# Isogenies¶

An isogeny $$\varphi: E_1\to E_2$$ between two elliptic curves $$E_1$$ and $$E_2$$ is a morphism of curves that sends the origin of $$E_1$$ to the origin of $$E_2$$. Such a morphism is automatically a morphism of group schemes and the kernel is a finite subgroup scheme of $$E_1$$. Such a subscheme can either be given by a list of generators, which have to be torsion points, or by a polynomial in the coordinate $$x$$ of the Weierstrass equation of $$E_1$$.

The usual way to create and work with isogenies is illustrated with the following example:

sage: k = GF(11)
sage: E = EllipticCurve(k,[1,1])
sage: Q = E(6,5)
sage: phi = E.isogeny(Q)
sage: phi
Isogeny of degree 7 from Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field of size 11 to Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 11
sage: P = E(4,5)
sage: phi(P)
(10 : 0 : 1)
sage: phi.codomain()
Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 11
sage: phi.rational_maps()
((x^7 + 4*x^6 - 3*x^5 - 2*x^4 - 3*x^3 + 3*x^2 + x - 2)/(x^6 + 4*x^5 - 4*x^4 - 5*x^3 + 5*x^2), (x^9*y - 5*x^8*y - x^7*y + x^5*y - x^4*y - 5*x^3*y - 5*x^2*y - 2*x*y - 5*y)/(x^9 - 5*x^8 + 4*x^6 - 3*x^4 + 2*x^3))


The functions directly accessible from an elliptic curve E over a field are isogeny and isogeny_codomain.

The most useful functions that apply to isogenies are

• codomain

• degree

• domain

• dual

• rational_maps

• kernel_polynomial

Warning

Only cyclic, separable isogenies are implemented (except for ). Some algorithms may need the isogeny to be normalized.

AUTHORS:

• Daniel Shumow <shumow@gmail.com>: 2009-04-19: initial version

• Chris Wuthrich : 7/09: changes: add check of input, not the full list is needed. 10/09: eliminating some bugs.

• John Cremona 2014-08-08: tidying of code and docstrings, systematic use of univariate vs. bivariate polynomials and rational functions.

class sage.schemes.elliptic_curves.ell_curve_isogeny.EllipticCurveIsogeny(E, kernel, codomain=None, degree=None, model=None, check=True)

Class Implementing Isogenies of Elliptic Curves

This class implements cyclic, separable, normalized isogenies of elliptic curves.

Several different algorithms for computing isogenies are available. These include:

• Velu’s Formulas: Velu’s original formulas for computing isogenies. This algorithm is selected by giving as the kernel parameter a list of points which generate a finite subgroup.

• Kohel’s Formulas: Kohel’s original formulas for computing isogenies. This algorithm is selected by giving as the kernel parameter a monic polynomial (or a coefficient list (little endian)) which will define the kernel of the isogeny.

INPUT:

• E – an elliptic curve, the domain of the isogeny to initialize.

• kernel – a kernel, either a point in E, a list of points in E, a monic kernel polynomial, or None. If initializing from a domain/codomain, this must be set to None.

• codomain – an elliptic curve (default:None). If kernel is None, then this must be the codomain of a cyclic, separable, normalized isogeny, furthermore, degree must be the degree of the isogeny from E to codomain. If kernel is not None, then this must be isomorphic to the codomain of the cyclic normalized separable isogeny defined by kernel, in this case, the isogeny is post composed with an isomorphism so that this parameter is the codomain.

• degree – an integer (default:None). If kernel is None, then this is the degree of the isogeny from E to codomain. If kernel is not None, then this is used to determine whether or not to skip a gcd of the kernel polynomial with the two torsion polynomial of E.

• model – a string (default:None). Only supported variable is minimal, in which case if E is a curve over the rationals or over a number field, then the codomain is a global minimum model where this exists.

• check (default: True) checks if the input is valid to define an isogeny

EXAMPLES:

A simple example of creating an isogeny of a field of small characteristic:

sage: E = EllipticCurve(GF(7), [0,0,0,1,0])
sage: phi = EllipticCurveIsogeny(E, E((0,0)) ); phi
Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 7 to Elliptic Curve defined by y^2 = x^3 + 3*x over Finite Field of size 7
sage: phi.degree() == 2
True
sage: phi.kernel_polynomial()
x
sage: phi.rational_maps()
((x^2 + 1)/x, (x^2*y - y)/x^2)
sage: phi == loads(dumps(phi))  # known bug
True


A more complicated example of a characteristic 2 field:

sage: E = EllipticCurve(GF(2^4,'alpha'), [0,0,1,0,1])
sage: P = E((1,1))
sage: phi_v = EllipticCurveIsogeny(E, P); phi_v
Isogeny of degree 3 from Elliptic Curve defined by y^2 + y = x^3 + 1 over Finite Field in alpha of size 2^4 to Elliptic Curve defined by y^2 + y = x^3 over Finite Field in alpha of size 2^4
sage: phi_ker_poly = phi_v.kernel_polynomial()
sage: phi_ker_poly
x + 1
sage: ker_poly_list = phi_ker_poly.list()
sage: phi_k = EllipticCurveIsogeny(E, ker_poly_list)
sage: phi_k == phi_v
True
sage: phi_k.rational_maps()
((x^3 + x + 1)/(x^2 + 1), (x^3*y + x^2*y + x*y + x + y)/(x^3 + x^2 + x + 1))
sage: phi_v.rational_maps()
((x^3 + x + 1)/(x^2 + 1), (x^3*y + x^2*y + x*y + x + y)/(x^3 + x^2 + x + 1))
sage: phi_k.degree() == phi_v.degree() == 3
True
sage: phi_k.is_separable()
True
sage: phi_v(E(0))
(0 : 1 : 0)
sage: alpha = E.base_field().gen()
sage: Q = E((0, alpha*(alpha + 1)))
sage: phi_v(Q)
(1 : alpha^2 + alpha : 1)
sage: phi_v(P) == phi_k(P)
True
sage: phi_k(P) == phi_v.codomain()(0)
True


We can create an isogeny that has kernel equal to the full 2 torsion:

sage: E = EllipticCurve(GF(3), [0,0,0,1,1])
sage: ker_list = E.division_polynomial(2).list()
sage: phi = EllipticCurveIsogeny(E, ker_list); phi
Isogeny of degree 4 from Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field of size 3 to Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field of size 3
sage: phi(E(0))
(0 : 1 : 0)
sage: phi(E((0,1)))
(1 : 0 : 1)
sage: phi(E((0,2)))
(1 : 0 : 1)
sage: phi(E((1,0)))
(0 : 1 : 0)
sage: phi.degree()
4


We can also create trivial isogenies with the trivial kernel:

sage: E = EllipticCurve(GF(17), [11, 11, 4, 12, 10])
sage: phi_v = EllipticCurveIsogeny(E, E(0))
sage: phi_v.degree()
1
sage: phi_v.rational_maps()
(x, y)
sage: E == phi_v.codomain()
True
sage: P = E.random_point()
sage: phi_v(P) == P
True

sage: E = EllipticCurve(GF(31), [23, 1, 22, 7, 18])
sage: phi_k = EllipticCurveIsogeny(E, ); phi_k
Isogeny of degree 1 from Elliptic Curve defined by y^2 + 23*x*y + 22*y = x^3 + x^2 + 7*x + 18 over Finite Field of size 31 to Elliptic Curve defined by y^2 + 23*x*y + 22*y = x^3 + x^2 + 7*x + 18 over Finite Field of size 31
sage: phi_k.degree()
1
sage: phi_k.rational_maps()
(x, y)
sage: phi_k.codomain() == E
True
sage: phi_k.kernel_polynomial()
1
sage: P = E.random_point(); P == phi_k(P)
True


