage.schemes.elliptic_curves.hom_frobenius.EllipticCurveHom_frobenius.dual()

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

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)

Python

>>> from sage.all import *
>>> E = EllipticCurve(GF(Integer(13)),[Integer(1),Integer(7)])
>>> phi = E.isogeny(E(Integer(10),Integer(4)))
>>> phi.formal()
t + 12*t^13 + 2*t^17 + 8*t^19 + 2*t^21 + O(t^23)

Sage Live

E = EllipticCurve(GF(13),[1,7])
phi = E.isogeny(E(10,4))
phi.formal()

Sage

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)

Python

>>> from sage.all import *
>>> E = EllipticCurve([Integer(0),Integer(1)])
>>> phi = E.isogeny(E(Integer(2),Integer(3)))
>>> phi.formal(prec=Integer(10))
t + 54*t^5 + 255*t^7 + 2430*t^9 + 19278*t^11 + O(t^13)

Sage Live

E = EllipticCurve([0,1])
phi = E.isogeny(E(2,3))
phi.formal(prec=10)

Sage

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)

Python

>>> from sage.all import *
>>> E = EllipticCurve('11a2')
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> phi = E.isogeny(x**Integer(2) + Integer(101)*x + Integer(12751)/Integer(5))
>>> phi.formal(prec=Integer(7))
t - 2724/5*t^5 + 209046/5*t^7 - 4767/5*t^8 + 29200946/5*t^9 + O(t^10)

Sage Live

E = EllipticCurve('11a2')
R.<x> = QQ[]
phi = E.isogeny(x^2 + 101*x + 12751/5)
phi.formal(prec=7)

inseparable_degree()

Return the inseparable degree of this isogeny.

Implemented by child classes. For examples, see:

• EllipticCurveIsogeny.inseparable_degree()

• sage.schemes.elliptic_curves.weierstrass_morphism.WeierstrassIsomorphism.inseparable_degree()

• sage.schemes.elliptic_curves.hom_composite.EllipticCurveHom_composite.inseparable_degree()

• sage.schemes.elliptic_curves.hom_sum.EllipticCurveHom_sum.inseparable_degree()

• sage.schemes.elliptic_curves.hom_scalar.EllipticCurveHom_scalar.inseparable_degree()

• sage.schemes.elliptic_curves.hom_frobenius.EllipticCurveHom_frobenius.inseparable_degree()

is_injective()

Determine whether or not this morphism has trivial kernel.

The kernel is trivial if and only if this morphism is a purely inseparable isogeny.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> E = EllipticCurve('11a1')
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> f = x**Integer(2) + x - Integer(29)/Integer(5)
>>> phi = EllipticCurveIsogeny(E, f)
>>> phi.is_injective()
False
>>> phi = EllipticCurveIsogeny(E, R(Integer(1)))
>>> phi.is_injective()
True

Sage Live

E = EllipticCurve('11a1')
R.<x> = QQ[]
f = x^2 + x - 29/5
phi = EllipticCurveIsogeny(E, f)
phi.is_injective()
phi = EllipticCurveIsogeny(E, R(1))
phi.is_injective()

Sage

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

Python

>>> from sage.all import *
>>> F = GF(Integer(7))
>>> E = EllipticCurve(j=F(Integer(0)))
>>> phi = EllipticCurveIsogeny(E, [ E((Integer(0),-Integer(1))), E((Integer(0),Integer(1)))])
>>> phi.is_injective()
False
>>> phi = EllipticCurveIsogeny(E, E(Integer(0)))
>>> phi.is_injective()
True

Sage Live

F = GF(7)
E = EllipticCurve(j=F(0))
phi = EllipticCurveIsogeny(E, [ E((0,-1)), E((0,1))])
phi.is_injective()
phi = EllipticCurveIsogeny(E, E(0))
phi.is_injective()

Sage

sage: from sage.schemes.elliptic_curves.hom_composite import EllipticCurveHom_composite
sage: E = EllipticCurve([1,0])
sage: phi = EllipticCurveHom_composite(E, E(0,0))
sage: phi.is_injective()
False
sage: E = EllipticCurve_from_j(GF(3).algebraic_closure()(0))
sage: nu = EllipticCurveHom_composite.from_factors(E.automorphisms())
sage: nu
Composite morphism of degree 1 = 1^12:
  From: Elliptic Curve defined by y^2 = x^3 + x
        over Algebraic closure of Finite Field of size 3
  To:   Elliptic Curve defined by y^2 = x^3 + x
        over Algebraic closure of Finite Field of size 3
sage: nu.is_injective()
True

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.hom_composite import EllipticCurveHom_composite
>>> E = EllipticCurve([Integer(1),Integer(0)])
>>> phi = EllipticCurveHom_composite(E, E(Integer(0),Integer(0)))
>>> phi.is_injective()
False
>>> E = EllipticCurve_from_j(GF(Integer(3)).algebraic_closure()(Integer(0)))
>>> nu = EllipticCurveHom_composite.from_factors(E.automorphisms())
>>> nu
Composite morphism of degree 1 = 1^12:
  From: Elliptic Curve defined by y^2 = x^3 + x
        over Algebraic closure of Finite Field of size 3
  To:   Elliptic Curve defined by y^2 = x^3 + x
        over Algebraic closure of Finite Field of size 3
>>> nu.is_injective()
True

Sage Live

from sage.schemes.elliptic_curves.hom_composite import EllipticCurveHom_composite
E = EllipticCurve([1,0])
phi = EllipticCurveHom_composite(E, E(0,0))
phi.is_injective()
E = EllipticCurve_from_j(GF(3).algebraic_closure()(0))
nu = EllipticCurveHom_composite.from_factors(E.automorphisms())
nu
nu.is_injective()

