Frobenius isogenies of elliptic curves¶

Frobenius isogenies only exist in positive characteristic \(p\). They are given by \(\pi_n:(x,y)\mapsto (x^{p^n},y^{p^n})\).

This class implements \(\pi_n\) for \(n \geq 0\). Together with existing tools for composing isogenies (see EllipticCurveHom_composite), we can therefore represent arbitrary inseparable isogenies in Sage.

EXAMPLES:

Constructing a Frobenius isogeny is straightforward:

Sage

sage: from sage.schemes.elliptic_curves.hom_frobenius import EllipticCurveHom_frobenius
sage: z5, = GF(17^5).gens()
sage: E = EllipticCurve([z5,1])
sage: pi = EllipticCurveHom_frobenius(E); pi
Frobenius isogeny of degree 17:
  From: Elliptic Curve defined by y^2 = x^3 + z5*x + 1
        over Finite Field in z5 of size 17^5
  To:   Elliptic Curve defined by y^2 = x^3 + (9*z5^4+7*z5^3+10*z5^2+z5+14)*x + 1
        over Finite Field in z5 of size 17^5

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.hom_frobenius import EllipticCurveHom_frobenius
>>> z5, = GF(Integer(17)**Integer(5)).gens()
>>> E = EllipticCurve([z5,Integer(1)])
>>> pi = EllipticCurveHom_frobenius(E); pi
Frobenius isogeny of degree 17:
  From: Elliptic Curve defined by y^2 = x^3 + z5*x + 1
        over Finite Field in z5 of size 17^5
  To:   Elliptic Curve defined by y^2 = x^3 + (9*z5^4+7*z5^3+10*z5^2+z5+14)*x + 1
        over Finite Field in z5 of size 17^5

Sage Live

from sage.schemes.elliptic_curves.hom_frobenius import EllipticCurveHom_frobenius
z5, = GF(17^5).gens()
E = EllipticCurve([z5,1])
pi = EllipticCurveHom_frobenius(E); pi

By passing \(n\), we can also construct higher-power Frobenius maps, such as the Frobenius endomorphism:

Sage

sage: z5, = GF(7^5).gens()
sage: E = EllipticCurve([z5,1])
sage: pi = EllipticCurveHom_frobenius(E, 5); pi
Frobenius endomorphism of degree 16807 = 7^5:
  From: Elliptic Curve defined by y^2 = x^3 + z5*x + 1
        over Finite Field in z5 of size 7^5
  To:   Elliptic Curve defined by y^2 = x^3 + z5*x + 1
        over Finite Field in z5 of size 7^5

Python

>>> from sage.all import *
>>> z5, = GF(Integer(7)**Integer(5)).gens()
>>> E = EllipticCurve([z5,Integer(1)])
>>> pi = EllipticCurveHom_frobenius(E, Integer(5)); pi
Frobenius endomorphism of degree 16807 = 7^5:
  From: Elliptic Curve defined by y^2 = x^3 + z5*x + 1
        over Finite Field in z5 of size 7^5
  To:   Elliptic Curve defined by y^2 = x^3 + z5*x + 1
        over Finite Field in z5 of size 7^5

Sage Live

z5, = GF(7^5).gens()
E = EllipticCurve([z5,1])
pi = EllipticCurveHom_frobenius(E, 5); pi

The usual EllipticCurveHom methods are supported:

Sage

sage: z5, = GF(7^5).gens()
sage: E = EllipticCurve([z5,1])
sage: pi = EllipticCurveHom_frobenius(E,5)
sage: pi.degree()
16807
sage: pi.rational_maps()
(x^16807, y^16807)
sage: pi.formal()                   # known bug
...
sage: pi.is_normalized()            # known bug
...
sage: pi.is_separable()
False
sage: pi.is_injective()
True
sage: pi.is_surjective()
True

Python

>>> from sage.all import *
>>> z5, = GF(Integer(7)**Integer(5)).gens()
>>> E = EllipticCurve([z5,Integer(1)])
>>> pi = EllipticCurveHom_frobenius(E,Integer(5))
>>> pi.degree()
16807
>>> pi.rational_maps()
(x^16807, y^16807)
>>> pi.formal()                   # known bug
...
>>> pi.is_normalized()            # known bug
...
>>> pi.is_separable()
False
>>> pi.is_injective()
True
>>> pi.is_surjective()
True

Sage Live

z5, = GF(7^5).gens()
E = EllipticCurve([z5,1])
pi = EllipticCurveHom_frobenius(E,5)
pi.degree()
pi.rational_maps()
pi.formal()                   # known bug
pi.is_normalized()            # known bug
pi.is_separable()
pi.is_injective()
pi.is_surjective()

Computing the dual of Frobenius is supported as well:

Sage

sage: E = EllipticCurve([GF(17^6).gen(), 0])
sage: pi = EllipticCurveHom_frobenius(E)
sage: pihat = pi.dual(); pihat
Isogeny of degree 17
 from Elliptic Curve defined by y^2 = x^3 + (15*z6^5+5*z6^4+8*z6^3+12*z6^2+11*z6+7)*x
      over Finite Field in z6 of size 17^6
   to Elliptic Curve defined by y^2 = x^3 + z6*x
      over Finite Field in z6 of size 17^6
sage: pihat.is_separable()
True
sage: pihat * pi == EllipticCurveHom_scalar(E,17)   # known bug -- #6413
True