Velu and Kohel also work in characteristic 0:

sage: E = EllipticCurve(QQ, [0,0,0,3,4])
sage: P_list = E.torsion_points()
sage: phi = EllipticCurveIsogeny(E, P_list); phi
Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 3*x + 4 over Rational Field to Elliptic Curve defined by y^2 = x^3 - 27*x + 46 over Rational Field
sage: P = E((0,2))
sage: phi(P)
(6 : -10 : 1)
sage: phi_ker_poly = phi.kernel_polynomial()
sage: phi_ker_poly
x + 1
sage: ker_poly_list = phi_ker_poly.list()
sage: phi_k = EllipticCurveIsogeny(E, ker_poly_list); phi_k
Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 3*x + 4 over Rational Field to Elliptic Curve defined by y^2 = x^3 - 27*x + 46 over Rational Field
sage: phi_k(P) == phi(P)
True
sage: phi_k == phi
True
sage: phi_k.degree()
2
sage: phi_k.is_separable()
True


A more complicated example over the rationals (of odd degree):

sage: E = EllipticCurve('11a1')
sage: P_list = E.torsion_points()
sage: phi_v = EllipticCurveIsogeny(E, P_list); phi_v
Isogeny of degree 5 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 - 7820*x - 263580 over Rational Field
sage: P = E((16,-61))
sage: phi_v(P)
(0 : 1 : 0)
sage: ker_poly = phi_v.kernel_polynomial(); ker_poly
x^2 - 21*x + 80
sage: ker_poly_list = ker_poly.list()
sage: phi_k = EllipticCurveIsogeny(E, ker_poly_list); phi_k
Isogeny of degree 5 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 - 7820*x - 263580 over Rational Field
sage: phi_k == phi_v
True
sage: phi_v(P) == phi_k(P)
True
sage: phi_k.is_separable()
True


We can also do this same example over the number field defined by the irreducible two torsion polynomial of $$E$$:

sage: E = EllipticCurve('11a1')
sage: P_list = E.torsion_points()
sage: K.<alpha> = NumberField(x^3 - 2* x^2 - 40*x - 158)
sage: EK = E.change_ring(K)
sage: P_list = [EK(P) for P in P_list]
sage: phi_v = EllipticCurveIsogeny(EK, P_list); phi_v
Isogeny of degree 5 from Elliptic Curve defined by y^2 + y = x^3 + (-1)*x^2 + (-10)*x + (-20) over Number Field in alpha with defining polynomial x^3 - 2*x^2 - 40*x - 158 to Elliptic Curve defined by y^2 + y = x^3 + (-1)*x^2 + (-7820)*x + (-263580) over Number Field in alpha with defining polynomial x^3 - 2*x^2 - 40*x - 158
sage: P = EK((alpha/2,-1/2))
sage: phi_v(P)
(122/121*alpha^2 + 1633/242*alpha - 3920/121 : -1/2 : 1)
sage: ker_poly = phi_v.kernel_polynomial()
sage: ker_poly
x^2 - 21*x + 80
sage: ker_poly_list = ker_poly.list()
sage: phi_k = EllipticCurveIsogeny(EK, ker_poly_list)
sage: phi_k
Isogeny of degree 5 from Elliptic Curve defined by y^2 + y = x^3 + (-1)*x^2 + (-10)*x + (-20) over Number Field in alpha with defining polynomial x^3 - 2*x^2 - 40*x - 158 to Elliptic Curve defined by y^2 + y = x^3 + (-1)*x^2 + (-7820)*x + (-263580) over Number Field in alpha with defining polynomial x^3 - 2*x^2 - 40*x - 158
sage: phi_v == phi_k
True
sage: phi_k(P) == phi_v(P)
True
sage: phi_k == phi_v
True
sage: phi_k.degree()
5
sage: phi_v.is_separable()
True


The following example shows how to specify an isogeny from domain and codomain:

sage: E = EllipticCurve('11a1')
sage: R.<x> = QQ[]
sage: f = x^2 - 21*x + 80
sage: phi = E.isogeny(f)
sage: E2 = phi.codomain()
sage: phi_s = EllipticCurveIsogeny(E, None, E2, 5)
sage: phi_s
Isogeny of degree 5 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 - 7820*x - 263580 over Rational Field
sage: phi_s == phi
True
sage: phi_s.rational_maps() == phi.rational_maps()
True


However only cyclic normalized isogenies can be constructed this way. So it won’t find the isogeny :

sage: E.isogeny(None, codomain=E,degree=9)
Traceback (most recent call last):
...
ValueError: The two curves are not linked by a cyclic normalized isogeny of degree 9


Also the presumed isogeny between the domain and codomain must be normalized:

sage: E2.isogeny(None,codomain=E,degree=5)
Traceback (most recent call last):
...
ValueError: The two curves are not linked by a cyclic normalized isogeny of degree 5
sage: phihat = phi.dual(); phihat
Isogeny of degree 5 from Elliptic Curve defined by y^2 + y = x^3 - x^2 - 7820*x - 263580 over Rational Field to Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
sage: phihat.is_normalized()
False


Here an example of a construction of a endomorphisms with cyclic kernel on a CM-curve:

sage: K.<i> = NumberField(x^2+1)
sage: E = EllipticCurve(K, [1,0])
sage: RK.<X> = K[]
sage: f = X^2 - 2/5*i + 1/5
sage: phi= E.isogeny(f)
sage: isom = phi.codomain().isomorphism_to(E)
sage: phi.set_post_isomorphism(isom)
sage: phi.codomain() == phi.domain()
True
sage: phi.rational_maps()
(((4/25*i + 3/25)*x^5 + (4/5*i - 2/5)*x^3 - x)/(x^4 + (-4/5*i + 2/5)*x^2 + (-4/25*i - 3/25)), ((11/125*i + 2/125)*x^6*y + (-23/125*i + 64/125)*x^4*y + (141/125*i + 162/125)*x^2*y + (3/25*i - 4/25)*y)/(x^6 + (-6/5*i + 3/5)*x^4 + (-12/25*i - 9/25)*x^2 + (2/125*i - 11/125)))


Domain and codomain tests (see trac ticket #12880):

sage: E = EllipticCurve(QQ, [0,0,0,1,0])
sage: phi = EllipticCurveIsogeny(E,  E(0,0))
sage: phi.domain() == E
True
sage: phi.codomain()
Elliptic Curve defined by y^2 = x^3 - 4*x over Rational Field

sage: E = EllipticCurve(GF(31), [1,0,0,1,2])
sage: phi = EllipticCurveIsogeny(E, [17, 1])
sage: phi.domain()
Elliptic Curve defined by y^2 + x*y = x^3 + x + 2 over Finite Field of size 31
sage: phi.codomain()
Elliptic Curve defined by y^2 + x*y = x^3 + 24*x + 6 over Finite Field of size 31


Composition tests (see trac ticket #16245):

sage: E = EllipticCurve(j=GF(7)(0))
sage: phi = E.isogeny([E(0), E((0,1)), E((0,-1))]); phi
Isogeny of degree 3 from Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field of size 7 to Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field of size 7
sage: phi2 = phi * phi; phi2
Composite map:
From: Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field of size 7
To:   Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field of size 7
Defn:   Isogeny of degree 3 from Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field of size 7 to Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field of size 7
then
Isogeny of degree 3 from Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field of size 7 to Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field of size 7


Examples over relative number fields used not to work (see trac ticket #16779):

sage: pol26 = hilbert_class_polynomial(-4*26)
sage: pol = NumberField(pol26,'a').optimized_representation().polynomial()
sage: K.<a> = NumberField(pol)
sage: j = pol26.roots(K)
sage: E = EllipticCurve(j=j)
sage: L.<b> = K.extension(x^2+26)
sage: EL = E.change_ring(L)
sage: iso2 = EL.isogenies_prime_degree(2); len(iso2)
1
sage: iso3 = EL.isogenies_prime_degree(3); len(iso3)
2


Examples over function fields used not to work (see trac ticket #11327):

sage: F.<t> = FunctionField(QQ)
sage: E = EllipticCurve([0,0,0,-t^2,0])
sage: isogs = E.isogenies_prime_degree(2)
sage: isogs
Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + (-t^2)*x over Rational function field in t over Rational Field to Elliptic Curve defined by y^2 = x^3 + 4*t^2*x over Rational function field in t over Rational Field
sage: isogs.rational_maps()
((x^2 - t^2)/x, (x^2*y + t^2*y)/x^2)
sage: duals = [phi.dual() for phi in isogs]
sage: duals
Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 4*t^2*x over Rational function field in t over Rational Field to Elliptic Curve defined by y^2 = x^3 + (-t^2)*x over Rational function field in t over Rational Field
sage: duals.rational_maps()
((1/4*x^2 + t^2)/x, (1/8*x^2*y + (-1/2*t^2)*y)/x^2)
sage: duals
Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 4*t^2*x over Rational function field in t over Rational Field to Elliptic Curve defined by y^2 = x^3 + (-t^2)*x over Rational function field in t over Rational Field

degree()

Return the degree of this isogeny.