Sage

sage: E = EllipticCurve(GF(23), [1,0])
sage: E.scalar_multiplication(4).is_injective()
False
sage: E.scalar_multiplication(5).is_injective()
False
sage: E.scalar_multiplication(1).is_injective()
True
sage: E.scalar_multiplication(-1).is_injective()
True
sage: E.scalar_multiplication(23).is_injective()
True
sage: E.scalar_multiplication(-23).is_injective()
True
sage: E.scalar_multiplication(0).is_injective()
False

Python

>>> from sage.all import *
>>> E = EllipticCurve(GF(Integer(23)), [Integer(1),Integer(0)])
>>> E.scalar_multiplication(Integer(4)).is_injective()
False
>>> E.scalar_multiplication(Integer(5)).is_injective()
False
>>> E.scalar_multiplication(Integer(1)).is_injective()
True
>>> E.scalar_multiplication(-Integer(1)).is_injective()
True
>>> E.scalar_multiplication(Integer(23)).is_injective()
True
>>> E.scalar_multiplication(-Integer(23)).is_injective()
True
>>> E.scalar_multiplication(Integer(0)).is_injective()
False

Sage Live

E = EllipticCurve(GF(23), [1,0])
E.scalar_multiplication(4).is_injective()
E.scalar_multiplication(5).is_injective()
E.scalar_multiplication(1).is_injective()
E.scalar_multiplication(-1).is_injective()
E.scalar_multiplication(23).is_injective()
E.scalar_multiplication(-23).is_injective()
E.scalar_multiplication(0).is_injective()

Sage

sage: from sage.schemes.elliptic_curves.hom_frobenius import EllipticCurveHom_frobenius
sage: E = EllipticCurve(GF(11), [1,1])
sage: pi = EllipticCurveHom_frobenius(E, 5)
sage: pi.is_injective()
True

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.hom_frobenius import EllipticCurveHom_frobenius
>>> E = EllipticCurve(GF(Integer(11)), [Integer(1),Integer(1)])
>>> pi = EllipticCurveHom_frobenius(E, Integer(5))
>>> pi.is_injective()
True

Sage Live

from sage.schemes.elliptic_curves.hom_frobenius import EllipticCurveHom_frobenius
E = EllipticCurve(GF(11), [1,1])
pi = EllipticCurveHom_frobenius(E, 5)
pi.is_injective()

is_normalized()

Determine whether this morphism is a normalized isogeny.

Note

An isogeny $\phi :E_1\to E_2$ between two given Weierstrass equations is said to be normalized if the ${\phi }^{\ast }({\omega }_2)={\omega }_1$, where ${\omega }_1$ and ${\omega }_2$ are the invariant differentials on $E_1$ and $E_2$ corresponding to the given equation.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.weierstrass_morphism import WeierstrassIsomorphism
>>> E = EllipticCurve(GF(Integer(7)), [Integer(0),Integer(0),Integer(0),Integer(1),Integer(0)])
>>> R = GF(Integer(7))['x']; (x,) = R._first_ngens(1)
>>> phi = EllipticCurveIsogeny(E, x)
>>> phi.is_normalized()
True
>>> isom = WeierstrassIsomorphism(phi.codomain(), (Integer(3), Integer(0), Integer(0), Integer(0)))
>>> phi = isom * phi
>>> phi.is_normalized()
False
>>> isom = WeierstrassIsomorphism(phi.codomain(), (Integer(5), Integer(0), Integer(0), Integer(0)))
>>> phi = isom * phi
>>> phi.is_normalized()
True
>>> isom = WeierstrassIsomorphism(phi.codomain(), (Integer(1), Integer(1), Integer(1), Integer(1)))
>>> phi = isom * phi
>>> phi.is_normalized()
True

Sage Live

from sage.schemes.elliptic_curves.weierstrass_morphism import WeierstrassIsomorphism
E = EllipticCurve(GF(7), [0,0,0,1,0])
R.<x> = GF(7)[]
phi = EllipticCurveIsogeny(E, x)
phi.is_normalized()
isom = WeierstrassIsomorphism(phi.codomain(), (3, 0, 0, 0))
phi = isom * phi
phi.is_normalized()
isom = WeierstrassIsomorphism(phi.codomain(), (5, 0, 0, 0))
phi = isom * phi
phi.is_normalized()
isom = WeierstrassIsomorphism(phi.codomain(), (1, 1, 1, 1))
phi = isom * phi
phi.is_normalized()

Sage

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

Python

>>> from sage.all import *
>>> F = GF(Integer(2)**Integer(5), 'alpha'); alpha = F.gen()
>>> E = EllipticCurve(F, [Integer(1),Integer(0),Integer(1),Integer(1),Integer(1)])
>>> R = F['x']; (x,) = R._first_ngens(1)
>>> phi = EllipticCurveIsogeny(E, x+Integer(1))
>>> isom = WeierstrassIsomorphism(phi.codomain(), (alpha, Integer(0), Integer(0), Integer(0)))
>>> phi.is_normalized()
True
>>> phi = isom * phi
>>> phi.is_normalized()
False
>>> isom = WeierstrassIsomorphism(phi.codomain(), (Integer(1)/alpha, Integer(0), Integer(0), Integer(0)))
>>> phi = isom * phi
>>> phi.is_normalized()
True
>>> isom = WeierstrassIsomorphism(phi.codomain(), (Integer(1), Integer(1), Integer(1), Integer(1)))
>>> phi = isom * phi
>>> phi.is_normalized()
True

