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 methods directly accessible from an elliptic curve E over a field are isogeny() and isogeny_codomain().

The most useful methods that apply to isogenies are:

.domain()
.codomain()
degree()
dual()
rational_maps()
kernel_polynomial()

Warning

This class only implements separable isogenies. When using Kohel’s algorithm, only cyclic isogenies can be computed (except for \([2]\)).

Working with other kinds of isogenies may be possible using other child classes of EllipticCurveHom.

Some algorithms may need the isogeny to be normalized.

AUTHORS:

Daniel Shumow <shumow@gmail.com>: 2009-04-19: initial version
Chris Wuthrich: 7/09: 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.
Lorenz Panny (2022-04): major cleanup of code and documentation
Lorenz Panny (2022): inseparable duals

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

Bases: EllipticCurveHom

This class implements separable isogenies of elliptic curves.

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

Vélu’s Formulas: Vélu’s original formulas for computing isogenies. This algorithm is selected by giving as the kernel parameter a single point, or a list of points, generating 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) which will define the kernel of the isogeny. Kohel’s algorithm is currently only implemented for cyclic isogenies, with the exception of \([2]\).

INPUT:

E – an elliptic curve, the domain of the isogeny to initialize.
kernel – a kernel: either a point on E, a list of points on E, a monic kernel polynomial, or None. If initializing from a domain/codomain, this must be None.
codomain – an elliptic curve (default: None).
- If kernel is None, then degree must be given as well and the given codomain must be the codomain of a cyclic, separable, normalized isogeny of the given degree.
- If kernel is not None, then this must be isomorphic to the codomain of the separable isogeny defined by kernel; in this case, the isogeny is post-composed with an isomorphism so that the codomain equals the given curve.
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 given kernel polynomial with the two-torsion polynomial of E.
model – a string (default: None). Supported values (cf. compute_model()):
- "minimal": If E is a curve over the rationals or over a number field, then the codomain is a global minimal model where this exists.
- "short_weierstrass": The codomain is a short Weierstrass curve, assuming one exists.
- "montgomery": The codomain is an (untwisted) Montgomery curve, assuming one exists over this field.
check (default: True) – check whether the input is valid. Setting this to False can lead to significant speedups.

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: phi_k = EllipticCurveIsogeny(E, phi_ker_poly)
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 whose kernel equals the full 2-torsion:

sage: E = EllipticCurve(GF((3,2)), [0,0,0,1,1])
sage: ker_poly = E.division_polynomial(2)
sage: phi = EllipticCurveIsogeny(E, ker_poly); phi
Isogeny of degree 4 from Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field in z2 of size 3^2 to Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field in z2 of size 3^2
sage: P1,P2,P3 = filter(bool, E(0).division_points(2))
sage: phi(P1)
(0 : 1 : 0)
sage: phi(P2)
(0 : 1 : 0)
sage: phi(P3)
(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, [1]); 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

Vélu 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: phi_k = EllipticCurveIsogeny(E, phi_ker_poly); 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: phi_k = EllipticCurveIsogeny(E, ker_poly); 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: phi_k = EllipticCurveIsogeny(EK, ker_poly)
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. The non-cyclic multiplication-by-\(3\) isogeny won’t be found:

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

Non-normalized isogeny also won’t be found:

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 = isom * phi
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)))

dual()#

Return the isogeny dual to this isogeny.

Note

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

Inseparable duals should be computed correctly:

sage: z2 = GF(71^2).gen()
sage: E = EllipticCurve(j=57*z2+51)
sage: E.isogeny(3*E.lift_x(0)).dual()
Composite morphism of degree 71 = 71*1^2:
  From: Elliptic Curve defined by y^2 = x^3 + (32*z2+67)*x + (24*z2+37) over Finite Field in z2 of size 71^2
  To:   Elliptic Curve defined by y^2 = x^3 + (41*z2+56)*x + (18*z2+42) over Finite Field in z2 of size 71^2
sage: E.isogeny(E.lift_x(0)).dual()
Composite morphism of degree 213 = 71*3:
  From: Elliptic Curve defined by y^2 = x^3 + (58*z2+31)*x + (34*z2+58) over Finite Field in z2 of size 71^2
  To:   Elliptic Curve defined by y^2 = x^3 + (41*z2+56)*x + (18*z2+42) over Finite Field in z2 of size 71^2

…even if pre- or post-isomorphisms are present:

sage: from sage.schemes.elliptic_curves.weierstrass_morphism import WeierstrassIsomorphism
sage: phi = E.isogeny(E.lift_x(0))
sage: pre = ~WeierstrassIsomorphism(phi.domain(), (z2,2,3,4))
sage: post = WeierstrassIsomorphism(phi.codomain(), (5,6,7,8))
sage: phi = post * phi * pre
sage: phi.dual()
Composite morphism of degree 213 = 71*3:
  From: Elliptic Curve defined by y^2 + 17*x*y + 45*y = x^3 + 30*x^2 + (6*z2+64)*x + (48*z2+65) over Finite Field in z2 of size 71^2
  To:   Elliptic Curve defined by y^2 + (60*z2+22)*x*y + (69*z2+37)*y = x^3 + (32*z2+48)*x^2 + (19*z2+58)*x + (56*z2+22) over Finite Field in z2 of size 71^2

is_separable()#

Determine whether or not this isogeny is separable.