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

dual()

Return the isogeny dual to this isogeny.

Note

If $$\varphi\colon E \to E_2$$ is the given isogeny and $$n$$ is its degree, then the dual is by definition the unique isogeny $$\hat\varphi\colon E_2\to E$$ such that the compositions $$\hat\varphi\circ\varphi$$ and $$\varphi\circ\hat\varphi$$ are the multiplication-by-$$n$$ maps on $$E$$ and $$E_2$$, respectively.

EXAMPLES:

sage: E = EllipticCurve('11a1')
sage: R.<x> = QQ[]
sage: f = x^2 - 21*x + 80
sage: phi = EllipticCurveIsogeny(E, f)
sage: phi_hat = phi.dual()
sage: phi_hat.domain() == phi.codomain()
True
sage: phi_hat.codomain() == phi.domain()
True
sage: (X, Y) = phi.rational_maps()
sage: (Xhat, Yhat) = phi_hat.rational_maps()
sage: Xm = Xhat.subs(x=X, y=Y)
sage: Ym = Yhat.subs(x=X, y=Y)
sage: (Xm, Ym) == E.multiplication_by_m(5)
True

sage: E = EllipticCurve(GF(37), [0,0,0,1,8])
sage: R.<x> = GF(37)[]
sage: f = x^3 + x^2 + 28*x + 33
sage: phi = EllipticCurveIsogeny(E, f)
sage: phi_hat = phi.dual()
sage: phi_hat.codomain() == phi.domain()
True
sage: phi_hat.domain() == phi.codomain()
True
sage: (X, Y) = phi.rational_maps()
sage: (Xhat, Yhat) = phi_hat.rational_maps()
sage: Xm = Xhat.subs(x=X, y=Y)
sage: Ym = Yhat.subs(x=X, y=Y)
sage: (Xm, Ym) == E.multiplication_by_m(7)
True

sage: E = EllipticCurve(GF(31), [0,0,0,1,8])
sage: R.<x> = GF(31)[]
sage: f = x^2 + 17*x + 29
sage: phi = EllipticCurveIsogeny(E, f)
sage: phi_hat = phi.dual()
sage: phi_hat.codomain() == phi.domain()
True
sage: phi_hat.domain() == phi.codomain()
True
sage: (X, Y) = phi.rational_maps()
sage: (Xhat, Yhat) = phi_hat.rational_maps()
sage: Xm = Xhat.subs(x=X, y=Y)
sage: Ym = Yhat.subs(x=X, y=Y)
sage: (Xm, Ym) == E.multiplication_by_m(5)
True


Test for trac ticket #23928:

sage: E = EllipticCurve(j=GF(431**2)(4))
sage: phi = E.isogeny(E.lift_x(0))
sage: phi.dual()
Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 427*x over Finite Field in z2 of size 431^2 to Elliptic Curve defined by y^2 = x^3 + x over Finite Field in z2 of size 431^2


Test (for trac ticket #7096):

sage: E = EllipticCurve('11a1')
sage: phi = E.isogeny(E(5,5))
sage: phi.dual().dual() == phi
True

sage: k = GF(103)
sage: E = EllipticCurve(k,[11,11])
sage: phi = E.isogeny(E(4,4))
sage: phi
Isogeny of degree 5 from Elliptic Curve defined by y^2 = x^3 + 11*x + 11 over Finite Field of size 103 to Elliptic Curve defined by y^2 = x^3 + 25*x + 80 over Finite Field of size 103
sage: from sage.schemes.elliptic_curves.weierstrass_morphism import WeierstrassIsomorphism
sage: phi.set_post_isomorphism(WeierstrassIsomorphism(phi.codomain(),(5,0,1,2)))
sage: phi.dual().dual() == phi
True

sage: E = EllipticCurve(GF(103),[1,0,0,1,-1])
sage: phi = E.isogeny(E(60,85))
sage: phi.dual()
Isogeny of degree 7 from Elliptic Curve defined by y^2 + x*y = x^3 + 84*x + 34 over Finite Field of size 103 to Elliptic Curve defined by y^2 + x*y = x^3 + x + 102 over Finite Field of size 103


Check that trac ticket #17293 is fixed:

sage: k.<s> = QuadraticField(2) sage: E = EllipticCurve(k, [-3*s*(4 + 5*s), 2*s*(2 + 14*s + 11*s^2)]) sage: phi = E.isogenies_prime_degree(3) sage: (-phi).dual() == -(phi.dual()) True sage: phi._EllipticCurveIsogeny__clear_cached_values() # forget the dual sage: -(phi.dual()) == (-phi).dual() True

formal(prec=20)

Return the formal isogeny 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)

get_post_isomorphism()

Return the post-isomorphism of this isogeny, or None.

EXAMPLES:

sage: E = EllipticCurve(j=GF(31)(0))
sage: R.<x> = GF(31)[]
sage: phi = EllipticCurveIsogeny(E, x+18)
sage: phi.get_post_isomorphism()
sage: from sage.schemes.elliptic_curves.weierstrass_morphism import WeierstrassIsomorphism
sage: isom = WeierstrassIsomorphism(phi.codomain(), (6,8,10,12))
sage: phi.set_post_isomorphism(isom)
sage: isom == phi.get_post_isomorphism()
True

sage: E = EllipticCurve(GF(83), [1,0,1,1,0])
sage: R.<x> = GF(83)[]; f = x+24
sage: phi = EllipticCurveIsogeny(E, f)
sage: E2 = phi.codomain()
sage: phi2 = EllipticCurveIsogeny(E, None, E2, 2)
sage: phi2.get_post_isomorphism()
Generic morphism:
From: Abelian group of points on Elliptic Curve defined by y^2 = x^3 + 65*x + 69 over Finite Field of size 83
To:   Abelian group of points on Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x + 16 over Finite Field of size 83
Via:  (u,r,s,t) = (1, 7, 42, 42)

get_pre_isomorphism()

Return the pre-isomorphism of this isogeny, or None.

EXAMPLES:

sage: E = EllipticCurve(GF(31), [1,1,0,1,-1])
sage: R.<x> = GF(31)[]
sage: f = x^3 + 9*x^2 + x + 30
sage: phi = EllipticCurveIsogeny(E, f)
sage: phi.get_post_isomorphism()
sage: Epr = E.short_weierstrass_model()
sage: isom = Epr.isomorphism_to(E)
sage: phi.set_pre_isomorphism(isom)
sage: isom == phi.get_pre_isomorphism()
True

sage: E = EllipticCurve(GF(83), [1,0,1,1,0])
sage: R.<x> = GF(83)[]; f = x+24
sage: phi = EllipticCurveIsogeny(E, f)
sage: E2 = phi.codomain()
sage: phi2 = EllipticCurveIsogeny(E, None, E2, 2)
sage: phi2.get_pre_isomorphism()
Generic morphism:
From: Abelian group of points on Elliptic Curve defined by y^2 + x*y + y = x^3 + x over Finite Field of size 83
To:   Abelian group of points on Elliptic Curve defined by y^2 = x^3 + 62*x + 74 over Finite Field of size 83
Via:  (u,r,s,t) = (1, 76, 41, 3)

is_injective()

Return True if and only if this isogeny 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(via_formal=True, check_by_pullback=True)

Return whether this isogeny is normalized.

Note

An isogeny $$\varphi\colon E\to E_2$$ between two given Weierstrass equations is said to be normalized if the constant $$c$$ is $$1$$ in $$\varphi*(\omega_2) = c\cdot\omega$$, where $$\omega$$ and $$omega_2$$ are the invariant differentials on $$E$$ and $$E_2$$ corresponding to the given equation.