Sage Live

F = GF(2^5, 'alpha'); alpha = F.gen()
E = EllipticCurve(F, [1,0,1,1,1])
R.<x> = F[]
phi = EllipticCurveIsogeny(E, x+1)
isom = WeierstrassIsomorphism(phi.codomain(), (alpha, 0, 0, 0))
phi.is_normalized()
phi = isom * phi
phi.is_normalized()
isom = WeierstrassIsomorphism(phi.codomain(), (1/alpha, 0, 0, 0))
phi = isom * phi
phi.is_normalized()
isom = WeierstrassIsomorphism(phi.codomain(), (1, 1, 1, 1))
phi = isom * phi
phi.is_normalized()

Sage

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

Python

>>> from sage.all import *
>>> E = EllipticCurve('11a1')
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> f = x**Integer(3) - x**Integer(2) - Integer(10)*x - Integer(79)/Integer(4)
>>> phi = EllipticCurveIsogeny(E, f)
>>> isom = WeierstrassIsomorphism(phi.codomain(), (Integer(2), Integer(0), Integer(0), Integer(0)))
>>> phi.is_normalized()
True
>>> phi = isom * phi
>>> phi.is_normalized()
False
>>> isom = WeierstrassIsomorphism(phi.codomain(), (Integer(1)/Integer(2), Integer(0), Integer(0), Integer(0)))
>>> phi = isom * phi
>>> phi.is_normalized()
True
>>> isom = WeierstrassIsomorphism(phi.codomain(), (Integer(1), Integer(1), Integer(1), Integer(1)))
>>> phi = isom * phi
>>> phi.is_normalized()
True

Sage Live

E = EllipticCurve('11a1')
R.<x> = QQ[]
f = x^3 - x^2 - 10*x - 79/4
phi = EllipticCurveIsogeny(E, f)
isom = WeierstrassIsomorphism(phi.codomain(), (2, 0, 0, 0))
phi.is_normalized()
phi = isom * phi
phi.is_normalized()
isom = WeierstrassIsomorphism(phi.codomain(), (1/2, 0, 0, 0))
phi = isom * phi
phi.is_normalized()
isom = WeierstrassIsomorphism(phi.codomain(), (1, 1, 1, 1))
phi = isom * phi
phi.is_normalized()

ALGORITHM: We check if scaling_factor() returns $1$.

is_separable()

Determine whether or not this morphism is a separable isogeny.

EXAMPLES:

Sage

sage: E = EllipticCurve(GF(17), [0,0,0,3,0])
sage: phi = EllipticCurveIsogeny(E,  E((0,0)))
sage: phi.is_separable()
True

Python

>>> from sage.all import *
>>> E = EllipticCurve(GF(Integer(17)), [Integer(0),Integer(0),Integer(0),Integer(3),Integer(0)])
>>> phi = EllipticCurveIsogeny(E,  E((Integer(0),Integer(0))))
>>> phi.is_separable()
True

Sage Live

E = EllipticCurve(GF(17), [0,0,0,3,0])
phi = EllipticCurveIsogeny(E,  E((0,0)))
phi.is_separable()

Sage

sage: E = EllipticCurve('11a1')
sage: phi = EllipticCurveIsogeny(E, E.torsion_points())
sage: phi.is_separable()
True

Python

>>> from sage.all import *
>>> E = EllipticCurve('11a1')
>>> phi = EllipticCurveIsogeny(E, E.torsion_points())
>>> phi.is_separable()
True

Sage Live

E = EllipticCurve('11a1')
phi = EllipticCurveIsogeny(E, E.torsion_points())
phi.is_separable()

Sage

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

Python

>>> from sage.all import *
>>> E = EllipticCurve(GF(Integer(31337)), [Integer(0),Integer(1)])                                   # needs sage.rings.finite_rings
>>> {f.is_separable() for f in E.automorphisms()}                         # needs sage.rings.finite_rings
{True}

Sage Live

E = EllipticCurve(GF(31337), [0,1])                                   # needs sage.rings.finite_rings
{f.is_separable() for f in E.automorphisms()}                         # needs sage.rings.finite_rings

Sage

sage: # needs sage.rings.finite_rings
sage: from sage.schemes.elliptic_curves.hom_composite import EllipticCurveHom_composite
sage: E = EllipticCurve(GF(7^2), [3,2])
sage: P = E.lift_x(1)
sage: phi = EllipticCurveHom_composite(E, P); phi
Composite morphism of degree 7:
  From: Elliptic Curve defined by y^2 = x^3 + 3*x + 2 over Finite Field in z2 of size 7^2
  To:   Elliptic Curve defined by y^2 = x^3 + 3*x + 2 over Finite Field in z2 of size 7^2
sage: phi.is_separable()
True

Python

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> from sage.schemes.elliptic_curves.hom_composite import EllipticCurveHom_composite
>>> E = EllipticCurve(GF(Integer(7)**Integer(2)), [Integer(3),Integer(2)])
>>> P = E.lift_x(Integer(1))
>>> phi = EllipticCurveHom_composite(E, P); phi
Composite morphism of degree 7:
  From: Elliptic Curve defined by y^2 = x^3 + 3*x + 2 over Finite Field in z2 of size 7^2
  To:   Elliptic Curve defined by y^2 = x^3 + 3*x + 2 over Finite Field in z2 of size 7^2
>>> phi.is_separable()
True

Sage Live

# needs sage.rings.finite_rings
from sage.schemes.elliptic_curves.hom_composite import EllipticCurveHom_composite
E = EllipticCurve(GF(7^2), [3,2])
P = E.lift_x(1)
phi = EllipticCurveHom_composite(E, P); phi
phi.is_separable()