Since EllipticCurveIsogeny only implements separable isogenies, this method always returns True.

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

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, [1])
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

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, [1])
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)

scaling_factor()#

Return the Weierstrass scaling factor associated to this elliptic-curve isogeny.

The scaling factor is the constant \(u\) (in the base field) such that \(\varphi^* \omega_2 = u \omega_1\), where \(\varphi: E_1\to E_2\) is this isogeny and \(\omega_i\) are the standard Weierstrass differentials on \(E_i\) defined by \(\mathrm dx/(2y+a_1x+a_3)\).

EXAMPLES:

sage: E = EllipticCurve(GF(257^2), [0,1])
sage: phi = E.isogeny(E.lift_x(240))
sage: phi.degree()
43
sage: phi.scaling_factor()
1
sage: phi.dual().scaling_factor()
43

ALGORITHM: The “inner” isogeny is normalized by construction, so we only need to account for the scaling factors of a pre- and post-isomorphism.

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, [1])
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 Vélu/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 invocation of the EllipticCurveIsogeny constructor:

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)#

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

INPUT:

E – domain elliptic curve
kernel (polynomial or list) – the kernel polynomial, or a list of its coefficients

OUTPUT:

(elliptic curve) The codomain elliptic curve of the isogeny defined by 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])
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)
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)
Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 3*x + 15 over Finite Field of size 19

ALGORITHM:

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 to compute \(\wp\) functions from the short Weierstrass model more easily.

Warning

The base field must be of characteristic not equal to \(2\) or \(3\).

INPUT:

E1, E2 – elliptic curves

OUTPUT:

A tuple (pre_isomorphism, post_isomorphism, intermediate_domain, intermediate_codomain) where:

intermediate_domain is a short Weierstrass curve isomorphic to E1;
intermediate_codomain is a short Weierstrass curve isomorphic to E2;
pre_isomorphism is a normalized isomorphism from E1 to intermediate_domain;
post_isomorphism is a 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,
 Elliptic-curve morphism:
  From: Elliptic Curve defined by y^2 + x*y + y = x^3 + x over Finite Field of size 83
  To:   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),
 Elliptic-curve morphism:
  From: Elliptic Curve defined by y^2 = x^3 + 65*x + 69 over Finite Field of size 83
  To:   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,
 Elliptic-curve endomorphism of 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),
 Elliptic-curve endomorphism of 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 from E1 to E2.

INPUT:

E1 – domain elliptic curve in short Weierstrass form
E2 – codomain elliptic curve in short Weierstrass form
ell – the degree of an isogeny from E1 to E2
algorithm – currently only "starks" (default) is implemented

OUTPUT:

The kernel polynomial of an isogeny from E1 to E2.

Note

If there is no degree-ell, cyclic, separable, normalized isogeny from E1 to E2, a ValueError 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 polynomial of an isogeny of degree ell from E1 to E2.

INPUT:

E1 – domain elliptic curve in short Weierstrass form
E2 – codomain elliptic curve in short Weierstrass form
ell – the degree of an isogeny from E1 to E2

OUTPUT:

The kernel polynomial of an isogeny from E1 to E2.

Note

If there is no degree-ell, cyclic, separable, normalized isogeny from E1 to E2, a ValueError 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:

A tuple (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 normalized isomorphism from E1 to E1pr;
post_isom is a normalized 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)
(Elliptic-curve morphism:
  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 = x^3 - 31/3*x - 2501/108 over Rational Field
  Via:  (u,r,s,t) = (1, 1/3, 0, -1/2),
 Elliptic-curve morphism:
  From: Elliptic Curve defined by y^2 = x^3 - 23461/3*x - 28748141/108 over Rational Field
  To:   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)
(Elliptic-curve endomorphism of 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),
 Elliptic-curve endomorphism of 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)
(Elliptic-curve morphism:
  From: Elliptic Curve defined by y^2 + x*y + y = x^3 + x over Finite Field of size 97
  To:   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),
 Elliptic-curve morphism:
  From: Elliptic Curve defined by y^2 = x^3 + 41*x + 66 over Finite Field of size 97
  To:   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 Vélu’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) Vélu’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.a_invariants()
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 Vélu’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) Vélu’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 Vélu’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 (integer) – the degree

OUTPUT:

(tuple) Vélu’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 \(0\)

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 – domain elliptic curve
kernel – either a list of points in the kernel of the isogeny,
or a kernel polynomial (specified as either a univariate polynomial or a coefficient list)

OUTPUT:

(elliptic curve) The codomain of the separable normalized isogeny defined by 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])
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])
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.split_kernel_polynomial(poly)#

Obsolete internal helper function formerly used by compute_isogeny_kernel_polynomial().

Use radical() instead.

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:

Check that this behaves identically to .radical():

sage: from sage.schemes.elliptic_curves.ell_curve_isogeny import split_kernel_polynomial
sage: q = next_prime(randrange(3,10^3))
sage: e = randrange(1,5)
sage: R = GF(q^e,'a')['x']
sage: f = R.random_element(randrange(10,100)).monic()
sage: split_kernel_polynomial(f) == f.radical()
doctest:warning ...
DeprecationWarning: ...
True

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