INPUT:

• via_formal - (default: True) If True it simply checks if the leading term of the formal series is 1. Otherwise it uses a deprecated algorithm involving the second optional argument.

• check_by_pullback - (default:True) Deprecated.

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.set_post_isomorphism(isom)
sage: phi.is_normalized()
False
sage: isom = WeierstrassIsomorphism(phi.codomain(), (5, 0, 0, 0))
sage: phi.set_post_isomorphism(isom)
sage: phi.is_normalized()
True
sage: isom = WeierstrassIsomorphism(phi.codomain(), (1, 1, 1, 1))
sage: phi.set_post_isomorphism(isom)
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.set_post_isomorphism(isom)
sage: phi.is_normalized()
False
sage: isom = WeierstrassIsomorphism(phi.codomain(), (1/alpha, 0, 0, 0))
sage: phi.set_post_isomorphism(isom)
sage: phi.is_normalized()
True
sage: isom = WeierstrassIsomorphism(phi.codomain(), (1, 1, 1, 1))
sage: phi.set_post_isomorphism(isom)
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.set_post_isomorphism(isom)
sage: phi.is_normalized()
False
sage: isom = WeierstrassIsomorphism(phi.codomain(), (1/2, 0, 0, 0))
sage: phi.set_post_isomorphism(isom)
sage: phi.is_normalized()
True
sage: isom = WeierstrassIsomorphism(phi.codomain(), (1, 1, 1, 1))
sage: phi.set_post_isomorphism(isom)
sage: phi.is_normalized()
True

is_separable()

Return whether or not this isogeny is separable.

Note

This function always returns True as currently this class only implements separable isogenies.

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()

Return True if and only if this isogeny is surjective.

Note

This function always returns True, as a non-constant map of algebraic curves must be surjective, and this class does not model the constant $$0$$ isogeny.

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()

Return whether this isogeny is zero.

Note

Currently this class does not allow zero isogenies, so this function will always return True.

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 isogeny.

EXAMPLES:

sage: E = EllipticCurve(QQ, [0,0,0,2,0])
sage: phi = EllipticCurveIsogeny(E,  E((0,0)))
sage: phi.kernel_polynomial()
x

sage: E = EllipticCurve('11a1')
sage: phi = EllipticCurveIsogeny(E, E.torsion_points())
sage: phi.kernel_polynomial()
x^2 - 21*x + 80

sage: E = EllipticCurve(GF(17), [1,-1,1,-1,1])
sage: phi = EllipticCurveIsogeny(E, )
sage: phi.kernel_polynomial()
1

sage: E = EllipticCurve(GF(31), [0,0,0,3,0])
sage: phi = EllipticCurveIsogeny(E, [0,3,0,1])
sage: phi.kernel_polynomial()
x^3 + 3*x

n()

Numerical Approximation inherited from Map (through morphism), nonsensical for isogenies.

EXAMPLES:

sage: E = EllipticCurve(j=GF(7)(0))
sage: phi = EllipticCurveIsogeny(E, [ E((0,1)), E((0,-1))])
sage: phi.n()
Traceback (most recent call last):
...
NotImplementedError: Numerical approximations do not make sense for Elliptic Curve Isogenies

post_compose(left)

Return the post-composition of this isogeny with left.

EXAMPLES:

sage: E = EllipticCurve(j=GF(7)(0))
sage: phi = EllipticCurveIsogeny(E, [ E((0,1)), E((0,-1))])
sage: phi.post_compose(phi)
Traceback (most recent call last):
...
NotImplementedError: post-composition of isogenies not yet implemented

pre_compose(right)

Return the pre-composition of this isogeny with right.

EXAMPLES:

sage: E = EllipticCurve(j=GF(7)(0))
sage: phi = EllipticCurveIsogeny(E, [ E((0,1)), E((0,-1))])
sage: phi.pre_compose(phi)
Traceback (most recent call last):
...
NotImplementedError: pre-composition of isogenies not yet implemented

rational_maps()

Return the pair of rational maps defining this isogeny.

Note

Both components are returned as elements of the function field $$F(x,y)$$ in two variables over the base field $$F$$, though the first only involves $$x$$. To obtain the $$x$$-coordinate function as a rational function in $$F(x)$$, use x_rational_map().

EXAMPLES:

sage: E = EllipticCurve(QQ, [0,2,0,1,-1])
sage: phi = EllipticCurveIsogeny(E, )
sage: phi.rational_maps()
(x, y)

sage: E = EllipticCurve(GF(17), [0,0,0,3,0])
sage: phi = EllipticCurveIsogeny(E,  E((0,0)))
sage: phi.rational_maps()
((x^2 + 3)/x, (x^2*y - 3*y)/x^2)

set_post_isomorphism(postWI)

Modify this isogeny by postcomposing with a Weierstrass isomorphism.

EXAMPLES:

sage: E = EllipticCurve(j=GF(31)(0))
sage: R.<x> = GF(31)[]
sage: phi = EllipticCurveIsogeny(E, x+18)
sage: from sage.schemes.elliptic_curves.weierstrass_morphism import WeierstrassIsomorphism
sage: phi.set_post_isomorphism(WeierstrassIsomorphism(phi.codomain(), (6,8,10,12)))
sage: phi
Isogeny of degree 3 from Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field of size 31 to Elliptic Curve defined by y^2 + 24*x*y + 7*y = x^3 + 22*x^2 + 16*x + 20 over Finite Field of size 31

sage: E = EllipticCurve(j=GF(47)(0))
sage: f = E.torsion_polynomial(3)/3
sage: phi = EllipticCurveIsogeny(E, f)
sage: E2 = phi.codomain()
sage: post_isom = E2.isomorphism_to(E)
sage: phi.set_post_isomorphism(post_isom)
sage: phi.rational_maps() == E.multiplication_by_m(3)
False
sage: phi.switch_sign()
sage: phi.rational_maps() == E.multiplication_by_m(3)
True


Example over a number field:

sage: R.<x> = QQ[]
sage: K.<a> = NumberField(x^2 + 2)
sage: E = EllipticCurve(j=K(1728))
sage: ker_list = E.torsion_points()
sage: phi = EllipticCurveIsogeny(E, ker_list)
sage: from sage.schemes.elliptic_curves.weierstrass_morphism import WeierstrassIsomorphism
sage: post_isom = WeierstrassIsomorphism(phi.codomain(), (a,2,3,5))
sage: phi
Isogeny of degree 4 from Elliptic Curve defined by y^2 = x^3 + x over Number Field in a with defining polynomial x^2 + 2 to Elliptic Curve defined by y^2 = x^3 + (-44)*x + 112 over Number Field in a with defining polynomial x^2 + 2

set_pre_isomorphism(preWI)

Modify this isogeny by precomposing with a Weierstrass isomorphism.

EXAMPLES:

sage: E = EllipticCurve(GF(31), [1,1,0,1,-1])
sage: R.<x> = GF(31)[]
sage: f = x^3 + 9*x^2 + x + 30
sage: phi = EllipticCurveIsogeny(E, f)
sage: Epr = E.short_weierstrass_model()
sage: isom = Epr.isomorphism_to(E)
sage: phi.set_pre_isomorphism(isom)
sage: phi.rational_maps()
((-6*x^4 - 3*x^3 + 12*x^2 + 10*x - 1)/(x^3 + x - 12), (3*x^7 + x^6*y - 14*x^6 - 3*x^5 + 5*x^4*y + 7*x^4 + 8*x^3*y - 8*x^3 - 5*x^2*y + 5*x^2 - 14*x*y + 14*x - 6*y - 6)/(x^6 + 2*x^4 + 7*x^3 + x^2 + 7*x - 11))
sage: phi(Epr((0,22)))
(13 : 21 : 1)
sage: phi(Epr((3,7)))
(14 : 17 : 1)