Sage

sage: E = EllipticCurve(GF(11), [4,4])
sage: E.scalar_multiplication(11).is_separable()
False
sage: E.scalar_multiplication(-11).is_separable()
False
sage: E.scalar_multiplication(777).is_separable()
True
sage: E.scalar_multiplication(-1).is_separable()
True
sage: E.scalar_multiplication(77).is_separable()
False
sage: E.scalar_multiplication(121).is_separable()
False

Python

>>> from sage.all import *
>>> E = EllipticCurve(GF(Integer(11)), [Integer(4),Integer(4)])
>>> E.scalar_multiplication(Integer(11)).is_separable()
False
>>> E.scalar_multiplication(-Integer(11)).is_separable()
False
>>> E.scalar_multiplication(Integer(777)).is_separable()
True
>>> E.scalar_multiplication(-Integer(1)).is_separable()
True
>>> E.scalar_multiplication(Integer(77)).is_separable()
False
>>> E.scalar_multiplication(Integer(121)).is_separable()
False

Sage Live

E = EllipticCurve(GF(11), [4,4])
E.scalar_multiplication(11).is_separable()
E.scalar_multiplication(-11).is_separable()
E.scalar_multiplication(777).is_separable()
E.scalar_multiplication(-1).is_separable()
E.scalar_multiplication(77).is_separable()
E.scalar_multiplication(121).is_separable()

Sage

sage: from sage.schemes.elliptic_curves.hom_frobenius import EllipticCurveHom_frobenius
sage: E = EllipticCurve(GF(11), [1,1])
sage: pi = EllipticCurveHom_frobenius(E)
sage: pi.degree()
11
sage: pi.is_separable()
False
sage: pi = EllipticCurveHom_frobenius(E, 0)
sage: pi.degree()
1
sage: pi.is_separable()
True

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.hom_frobenius import EllipticCurveHom_frobenius
>>> E = EllipticCurve(GF(Integer(11)), [Integer(1),Integer(1)])
>>> pi = EllipticCurveHom_frobenius(E)
>>> pi.degree()
11
>>> pi.is_separable()
False
>>> pi = EllipticCurveHom_frobenius(E, Integer(0))
>>> pi.degree()
1
>>> pi.is_separable()
True

Sage Live

from sage.schemes.elliptic_curves.hom_frobenius import EllipticCurveHom_frobenius
E = EllipticCurve(GF(11), [1,1])
pi = EllipticCurveHom_frobenius(E)
pi.degree()
pi.is_separable()
pi = EllipticCurveHom_frobenius(E, 0)
pi.degree()
pi.is_separable()

Sage

sage: E = EllipticCurve(GF(17), [0,0,0,3,0])
sage: phi = E.isogeny(E((1,2)), algorithm='velusqrt')
sage: phi.is_separable()
True

Python

>>> from sage.all import *
>>> E = EllipticCurve(GF(Integer(17)), [Integer(0),Integer(0),Integer(0),Integer(3),Integer(0)])
>>> phi = E.isogeny(E((Integer(1),Integer(2))), algorithm='velusqrt')
>>> phi.is_separable()
True

Sage Live

E = EllipticCurve(GF(17), [0,0,0,3,0])
phi = E.isogeny(E((1,2)), algorithm='velusqrt')
phi.is_separable()

is_surjective()

Determine whether or not this morphism is surjective.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> E = EllipticCurve('11a1')
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> f = x**Integer(2) + x - Integer(29)/Integer(5)
>>> phi = EllipticCurveIsogeny(E, f)
>>> phi.is_surjective()
True

Sage Live

E = EllipticCurve('11a1')
R.<x> = QQ[]
f = x^2 + x - 29/5
phi = EllipticCurveIsogeny(E, f)
phi.is_surjective()

Sage

sage: E = EllipticCurve(GF(7), [0,0,0,1,0])
sage: phi = EllipticCurveIsogeny(E,  E((0,0)))
sage: phi.is_surjective()
True

Python

>>> from sage.all import *
>>> E = EllipticCurve(GF(Integer(7)), [Integer(0),Integer(0),Integer(0),Integer(1),Integer(0)])
>>> phi = EllipticCurveIsogeny(E,  E((Integer(0),Integer(0))))
>>> phi.is_surjective()
True

Sage Live

E = EllipticCurve(GF(7), [0,0,0,1,0])
phi = EllipticCurveIsogeny(E,  E((0,0)))
phi.is_surjective()

Sage

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

Python

>>> from sage.all import *
>>> F = GF(Integer(2)**Integer(5), 'omega')
>>> E = EllipticCurve(j=F(Integer(0)))
>>> R = F['x']; (x,) = R._first_ngens(1)
>>> phi = EllipticCurveIsogeny(E, x)
>>> phi.is_surjective()
True

Sage Live

F = GF(2^5, 'omega')
E = EllipticCurve(j=F(0))
R.<x> = F[]
phi = EllipticCurveIsogeny(E, x)
phi.is_surjective()

is_zero()

Check whether this elliptic-curve morphism is the zero map.

EXAMPLES:

Sage

sage: E = EllipticCurve(j=GF(7)(0))
sage: phi = EllipticCurveIsogeny(E, [E(0,1), E(0,-1)])
sage: phi.is_zero()
False

Python

>>> from sage.all import *
>>> E = EllipticCurve(j=GF(Integer(7))(Integer(0)))
>>> phi = EllipticCurveIsogeny(E, [E(Integer(0),Integer(1)), E(Integer(0),-Integer(1))])
>>> phi.is_zero()
False

Sage Live

E = EllipticCurve(j=GF(7)(0))
phi = EllipticCurveIsogeny(E, [E(0,1), E(0,-1)])
phi.is_zero()

kernel_polynomial()

Return the kernel polynomial of this elliptic-curve morphism.