sage: E = EllipticCurve(GF(29), [0,0,0,1,0])
sage: R.<x> = GF(29)[]
sage: f = x^2 + 5
sage: phi = EllipticCurveIsogeny(E, f)
sage: phi
Isogeny of degree 5 from Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 29 to Elliptic Curve defined by y^2 = x^3 + 20*x over Finite Field of size 29
sage: from sage.schemes.elliptic_curves.weierstrass_morphism import WeierstrassIsomorphism
sage: inv_isom = WeierstrassIsomorphism(E, (1,-2,5,10))
sage: Epr = inv_isom.codomain().codomain()
sage: isom = Epr.isomorphism_to(E)
sage: phi.set_pre_isomorphism(isom); phi
Isogeny of degree 5 from Elliptic Curve defined by y^2 + 10*x*y + 20*y = x^3 + 27*x^2 + 6 over Finite Field of size 29 to Elliptic Curve defined by y^2 = x^3 + 20*x over Finite Field of size 29
sage: phi(Epr((12,1)))
(26 : 0 : 1)
sage: phi(Epr((2,9)))
(0 : 0 : 1)
sage: phi(Epr((21,12)))
(3 : 0 : 1)
sage: phi.rational_maps()
(x^5 - 10*x^4 - 6*x^3 - 7*x^2 - x + 3)/(x^4 - 8*x^3 + 5*x^2 - 14*x - 6)

sage: E = EllipticCurve('11a1')
sage: R.<x> = QQ[]
sage: f = x^2 - 21*x + 80
sage: phi = EllipticCurveIsogeny(E, f); phi
Isogeny of degree 5 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 - 7820*x - 263580 over Rational Field
sage: from sage.schemes.elliptic_curves.weierstrass_morphism import WeierstrassIsomorphism
sage: Epr = E.short_weierstrass_model()
sage: isom = Epr.isomorphism_to(E)
sage: phi.set_pre_isomorphism(isom)
sage: phi
Isogeny of degree 5 from Elliptic Curve defined by y^2 = x^3 - 13392*x - 1080432 over Rational Field to Elliptic Curve defined by y^2 + y = x^3 - x^2 - 7820*x - 263580 over Rational Field
sage: phi(Epr((168,1188)))
(0 : 1 : 0)

switch_sign()

Compose this isogeny with $$[-1]$$ (negation).

EXAMPLES:

sage: E = EllipticCurve(GF(23), [0,0,0,1,0])
sage: f = E.torsion_polynomial(3)/3
sage: phi = EllipticCurveIsogeny(E, f, E)
sage: phi.rational_maps() == E.multiplication_by_m(3)
False
sage: phi.switch_sign()
sage: phi.rational_maps() == E.multiplication_by_m(3)
True

sage: E = EllipticCurve(GF(17), [-2, 3, -5, 7, -11])
sage: R.<x> = GF(17)[]
sage: f = x+6
sage: phi = EllipticCurveIsogeny(E, f)
sage: phi
Isogeny of degree 2 from Elliptic Curve defined by y^2 + 15*x*y + 12*y = x^3 + 3*x^2 + 7*x + 6 over Finite Field of size 17 to Elliptic Curve defined by y^2 + 15*x*y + 12*y = x^3 + 3*x^2 + 4*x + 8 over Finite Field of size 17
sage: phi.rational_maps()
((x^2 + 6*x + 4)/(x + 6), (x^2*y - 5*x*y + 8*x - 2*y)/(x^2 - 5*x + 2))
sage: phi.switch_sign()
sage: phi
Isogeny of degree 2 from Elliptic Curve defined by y^2 + 15*x*y + 12*y = x^3 + 3*x^2 + 7*x + 6 over Finite Field of size 17 to Elliptic Curve defined by y^2 + 15*x*y + 12*y = x^3 + 3*x^2 + 4*x + 8 over Finite Field of size 17
sage: phi.rational_maps()
((x^2 + 6*x + 4)/(x + 6),
(2*x^3 - x^2*y - 5*x^2 + 5*x*y - 4*x + 2*y + 7)/(x^2 - 5*x + 2))

sage: E = EllipticCurve('11a1')
sage: R.<x> = QQ[]
sage: f = x^2 - 21*x + 80
sage: phi = EllipticCurveIsogeny(E, f)
sage: (xmap1, ymap1) = phi.rational_maps()
sage: phi.switch_sign()
sage: (xmap2, ymap2) = phi.rational_maps()
sage: xmap1 == xmap2
True
sage: ymap1 == -ymap2 - E.a1()*xmap2 - E.a3()
True

sage: K.<a> = NumberField(x^2 + 1)
sage: E = EllipticCurve(K, [0,0,0,1,0])
sage: R.<x> = K[]
sage: phi = EllipticCurveIsogeny(E, x-a)
sage: phi.rational_maps()
((x^2 + (-a)*x - 2)/(x + (-a)), (x^2*y + (-2*a)*x*y + y)/(x^2 + (-2*a)*x - 1))
sage: phi.switch_sign()
sage: phi.rational_maps()
((x^2 + (-a)*x - 2)/(x + (-a)), (-x^2*y + (2*a)*x*y - y)/(x^2 + (-2*a)*x - 1))

x_rational_map()

Return the rational map giving the $$x$$-coordinate of this isogeny.

Note

This function returns the $$x$$-coordinate component of the isogeny as a rational function in $$F(x)$$, where $$F$$ is the base field. To obtain both coordinate functions as elements of the function field $$F(x,y)$$ in two variables, use rational_maps().

EXAMPLES:

sage: E = EllipticCurve(QQ, [0,2,0,1,-1])
sage: phi = EllipticCurveIsogeny(E, )
sage: phi.x_rational_map()
x

sage: E = EllipticCurve(GF(17), [0,0,0,3,0])
sage: phi = EllipticCurveIsogeny(E,  E((0,0)))
sage: phi.x_rational_map()
(x^2 + 3)/x

sage.schemes.elliptic_curves.ell_curve_isogeny.compute_codomain_formula(E, v, w)

Compute the codomain curve given parameters $$v$$ and $$w$$ (as in Velu / Kohel / etc formulas).

INPUT:

• E – an elliptic curve

• v, w – elements of the base field of E

OUTPUT:

The elliptic curve with invariants $$[a_1,a_2,a_3,a_4-5v,a_6-(a_1^2+4a_2)v-7w]$$ where $$E=[a_1,a_2,a_3,a_4,a_6]$$.

EXAMPLES:

This formula is used by every Isogeny instantiation:

sage: E = EllipticCurve(GF(19), [1,2,3,4,5])
sage: phi = EllipticCurveIsogeny(E, E((1,2)) )
sage: phi.codomain()
Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 9*x + 13 over Finite Field of size 19
sage: from sage.schemes.elliptic_curves.ell_curve_isogeny import compute_codomain_formula
sage: v = phi._EllipticCurveIsogeny__v
sage: w = phi._EllipticCurveIsogeny__w
sage: compute_codomain_formula(E, v, w) == phi.codomain()
True

sage.schemes.elliptic_curves.ell_curve_isogeny.compute_codomain_kohel(E, kernel, degree)

Compute the codomain from the kernel polynomial using Kohel’s formulas.

INPUT:

• E – an elliptic curve

• kernel (polynomial or list) – the kernel polynomial, or a list of its coefficients

• degree (int) – degree of the isogeny

OUTPUT:

(elliptic curve) – the codomain elliptic curve E/kernel

EXAMPLES:

sage: from sage.schemes.elliptic_curves.ell_curve_isogeny import compute_codomain_kohel
sage: E = EllipticCurve(GF(19), [1,2,3,4,5])
sage: phi = EllipticCurveIsogeny(E, [9,1])
sage: phi.codomain() == isogeny_codomain_from_kernel(E, [9,1])
True
sage: compute_codomain_kohel(E, [9,1], 2)
Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 9*x + 8 over Finite Field of size 19
sage: R.<x> = GF(19)[]
sage: E = EllipticCurve(GF(19), [18,17,16,15,14])
sage: phi = EllipticCurveIsogeny(E, x^3 + 14*x^2 + 3*x + 11)
sage: phi.codomain() == isogeny_codomain_from_kernel(E, x^3 + 14*x^2 + 3*x + 11)
True
sage: compute_codomain_kohel(E, x^3 + 14*x^2 + 3*x + 11, 7)
Elliptic Curve defined by y^2 + 18*x*y + 16*y = x^3 + 17*x^2 + 18*x + 18 over Finite Field of size 19
sage: E = EllipticCurve(GF(19), [1,2,3,4,5])
sage: phi = EllipticCurveIsogeny(E, x^3 + 7*x^2 + 15*x + 12)
sage: isogeny_codomain_from_kernel(E, x^3 + 7*x^2 + 15*x + 12) == phi.codomain()
True
sage: compute_codomain_kohel(E, x^3 + 7*x^2 + 15*x + 12,4)
Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 3*x + 15 over Finite Field of size 19


Note

This function uses the formulas of Section 2.4 of [Koh1996].

sage.schemes.elliptic_curves.ell_curve_isogeny.compute_intermediate_curves(E1, E2)

Return intermediate curves and isomorphisms.

Note

This is used so we can compute $$\wp$$ functions from the short Weierstrass model more easily.

Warning

The base field must be of characteristic not equal to 2,3.

INPUT:

• E1 - an elliptic curve

• E2 - an elliptic curve

OUTPUT:

tuple (pre_isomorphism, post_isomorphism, intermediate_domain, intermediate_codomain):

• intermediate_domain: a short Weierstrass model isomorphic to E1

• intermediate_codomain: a short Weierstrass model isomorphic to E2

• pre_isomorphism: normalized isomorphism from E1 to intermediate_domain

• post_isomorphism: normalized isomorphism from intermediate_codomain to E2

EXAMPLES:

sage: from sage.schemes.elliptic_curves.ell_curve_isogeny import compute_intermediate_curves
sage: E = EllipticCurve(GF(83), [1,0,1,1,0])
sage: R.<x> = GF(83)[]; f = x+24
sage: phi = EllipticCurveIsogeny(E, f)
sage: E2 = phi.codomain()
sage: compute_intermediate_curves(E, E2)
(Elliptic Curve defined by y^2 = x^3 + 62*x + 74 over Finite Field of size 83,
Elliptic Curve defined by y^2 = x^3 + 65*x + 69 over Finite Field of size 83,
Generic morphism:
From: Abelian group of points on Elliptic Curve defined by y^2 + x*y + y = x^3 + x over Finite Field of size 83
To:   Abelian group of points on Elliptic Curve defined by y^2 = x^3 + 62*x + 74 over Finite Field of size 83
Via:  (u,r,s,t) = (1, 76, 41, 3),
Generic morphism:
From: Abelian group of points on Elliptic Curve defined by y^2 = x^3 + 65*x + 69 over Finite Field of size 83
To:   Abelian group of points on Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x + 16 over Finite Field of size 83
Via:  (u,r,s,t) = (1, 7, 42, 42))

sage: R.<x> = QQ[]
sage: K.<i> = NumberField(x^2 + 1)
sage: E = EllipticCurve(K, [0,0,0,1,0])
sage: E2 = EllipticCurve(K, [0,0,0,16,0])
sage: compute_intermediate_curves(E, E2)
(Elliptic Curve defined by y^2 = x^3 + x over Number Field in i with defining polynomial x^2 + 1,
Elliptic Curve defined by y^2 = x^3 + 16*x over Number Field in i with defining polynomial x^2 + 1,
Generic endomorphism of Abelian group of points on Elliptic Curve defined by y^2 = x^3 + x over Number Field in i with defining polynomial x^2 + 1
Via:  (u,r,s,t) = (1, 0, 0, 0),
Generic endomorphism of Abelian group of points on Elliptic Curve defined by y^2 = x^3 + 16*x over Number Field in i with defining polynomial x^2 + 1
Via:  (u,r,s,t) = (1, 0, 0, 0))

sage.schemes.elliptic_curves.ell_curve_isogeny.compute_isogeny_kernel_polynomial(E1, E2, ell, algorithm='starks')

Return the kernel polynomial of an isogeny of degree ell between E1 and E2.

INPUT:

• E1 - an elliptic curve in short Weierstrass form.

• E2 - an elliptic curve in short Weierstrass form.

• ell - the degree of the isogeny from E1 to E2.

• algorithm - currently only starks (default) is implemented.

OUTPUT:

polynomial over the field of definition of E1, E2, that is the kernel polynomial of the isogeny from E1 to E2.

Note

If there is no degree ell, cyclic, separable, normalized isogeny from E1 to E2 then an error will be raised.

EXAMPLES:

sage: from sage.schemes.elliptic_curves.ell_curve_isogeny import compute_isogeny_kernel_polynomial

sage: E = EllipticCurve(GF(37), [0,0,0,1,8])
sage: R.<x> = GF(37)[]
sage: f = (x + 14) * (x + 30)
sage: phi = EllipticCurveIsogeny(E, f)
sage: E2 = phi.codomain()
sage: compute_isogeny_kernel_polynomial(E, E2, 5)
x^2 + 7*x + 13
sage: f
x^2 + 7*x + 13

sage: R.<x> = QQ[]
sage: K.<i> = NumberField(x^2 + 1)
sage: E = EllipticCurve(K, [0,0,0,1,0])
sage: E2 = EllipticCurve(K, [0,0,0,16,0])
sage: compute_isogeny_kernel_polynomial(E, E2, 4)
x^3 + x

sage.schemes.elliptic_curves.ell_curve_isogeny.compute_isogeny_starks(E1, E2, ell)

Return the kernel polynomials of an isogeny of degree ell between E1 and E2.

INPUT:

• E1 - an elliptic curve in short Weierstrass form.

• E2 - an elliptic curve in short Weierstrass form.

• ell - the degree of the isogeny from E1 to E2.

OUTPUT:

polynomial over the field of definition of E1, E2, that is the kernel polynomial of the isogeny from E1 to E2.

Note

There must be a degree ell, separable, normalized cyclic isogeny from E1 to E2, or an error will be raised.

ALGORITHM:

This function uses Starks Algorithm as presented in section 6.2 of [BMSS2006].

Note

As published in [BMSS2006], the algorithm is incorrect, and a correct version (with slightly different notation) can be found in [Mo2009]. The algorithm originates in [Sta1973].

EXAMPLES:

sage: from sage.schemes.elliptic_curves.ell_curve_isogeny import compute_isogeny_starks, compute_sequence_of_maps

sage: E = EllipticCurve(GF(97), [1,0,1,1,0])
sage: R.<x> = GF(97)[]; f = x^5 + 27*x^4 + 61*x^3 + 58*x^2 + 28*x + 21
sage: phi = EllipticCurveIsogeny(E, f)
sage: E2 = phi.codomain()
sage: (isom1, isom2, E1pr, E2pr, ker_poly) = compute_sequence_of_maps(E, E2, 11)
sage: compute_isogeny_starks(E1pr, E2pr, 11)
x^10 + 37*x^9 + 53*x^8 + 66*x^7 + 66*x^6 + 17*x^5 + 57*x^4 + 6*x^3 + 89*x^2 + 53*x + 8

sage: E = EllipticCurve(GF(37), [0,0,0,1,8])
sage: R.<x> = GF(37)[]
sage: f = (x + 14) * (x + 30)
sage: phi = EllipticCurveIsogeny(E, f)
sage: E2 = phi.codomain()
sage: compute_isogeny_starks(E, E2, 5)
x^4 + 14*x^3 + x^2 + 34*x + 21
sage: f**2
x^4 + 14*x^3 + x^2 + 34*x + 21

sage: E = EllipticCurve(QQ, [0,0,0,1,0])
sage: R.<x> = QQ[]
sage: f = x
sage: phi = EllipticCurveIsogeny(E, f)
sage: E2 = phi.codomain()
sage: compute_isogeny_starks(E, E2, 2)
x

sage.schemes.elliptic_curves.ell_curve_isogeny.compute_sequence_of_maps(E1, E2, ell)

Return intermediate curves, isomorphisms and kernel polynomial.

INPUT:

• E1, E2 – elliptic curves.

• ell – a prime such that there is a degree ell separable normalized isogeny from E1 to E2.

OUTPUT:

(pre_isom, post_isom, E1pr, E2pr, ker_poly) where:

• E1pr is an elliptic curve in short Weierstrass form isomorphic to E1;

• E2pr is an elliptic curve in short Weierstrass form isomorphic to E2;

• pre_isom is a normalised isomorphism from E1 to E1pr;

• post_isom is a normalised isomorphism from E2pr to E2;

• ker_poly is the kernel polynomial of an ell-isogeny from E1pr to E2pr.