Implemented by child classes. For examples, see:

• EllipticCurveIsogeny.kernel_polynomial()

• sage.schemes.elliptic_curves.weierstrass_morphism.WeierstrassIsomorphism.kernel_polynomial()

• sage.schemes.elliptic_curves.hom_composite.EllipticCurveHom_composite.kernel_polynomial()

• sage.schemes.elliptic_curves.hom_sum.EllipticCurveHom_sum.kernel_polynomial()

• sage.schemes.elliptic_curves.hom_scalar.EllipticCurveHom_scalar.kernel_polynomial()

• sage.schemes.elliptic_curves.hom_frobenius.EllipticCurveHom_frobenius.kernel_polynomial()

matrix_on_subgroup(domain_gens, codomain_gens=None)

Return the matrix by which this isogeny acts on the $n$-torsion subgroup with respect to the given bases.

INPUT:

• domain_gens – basis $(P,Q)$ of some $n$-torsion subgroup on the domain of this elliptic-curve morphism

• codomain_gens – basis $(R,S)$ of the $n$-torsion on the codomain of this morphism, or (default) None if self is an endomorphism

OUTPUT:

A $2\times 2$ matrix $M$ over $\mathbf{Z}/n$, such that the image of any point $[a]P+[b]Q$ under this morphism equals $[c]R+[d]S$ where $(cd{)}^T=(ab)M$.

EXAMPLES:

Sage

sage: F.<i> = GF(419^2, modulus=[1,0,1])
sage: E = EllipticCurve(F, [1,0])
sage: P = E(3, 176*i)
sage: Q = E(i+7, 67*i+48)
sage: P.weil_pairing(Q, 420).multiplicative_order()
420
sage: iota = E.automorphisms()[2]; iota
Elliptic-curve endomorphism of Elliptic Curve defined by y^2 = x^3 + x over Finite Field in i of size 419^2
  Via:  (u,r,s,t) = (i, 0, 0, 0)
sage: iota^2 == E.scalar_multiplication(-1)
True
sage: mat = iota.matrix_on_subgroup((P,Q)); mat
[301 386]
[ 83 119]
sage: mat.parent()
Full MatrixSpace of 2 by 2 dense matrices over Ring of integers modulo 420
sage: iota(P) == 301*P + 386*Q
True
sage: iota(Q) == 83*P + 119*Q
True
sage: a,b = 123, 456
sage: c,d = vector((a,b)) * mat; (c,d)
(111, 102)
sage: iota(a*P + b*Q) == c*P + d*Q
True

Python

>>> from sage.all import *
>>> F = GF(Integer(419)**Integer(2), modulus=[Integer(1),Integer(0),Integer(1)], names=('i',)); (i,) = F._first_ngens(1)
>>> E = EllipticCurve(F, [Integer(1),Integer(0)])
>>> P = E(Integer(3), Integer(176)*i)
>>> Q = E(i+Integer(7), Integer(67)*i+Integer(48))
>>> P.weil_pairing(Q, Integer(420)).multiplicative_order()
420
>>> iota = E.automorphisms()[Integer(2)]; iota
Elliptic-curve endomorphism of Elliptic Curve defined by y^2 = x^3 + x over Finite Field in i of size 419^2
  Via:  (u,r,s,t) = (i, 0, 0, 0)
>>> iota**Integer(2) == E.scalar_multiplication(-Integer(1))
True
>>> mat = iota.matrix_on_subgroup((P,Q)); mat
[301 386]
[ 83 119]
>>> mat.parent()
Full MatrixSpace of 2 by 2 dense matrices over Ring of integers modulo 420
>>> iota(P) == Integer(301)*P + Integer(386)*Q
True
>>> iota(Q) == Integer(83)*P + Integer(119)*Q
True
>>> a,b = Integer(123), Integer(456)
>>> c,d = vector((a,b)) * mat; (c,d)
(111, 102)
>>> iota(a*P + b*Q) == c*P + d*Q
True

Sage Live

F.<i> = GF(419^2, modulus=[1,0,1])
E = EllipticCurve(F, [1,0])
P = E(3, 176*i)
Q = E(i+7, 67*i+48)
P.weil_pairing(Q, 420).multiplicative_order()
iota = E.automorphisms()[2]; iota
iota^2 == E.scalar_multiplication(-1)
mat = iota.matrix_on_subgroup((P,Q)); mat
mat.parent()
iota(P) == 301*P + 386*Q
iota(Q) == 83*P + 119*Q
a,b = 123, 456
c,d = vector((a,b)) * mat; (c,d)
iota(a*P + b*Q) == c*P + d*Q

One important application of this is to compute generators of the kernel subgroup of an isogeny, when the $n$-torsion subgroup containing the kernel is accessible:

Sage

sage: K = E(83*i-16, 9*i-147)
sage: K.order()
7
sage: phi = E.isogeny(K)
sage: R,S = phi.codomain().gens()
sage: mat = phi.matrix_on_subgroup((P,Q), (R,S))
sage: mat  # random -- depends on R,S
[124 263]
[115 141]
sage: kermat = mat.left_kernel_matrix(); kermat
[300  60]
sage: ker = [ZZ(v[0])*P + ZZ(v[1])*Q for v in kermat]
sage: {phi(T) for T in ker}
{(0 : 1 : 0)}
sage: phi == E.isogeny(ker)
True

Python

>>> from sage.all import *
>>> K = E(Integer(83)*i-Integer(16), Integer(9)*i-Integer(147))
>>> K.order()
7
>>> phi = E.isogeny(K)
>>> R,S = phi.codomain().gens()
>>> mat = phi.matrix_on_subgroup((P,Q), (R,S))
>>> mat  # random -- depends on R,S
[124 263]
[115 141]
>>> kermat = mat.left_kernel_matrix(); kermat
[300  60]
>>> ker = [ZZ(v[Integer(0)])*P + ZZ(v[Integer(1)])*Q for v in kermat]
>>> {phi(T) for T in ker}
{(0 : 1 : 0)}
>>> phi == E.isogeny(ker)
True

Sage Live

K = E(83*i-16, 9*i-147)
K.order()
phi = E.isogeny(K)
R,S = phi.codomain().gens()
mat = phi.matrix_on_subgroup((P,Q), (R,S))
mat  # random -- depends on R,S
kermat = mat.left_kernel_matrix(); kermat
ker = [ZZ(v[0])*P + ZZ(v[1])*Q for v in kermat]
{phi(T) for T in ker}
phi == E.isogeny(ker)

We can also compute the matrix of a Frobenius endomorphism (EllipticCurveHom_frobenius) on a large enough subgroup to verify point-counting results:

Sage

sage: F.<a> = GF((101, 36))
sage: E = EllipticCurve(GF(101), [1,1])
sage: EE = E.change_ring(F)
sage: P,Q = EE.torsion_basis(37)
sage: pi = EE.frobenius_isogeny()
sage: M = pi.matrix_on_subgroup((P,Q))
sage: M.parent()
Full MatrixSpace of 2 by 2 dense matrices over Ring of integers modulo 37
sage: M.trace()
34
sage: E.trace_of_frobenius()
-3

Python

>>> from sage.all import *
>>> F = GF((Integer(101), Integer(36)), names=('a',)); (a,) = F._first_ngens(1)
>>> E = EllipticCurve(GF(Integer(101)), [Integer(1),Integer(1)])
>>> EE = E.change_ring(F)
>>> P,Q = EE.torsion_basis(Integer(37))
>>> pi = EE.frobenius_isogeny()
>>> M = pi.matrix_on_subgroup((P,Q))
>>> M.parent()
Full MatrixSpace of 2 by 2 dense matrices over Ring of integers modulo 37
>>> M.trace()
34
>>> E.trace_of_frobenius()
-3

Sage Live

F.<a> = GF((101, 36))
E = EllipticCurve(GF(101), [1,1])
EE = E.change_ring(F)
P,Q = EE.torsion_basis(37)
pi = EE.frobenius_isogeny()
M = pi.matrix_on_subgroup((P,Q))
M.parent()
M.trace()
E.trace_of_frobenius()

See also

To compute a basis of the $n$-torsion, you may use torsion_basis().

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:

• EllipticCurveIsogeny.rational_maps()

• sage.schemes.elliptic_curves.weierstrass_morphism.WeierstrassIsomorphism.rational_maps()

• sage.schemes.elliptic_curves.hom_composite.EllipticCurveHom_composite.rational_maps()

• sage.schemes.elliptic_curves.hom_sum.EllipticCurveHom_sum.rational_maps()

• sage.schemes.elliptic_curves.hom_scalar.EllipticCurveHom_scalar.rational_maps()

• sage.schemes.elliptic_curves.hom_frobenius.EllipticCurveHom_frobenius.rational_maps()

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 ${\phi }^{\ast }{\omega }_2=u{\omega }_1$, where $\phi :E_1\to E_2$ is this morphism and ${\omega }_i$ are the standard Weierstrass differentials on $E_i$ defined by $\mathrm{d}x/(2y+a_{1}x+a_3)$.

Implemented by child classes. For examples, see:

• EllipticCurveIsogeny.scaling_factor()

• sage.schemes.elliptic_curves.weierstrass_morphism.WeierstrassIsomorphism.scaling_factor()

• sage.schemes.elliptic_curves.hom_composite.EllipticCurveHom_composite.scaling_factor()

• sage.schemes.elliptic_curves.hom_sum.EllipticCurveHom_sum.scaling_factor()

• sage.schemes.elliptic_curves.hom_scalar.EllipticCurveHom_scalar.scaling_factor()

separable_degree()

Return the separable degree of this isogeny.

The separable degree is the result of dividing the degree() by the inseparable_degree().

EXAMPLES:

Sage

sage: E = EllipticCurve(GF(11), [5,5])
sage: E.is_supersingular()
False
sage: E.scalar_multiplication(-77).separable_degree()
539
sage: E = EllipticCurve(GF(11), [5,0])
sage: E.is_supersingular()
True
sage: E.scalar_multiplication(-77).separable_degree()
49

Python

>>> from sage.all import *
>>> E = EllipticCurve(GF(Integer(11)), [Integer(5),Integer(5)])
>>> E.is_supersingular()
False
>>> E.scalar_multiplication(-Integer(77)).separable_degree()
539
>>> E = EllipticCurve(GF(Integer(11)), [Integer(5),Integer(0)])
>>> E.is_supersingular()
True
>>> E.scalar_multiplication(-Integer(77)).separable_degree()
49

Sage Live

E = EllipticCurve(GF(11), [5,5])
E.is_supersingular()
E.scalar_multiplication(-77).separable_degree()
E = EllipticCurve(GF(11), [5,0])
E.is_supersingular()
E.scalar_multiplication(-77).separable_degree()

trace()

Return the trace of this elliptic-curve morphism, which must be an endomorphism.