EXAMPLES:

sage: from sage.schemes.elliptic_curves.ell_curve_isogeny import compute_sequence_of_maps
sage: E = EllipticCurve('11a1')
sage: R.<x> = QQ[]; f = x^2 - 21*x + 80
sage: phi = EllipticCurveIsogeny(E, f)
sage: E2 = phi.codomain()
sage: compute_sequence_of_maps(E, E2, 5)
(Generic morphism:
From: Abelian group of points on Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
To:   Abelian group of points on Elliptic Curve defined by y^2 = x^3 - 31/3*x - 2501/108 over Rational Field
Via:  (u,r,s,t) = (1, 1/3, 0, -1/2),
Generic morphism:
From: Abelian group of points on Elliptic Curve defined by y^2 = x^3 - 23461/3*x - 28748141/108 over Rational Field
To:   Abelian group of points on Elliptic Curve defined by y^2 + y = x^3 - x^2 - 7820*x - 263580 over Rational Field
Via:  (u,r,s,t) = (1, -1/3, 0, 1/2),
Elliptic Curve defined by y^2 = x^3 - 31/3*x - 2501/108 over Rational Field,
Elliptic Curve defined by y^2 = x^3 - 23461/3*x - 28748141/108 over Rational Field,
x^2 - 61/3*x + 658/9)

sage: K.<i> = NumberField(x^2 + 1)
sage: E = EllipticCurve(K, [0,0,0,1,0])
sage: E2 = EllipticCurve(K, [0,0,0,16,0])
sage: compute_sequence_of_maps(E, E2, 4)
(Generic endomorphism of Abelian group of points on Elliptic Curve defined by y^2 = x^3 + x over Number Field in i with defining polynomial x^2 + 1
Via:  (u,r,s,t) = (1, 0, 0, 0),
Generic endomorphism of Abelian group of points on Elliptic Curve defined by y^2 = x^3 + 16*x over Number Field in i with defining polynomial x^2 + 1
Via:  (u,r,s,t) = (1, 0, 0, 0),
Elliptic Curve defined by y^2 = x^3 + x over Number Field in i with defining polynomial x^2 + 1,
Elliptic Curve defined by y^2 = x^3 + 16*x over Number Field in i with defining polynomial x^2 + 1,
x^3 + x)

sage: E = EllipticCurve(GF(97), [1,0,1,1,0])
sage: R.<x> = GF(97)[]; f = x^5 + 27*x^4 + 61*x^3 + 58*x^2 + 28*x + 21
sage: phi = EllipticCurveIsogeny(E, f)
sage: E2 = phi.codomain()
sage: compute_sequence_of_maps(E, E2, 11)
(Generic morphism:
From: Abelian group of points on Elliptic Curve defined by y^2 + x*y + y = x^3 + x over Finite Field of size 97
To:   Abelian group of points on Elliptic Curve defined by y^2 = x^3 + 52*x + 31 over Finite Field of size 97
Via:  (u,r,s,t) = (1, 8, 48, 44),
Generic morphism:
From: Abelian group of points on Elliptic Curve defined by y^2 = x^3 + 41*x + 66 over Finite Field of size 97
To:   Abelian group of points on Elliptic Curve defined by y^2 + x*y + y = x^3 + 87*x + 26 over Finite Field of size 97
Via:  (u,r,s,t) = (1, 89, 49, 49),
Elliptic Curve defined by y^2 = x^3 + 52*x + 31 over Finite Field of size 97,
Elliptic Curve defined by y^2 = x^3 + 41*x + 66 over Finite Field of size 97,
x^5 + 67*x^4 + 13*x^3 + 35*x^2 + 77*x + 69)

sage.schemes.elliptic_curves.ell_curve_isogeny.compute_vw_kohel_even_deg1(x0, y0, a1, a2, a4)

Compute Velu’s (v,w) using Kohel’s formulas for isogenies of degree exactly divisible by 2.

INPUT:

• x0, y0 – coordinates of a 2-torsion point on an elliptic curve E

• a1, a2, a4 – invariants of E

OUTPUT:

(tuple) Velu’s isogeny parameters (v,w).

EXAMPLES:

This function will be implicitly called by the following example:

sage: E = EllipticCurve(GF(19), [1,2,3,4,5])
sage: phi = EllipticCurveIsogeny(E, [9,1]); phi
Isogeny of degree 2 from Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Finite Field of size 19 to Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 9*x + 8 over Finite Field of size 19
sage: from sage.schemes.elliptic_curves.ell_curve_isogeny import compute_vw_kohel_even_deg1
sage: a1,a2,a3,a4,a6 = E.ainvs()
sage: x0 = -9
sage: y0 = -(a1*x0 + a3)/2
sage: compute_vw_kohel_even_deg1(x0, y0, a1, a2, a4)
(18, 9)

sage.schemes.elliptic_curves.ell_curve_isogeny.compute_vw_kohel_even_deg3(b2, b4, s1, s2, s3)

Compute Velu’s (v,w) using Kohel’s formulas for isogenies of degree divisible by 4.

INPUT:

• b2, b4 – invariants of an elliptic curve E

• s1, s2, s3 – signed coefficients of the 2-division polynomial of E

OUTPUT:

(tuple) Velu’s isogeny parameters (v,w).

EXAMPLES:

This function will be implicitly called by the following example:

sage: E = EllipticCurve(GF(19), [1,2,3,4,5])
sage: R.<x> = GF(19)[]
sage: phi = EllipticCurveIsogeny(E, x^3 + 7*x^2 + 15*x + 12); phi
Isogeny of degree 4 from Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Finite Field of size 19 to Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 3*x + 15 over Finite Field of size 19
sage: from sage.schemes.elliptic_curves.ell_curve_isogeny import compute_vw_kohel_even_deg3
sage: (b2,b4) = (E.b2(), E.b4())
sage: (s1, s2, s3) = (-7, 15, -12)
sage: compute_vw_kohel_even_deg3(b2, b4, s1, s2, s3)
(4, 7)

sage.schemes.elliptic_curves.ell_curve_isogeny.compute_vw_kohel_odd(b2, b4, b6, s1, s2, s3, n)

Compute Velu’s (v,w) using Kohel’s formulas for isogenies of odd degree.

INPUT:

• b2, b4, b6 – invariants of an elliptic curve E

• s1, s2, s3 – signed coefficients of lowest powers of x in the kernel polynomial.

• n (int) – the degree

OUTPUT:

(tuple) Velu’s isogeny parameters (v,w).

EXAMPLES:

This function will be implicitly called by the following example:

sage: E = EllipticCurve(GF(19), [18,17,16,15,14])
sage: R.<x> = GF(19)[]
sage: phi = EllipticCurveIsogeny(E, x^3 + 14*x^2 + 3*x + 11); phi
Isogeny of degree 7 from Elliptic Curve defined by y^2 + 18*x*y + 16*y = x^3 + 17*x^2 + 15*x + 14 over Finite Field of size 19 to Elliptic Curve defined by y^2 + 18*x*y + 16*y = x^3 + 17*x^2 + 18*x + 18 over Finite Field of size 19
sage: from sage.schemes.elliptic_curves.ell_curve_isogeny import compute_vw_kohel_odd
sage: (b2,b4,b6) = (E.b2(), E.b4(), E.b6())
sage: (s1,s2,s3) = (-14,3,-11)
sage: compute_vw_kohel_odd(b2,b4,b6,s1,s2,s3,3)
(7, 1)

sage.schemes.elliptic_curves.ell_curve_isogeny.fill_isogeny_matrix(M)

Return a filled isogeny matrix giving all degrees from one giving only prime degrees.

INPUT:

• M – a square symmetric matrix whose off-diagonal $$i$$, $$j$$ entry is either a prime $$l$$ (if the $$i$$’th and $$j$$’th curves have an $$l$$-isogeny between them), otherwise is 0.