ALGORITHM: compute_trace_generic()

EXAMPLES:

Sage

sage: E = EllipticCurve(QQ, [42, 42])
sage: m5 = E.scalar_multiplication(5)
sage: m5.trace()
10

Python

>>> from sage.all import *
>>> E = EllipticCurve(QQ, [Integer(42), Integer(42)])
>>> m5 = E.scalar_multiplication(Integer(5))
>>> m5.trace()
10

Sage Live

E = EllipticCurve(QQ, [42, 42])
m5 = E.scalar_multiplication(5)
m5.trace()

Sage

sage: E = EllipticCurve(GF(71^2), [45, 45])
sage: P = E.lift_x(27)
sage: P.order()
71
sage: tau = E.isogeny(P, codomain=E)
sage: tau.trace()
-1

Python

>>> from sage.all import *
>>> E = EllipticCurve(GF(Integer(71)**Integer(2)), [Integer(45), Integer(45)])
>>> P = E.lift_x(Integer(27))
>>> P.order()
71
>>> tau = E.isogeny(P, codomain=E)
>>> tau.trace()
-1

Sage Live

E = EllipticCurve(GF(71^2), [45, 45])
P = E.lift_x(27)
P.order()
tau = E.isogeny(P, codomain=E)
tau.trace()

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:

• EllipticCurveIsogeny.x_rational_map()

• sage.schemes.elliptic_curves.weierstrass_morphism.WeierstrassIsomorphism.x_rational_map()

• sage.schemes.elliptic_curves.hom_composite.EllipticCurveHom_composite.x_rational_map()

• sage.schemes.elliptic_curves.hom_sum.EllipticCurveHom_sum.x_rational_map()

• sage.schemes.elliptic_curves.hom_scalar.EllipticCurveHom_scalar.x_rational_map()

• sage.schemes.elliptic_curves.hom_frobenius.EllipticCurveHom_frobenius.x_rational_map()

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

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.scalar_multiplication(7))
True

Python

>>> from sage.all import *
>>> E = EllipticCurve(GF(Integer(83)), [Integer(1),Integer(0)])
>>> phi = E.isogeny(Integer(12)*E.gen(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
>>> 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
>>> from sage.schemes.elliptic_curves.hom_composite import EllipticCurveHom_composite
>>> mu = EllipticCurveHom_composite.from_factors([phi, psi])
>>> from sage.schemes.elliptic_curves.hom import compare_via_evaluation
>>> compare_via_evaluation(mu, E.scalar_multiplication(Integer(7)))
True

Sage Live

E = EllipticCurve(GF(83), [1,0])
phi = E.isogeny(12*E.0, model='montgomery'); phi
psi = phi.dual(); psi
from sage.schemes.elliptic_curves.hom_composite import EllipticCurveHom_composite
mu = EllipticCurveHom_composite.from_factors([phi, psi])
from sage.schemes.elliptic_curves.hom import compare_via_evaluation
compare_via_evaluation(mu, E.scalar_multiplication(7))

See also

• sage.schemes.elliptic_curves.hom_composite.EllipticCurveHom_composite._richcmp_()

sage.schemes.elliptic_curves.hom.compute_trace_generic(phi)

Compute the trace of the given elliptic-curve endomorphism.

ALGORITHM: Simple variant of Schoof’s algorithm. For enough small primes $\ell$, we find an order-$\ell$ point $P$ on $E$ and use a discrete-logarithm calculation to find the unique scalar $t_{\ell }\in \{0,...,\ell -1\}$ such that ${\phi }^2(P)+[\mathrm{deg}(\phi )]P=[t_{\ell }]\phi (P)$. Then $t_{\ell }$ equals the trace of $\phi$ modulo $\ell$, which can therefore be recovered using the Chinese remainder theorem.

EXAMPLES:

It works over finite fields:

Sage

sage: from sage.schemes.elliptic_curves.hom import compute_trace_generic
sage: E = EllipticCurve(GF(31337), [1,1])
sage: compute_trace_generic(E.frobenius_endomorphism())
314

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.hom import compute_trace_generic
>>> E = EllipticCurve(GF(Integer(31337)), [Integer(1),Integer(1)])
>>> compute_trace_generic(E.frobenius_endomorphism())
314

Sage Live

from sage.schemes.elliptic_curves.hom import compute_trace_generic
E = EllipticCurve(GF(31337), [1,1])
compute_trace_generic(E.frobenius_endomorphism())

It works over $\mathbf{Q}$:

Sage

sage: from sage.schemes.elliptic_curves.hom import compute_trace_generic
sage: E = EllipticCurve(QQ, [1,2,3,4,5])
sage: dbl = E.scalar_multiplication(2)
sage: compute_trace_generic(dbl)
4

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.hom import compute_trace_generic
>>> E = EllipticCurve(QQ, [Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)])
>>> dbl = E.scalar_multiplication(Integer(2))
>>> compute_trace_generic(dbl)
4

Sage Live

from sage.schemes.elliptic_curves.hom import compute_trace_generic
E = EllipticCurve(QQ, [1,2,3,4,5])
dbl = E.scalar_multiplication(2)
compute_trace_generic(dbl)

It works over number fields (for a CM curve):

Sage

sage: from sage.schemes.elliptic_curves.hom import compute_trace_generic
sage: x = polygen(QQ)
sage: K.<t> = NumberField(5*x^2 - 2*x + 1)
sage: E = EllipticCurve(K, [1,0])
sage: phi = E.isogeny([t,0,1], codomain=E)  # phi = 2 + i
sage: compute_trace_generic(phi)
4

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.hom import compute_trace_generic
>>> x = polygen(QQ)
>>> K = NumberField(Integer(5)*x**Integer(2) - Integer(2)*x + Integer(1), names=('t',)); (t,) = K._first_ngens(1)
>>> E = EllipticCurve(K, [Integer(1),Integer(0)])
>>> phi = E.isogeny([t,Integer(0),Integer(1)], codomain=E)  # phi = 2 + i
>>> compute_trace_generic(phi)
4

Sage Live

from sage.schemes.elliptic_curves.hom import compute_trace_generic
x = polygen(QQ)
K.<t> = NumberField(5*x^2 - 2*x + 1)
E = EllipticCurve(K, [1,0])
phi = E.isogeny([t,0,1], codomain=E)  # phi = 2 + i
compute_trace_generic(phi)

sage.schemes.elliptic_curves.hom.find_post_isomorphism(phi, psi)

Given two isogenies $\varphi :E\to E^{\prime }$ and $\psi :E\to E^{″}$ which are equal up to post-isomorphism defined over the same field, find that isomorphism.

In other words, this function computes an isomorphism $\alpha :E^{\prime }\to E^{″}$ such that $\alpha \circ \varphi =\psi$.

ALGORITHM:

Start with a list of all isomorphisms $E^{\prime }\to E^{″}$. Then repeatedly evaluate $\varphi$ and $\psi$ at random points $P$ to filter the list for isomorphisms $\alpha$ with $\alpha (\varphi (P))=\psi (P)$. Once only one candidate is left, return it. Periodically extend the base field to avoid getting stuck (say, if all candidate isomorphisms act the same on all rational points).

EXAMPLES:

Sage

sage: from sage.schemes.elliptic_curves.hom import find_post_isomorphism
sage: E = EllipticCurve(GF(7^2), [1,0])
sage: f = E.scalar_multiplication(1)
sage: g = choice(E.automorphisms())
sage: find_post_isomorphism(f, g) == g
True

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.hom import find_post_isomorphism
>>> E = EllipticCurve(GF(Integer(7)**Integer(2)), [Integer(1),Integer(0)])
>>> f = E.scalar_multiplication(Integer(1))
>>> g = choice(E.automorphisms())
>>> find_post_isomorphism(f, g) == g
True

Sage Live

from sage.schemes.elliptic_curves.hom import find_post_isomorphism
E = EllipticCurve(GF(7^2), [1,0])
f = E.scalar_multiplication(1)
g = choice(E.automorphisms())
find_post_isomorphism(f, g) == g

Sage

sage: from sage.schemes.elliptic_curves.weierstrass_morphism import WeierstrassIsomorphism
sage: from sage.schemes.elliptic_curves.hom_composite import EllipticCurveHom_composite
sage: x = polygen(ZZ, 'x')
sage: F.<i> = GF(883^2, modulus=x^2+1)
sage: E = EllipticCurve(F, [1,0])
sage: P = E.lift_x(117)
sage: Q = E.lift_x(774)
sage: w = WeierstrassIsomorphism(E, [i,0,0,0])
sage: phi = EllipticCurveHom_composite(E, [P,w(Q)]) * w
sage: psi = EllipticCurveHom_composite(E, [Q,w(P)])
sage: phi.kernel_polynomial() == psi.kernel_polynomial()
True
sage: find_post_isomorphism(phi, psi)
Elliptic-curve morphism:
  From: Elliptic Curve defined by y^2 = x^3 + 320*x + 482 over Finite Field in i of size 883^2
  To:   Elliptic Curve defined by y^2 = x^3 + 320*x + 401 over Finite Field in i of size 883^2
  Via:  (u,r,s,t) = (882*i, 0, 0, 0)

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.weierstrass_morphism import WeierstrassIsomorphism
>>> from sage.schemes.elliptic_curves.hom_composite import EllipticCurveHom_composite
>>> x = polygen(ZZ, 'x')
>>> F = GF(Integer(883)**Integer(2), modulus=x**Integer(2)+Integer(1), names=('i',)); (i,) = F._first_ngens(1)
>>> E = EllipticCurve(F, [Integer(1),Integer(0)])
>>> P = E.lift_x(Integer(117))
>>> Q = E.lift_x(Integer(774))
>>> w = WeierstrassIsomorphism(E, [i,Integer(0),Integer(0),Integer(0)])
>>> phi = EllipticCurveHom_composite(E, [P,w(Q)]) * w
>>> psi = EllipticCurveHom_composite(E, [Q,w(P)])
>>> phi.kernel_polynomial() == psi.kernel_polynomial()
True
>>> find_post_isomorphism(phi, psi)
Elliptic-curve morphism:
  From: Elliptic Curve defined by y^2 = x^3 + 320*x + 482 over Finite Field in i of size 883^2
  To:   Elliptic Curve defined by y^2 = x^3 + 320*x + 401 over Finite Field in i of size 883^2
  Via:  (u,r,s,t) = (882*i, 0, 0, 0)

Sage Live

from sage.schemes.elliptic_curves.weierstrass_morphism import WeierstrassIsomorphism
from sage.schemes.elliptic_curves.hom_composite import EllipticCurveHom_composite
x = polygen(ZZ, 'x')
F.<i> = GF(883^2, modulus=x^2+1)
E = EllipticCurve(F, [1,0])
P = E.lift_x(117)
Q = E.lift_x(774)
w = WeierstrassIsomorphism(E, [i,0,0,0])
phi = EllipticCurveHom_composite(E, [P,w(Q)]) * w
psi = EllipticCurveHom_composite(E, [Q,w(P)])
phi.kernel_polynomial() == psi.kernel_polynomial()
find_post_isomorphism(phi, psi)

Make live