OUTPUT:

(matrix) a square matrix with entries $$1$$ on the diagonal, and in general the $$i$$, $$j$$ entry is $$d>0$$ if $$d$$ is the minimal degree of an isogeny from the $$i$$’th to the $$j$$’th curve,

EXAMPLES:

sage: M = Matrix([[0, 2, 3, 3, 0, 0], [2, 0, 0, 0, 3, 3], [3, 0, 0, 0, 2, 0], [3, 0, 0, 0, 0, 2], [0, 3, 2, 0, 0, 0], [0, 3, 0, 2, 0, 0]]); M
[0 2 3 3 0 0]
[2 0 0 0 3 3]
[3 0 0 0 2 0]
[3 0 0 0 0 2]
[0 3 2 0 0 0]
[0 3 0 2 0 0]
sage: from sage.schemes.elliptic_curves.ell_curve_isogeny import fill_isogeny_matrix
sage: fill_isogeny_matrix(M)
[ 1  2  3  3  6  6]
[ 2  1  6  6  3  3]
[ 3  6  1  9  2 18]
[ 3  6  9  1 18  2]
[ 6  3  2 18  1  9]
[ 6  3 18  2  9  1]

sage.schemes.elliptic_curves.ell_curve_isogeny.isogeny_codomain_from_kernel(E, kernel, degree=None)

Compute the isogeny codomain given a kernel.

INPUT:

• E - The domain elliptic curve.

• kernel - Either a list of points in the kernel of the isogeny, or a

kernel polynomial (specified as a either a univariate polynomial or a coefficient list.)

• degree - an integer, (default:None) optionally specified degree

of the kernel.

OUTPUT:

(elliptic curve) the codomain of the separable normalized isogeny from this kernel

EXAMPLES:

sage: from sage.schemes.elliptic_curves.ell_curve_isogeny import isogeny_codomain_from_kernel
sage: E = EllipticCurve(GF(7), [1,0,1,0,1])
sage: R.<x> = GF(7)[]
sage: isogeny_codomain_from_kernel(E, [4,1], degree=3)
Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x + 6 over Finite Field of size 7
sage: EllipticCurveIsogeny(E, [4,1]).codomain() == isogeny_codomain_from_kernel(E, [4,1], degree=3)
True
sage: isogeny_codomain_from_kernel(E, x^3 + x^2 + 4*x + 3)
Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x + 6 over Finite Field of size 7
sage: isogeny_codomain_from_kernel(E, x^3 + 2*x^2 + 4*x + 3)
Elliptic Curve defined by y^2 + x*y + y = x^3 + 5*x + 2 over Finite Field of size 7

sage: E = EllipticCurve(GF(19), [1,2,3,4,5])
sage: kernel_list = [E((15,10)), E((10,3)),E((6,5))]
sage: isogeny_codomain_from_kernel(E, kernel_list)
Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 3*x + 15 over Finite Field of size 19

sage.schemes.elliptic_curves.ell_curve_isogeny.isogeny_determine_algorithm(E, kernel)

Helper function that allows the various isogeny functions to infer the algorithm type from the parameters passed in.

INPUT:

• E (elliptic curve) – an elliptic curve

• kernel – either a list of points on E, or a univariate polynomial or list of coefficients of a univariate polynomial.

OUTPUT:

(string) either ‘velu’ or ‘kohel’

If kernel is a list of points on the EllipticCurve $$E$$, then we will try to use Velu’s algorithm.

If kernel is a list of coefficients or a univariate polynomial, we will try to use the Kohel’s algorithms.

EXAMPLES:

This helper function will be implicitly called by the following examples:

sage: R.<x> = GF(5)[]
sage: E = EllipticCurve(GF(5), [0,0,0,1,0])


We can construct the same isogeny from a kernel polynomial:

sage: phi = EllipticCurveIsogeny(E, x+3)


or from a list of coefficients of a kernel polynomial:

sage: phi == EllipticCurveIsogeny(E, [3,1])
True


or from a rational point which generates the kernel:

sage: phi == EllipticCurveIsogeny(E,  E((2,0)) )
True


In the first two cases, Kohel’s algorithm will be used, while in the third case it is Velu:

sage: from sage.schemes.elliptic_curves.ell_curve_isogeny import isogeny_determine_algorithm
sage: isogeny_determine_algorithm(E, x+3)
'kohel'
sage: isogeny_determine_algorithm(E, [3, 1])
'kohel'
sage: isogeny_determine_algorithm(E, E((2,0)))
'velu'

sage.schemes.elliptic_curves.ell_curve_isogeny.split_kernel_polynomial(poly)

Internal helper function for compute_isogeny_kernel_polynomial.

INPUT:

• poly – a nonzero univariate polynomial.

OUTPUT:

The maximum separable divisor of poly. If the input is a full kernel polynomial where the roots which are $$x$$-coordinates of points of order greater than 2 have multiplicity 1, the output will be a polynomial with the same roots, all of multiplicity 1.

EXAMPLES:

The following example implicitly exercises this function:

sage: E = EllipticCurve(GF(37), [0,0,0,1,8])
sage: R.<x> = GF(37)[]
sage: f = (x + 10) * (x + 12) * (x + 16)
sage: phi = EllipticCurveIsogeny(E, f)
sage: E2 = phi.codomain()
sage: from sage.schemes.elliptic_curves.ell_curve_isogeny import compute_isogeny_starks
sage: from sage.schemes.elliptic_curves.ell_curve_isogeny import split_kernel_polynomial
sage: ker_poly = compute_isogeny_starks(E, E2, 7); ker_poly
x^6 + 2*x^5 + 20*x^4 + 11*x^3 + 36*x^2 + 35*x + 16
sage: ker_poly.factor()
(x + 10)^2 * (x + 12)^2 * (x + 16)^2
sage: poly = split_kernel_polynomial(ker_poly); poly
x^3 + x^2 + 28*x + 33
sage: poly.factor()
(x + 10) * (x + 12) * (x + 16)

sage.schemes.elliptic_curves.ell_curve_isogeny.two_torsion_part(E, psi)

Return the greatest common divisor of psi and the 2 torsion polynomial of $$E$$.

INPUT:

• E – an elliptic curve

• psi – a univariate polynomial over the base field of E

OUTPUT:

(polynomial) the gcd of psi and the 2-torsion polynomial of E.

EXAMPLES:

Every function that computes the kernel polynomial via Kohel’s formulas will call this function:

sage: E = EllipticCurve(GF(19), [1,2,3,4,5])
sage: R.<x> = GF(19)[]
sage: phi = EllipticCurveIsogeny(E, x + 13)
sage: isogeny_codomain_from_kernel(E, x + 13) == phi.codomain()
True
sage: from sage.schemes.elliptic_curves.ell_curve_isogeny import two_torsion_part
sage: two_torsion_part(E, x+13)
x + 13

sage.schemes.elliptic_curves.ell_curve_isogeny.unfill_isogeny_matrix(M)

Reverses the action of fill_isogeny_matrix.

INPUT:

• M – a square symmetric matrix of integers.

OUTPUT:

(matrix) a square symmetric matrix obtained from M by replacing non-prime entries with $$0$$.

EXAMPLES:

sage: M = Matrix([[0, 2, 3, 3, 0, 0], [2, 0, 0, 0, 3, 3], [3, 0, 0, 0, 2, 0], [3, 0, 0, 0, 0, 2], [0, 3, 2, 0, 0, 0], [0, 3, 0, 2, 0, 0]]); M
[0 2 3 3 0 0]
[2 0 0 0 3 3]
[3 0 0 0 2 0]
[3 0 0 0 0 2]
[0 3 2 0 0 0]
[0 3 0 2 0 0]
sage: from sage.schemes.elliptic_curves.ell_curve_isogeny import fill_isogeny_matrix, unfill_isogeny_matrix
sage: M1 = fill_isogeny_matrix(M); M1
[ 1  2  3  3  6  6]
[ 2  1  6  6  3  3]
[ 3  6  1  9  2 18]
[ 3  6  9  1 18  2]
[ 6  3  2 18  1  9]
[ 6  3 18  2  9  1]
sage: unfill_isogeny_matrix(M1)
[0 2 3 3 0 0]
[2 0 0 0 3 3]
[3 0 0 0 2 0]
[3 0 0 0 0 2]
[0 3 2 0 0 0]
[0 3 0 2 0 0]
sage: unfill_isogeny_matrix(M1) == M
True