Python

>>> from sage.all import *
>>> E = EllipticCurve([GF(Integer(17)**Integer(6)).gen(), Integer(0)])
>>> pi = EllipticCurveHom_frobenius(E)
>>> pihat = pi.dual(); pihat
Isogeny of degree 17
 from Elliptic Curve defined by y^2 = x^3 + (15*z6^5+5*z6^4+8*z6^3+12*z6^2+11*z6+7)*x
      over Finite Field in z6 of size 17^6
   to Elliptic Curve defined by y^2 = x^3 + z6*x
      over Finite Field in z6 of size 17^6
>>> pihat.is_separable()
True
>>> pihat * pi == EllipticCurveHom_scalar(E,Integer(17))   # known bug -- #6413
True

Sage Live

E = EllipticCurve([GF(17^6).gen(), 0])
pi = EllipticCurveHom_frobenius(E)
pihat = pi.dual(); pihat
pihat.is_separable()
pihat * pi == EllipticCurveHom_scalar(E,17)   # known bug -- #6413

A supersingular example (with purely inseparable dual):

Sage

sage: E = EllipticCurve([0, GF(17^6).gen()])
sage: E.is_supersingular()
True
sage: pi1 = EllipticCurveHom_frobenius(E)
sage: pi1hat = pi1.dual(); pi1hat
Composite morphism of degree 17 = 17*1:
  From: Elliptic Curve defined by y^2 = x^3 + (15*z6^5+5*z6^4+8*z6^3+12*z6^2+11*z6+7)
        over Finite Field in z6 of size 17^6
  To:   Elliptic Curve defined by y^2 = x^3 + z6
        over Finite Field in z6 of size 17^6
sage: pi6 = EllipticCurveHom_frobenius(E,6)
sage: pi6hat = pi6.dual(); pi6hat
Composite morphism of degree 24137569 = 24137569*1:
  From: Elliptic Curve defined by y^2 = x^3 + z6
        over Finite Field in z6 of size 17^6
  To:   Elliptic Curve defined by y^2 = x^3 + z6
        over Finite Field in z6 of size 17^6
sage: pi6hat.factors()
(Frobenius endomorphism of degree 24137569 = 17^6:
   From: Elliptic Curve defined by y^2 = x^3 + z6
         over Finite Field in z6 of size 17^6
   To:   Elliptic Curve defined by y^2 = x^3 + z6
         over Finite Field in z6 of size 17^6,
 Elliptic-curve endomorphism of
  Elliptic Curve defined by y^2 = x^3 + z6 over Finite Field in z6 of size 17^6
   Via:  (u,r,s,t) = (2*z6^5 + 10*z6^3 + z6^2 + 8, 0, 0, 0))

Python

>>> from sage.all import *
>>> E = EllipticCurve([Integer(0), GF(Integer(17)**Integer(6)).gen()])
>>> E.is_supersingular()
True
>>> pi1 = EllipticCurveHom_frobenius(E)
>>> pi1hat = pi1.dual(); pi1hat
Composite morphism of degree 17 = 17*1:
  From: Elliptic Curve defined by y^2 = x^3 + (15*z6^5+5*z6^4+8*z6^3+12*z6^2+11*z6+7)
        over Finite Field in z6 of size 17^6
  To:   Elliptic Curve defined by y^2 = x^3 + z6
        over Finite Field in z6 of size 17^6
>>> pi6 = EllipticCurveHom_frobenius(E,Integer(6))
>>> pi6hat = pi6.dual(); pi6hat
Composite morphism of degree 24137569 = 24137569*1:
  From: Elliptic Curve defined by y^2 = x^3 + z6
        over Finite Field in z6 of size 17^6
  To:   Elliptic Curve defined by y^2 = x^3 + z6
        over Finite Field in z6 of size 17^6
>>> pi6hat.factors()
(Frobenius endomorphism of degree 24137569 = 17^6:
   From: Elliptic Curve defined by y^2 = x^3 + z6
         over Finite Field in z6 of size 17^6
   To:   Elliptic Curve defined by y^2 = x^3 + z6
         over Finite Field in z6 of size 17^6,
 Elliptic-curve endomorphism of
  Elliptic Curve defined by y^2 = x^3 + z6 over Finite Field in z6 of size 17^6
   Via:  (u,r,s,t) = (2*z6^5 + 10*z6^3 + z6^2 + 8, 0, 0, 0))

Sage Live

E = EllipticCurve([0, GF(17^6).gen()])
E.is_supersingular()
pi1 = EllipticCurveHom_frobenius(E)
pi1hat = pi1.dual(); pi1hat
pi6 = EllipticCurveHom_frobenius(E,6)
pi6hat = pi6.dual(); pi6hat
pi6hat.factors()

AUTHORS:

Lorenz Panny (2021): implement EllipticCurveHom_frobenius
Mickaël Montessinos (2021): computing the dual of a Frobenius isogeny

class sage.schemes.elliptic_curves.hom_frobenius.EllipticCurveHom_frobenius(E, power=1)[source]¶

Bases: EllipticCurveHom

Construct a Frobenius isogeny on a given curve with a given power of the base-ring characteristic.

Writing \(n\) for the parameter power (default: \(1\)), the isogeny is defined by \((x,y) \to (x^{p^n}, y^{p^n})\) where \(p\) is the characteristic of the base ring.

EXAMPLES:

Sage

sage: from sage.schemes.elliptic_curves.hom_frobenius import EllipticCurveHom_frobenius
sage: E = EllipticCurve(j=GF(11^2).gen())
sage: EllipticCurveHom_frobenius(E)
Frobenius isogeny of degree 11:
  From: Elliptic Curve defined by y^2 = x^3 + (2*z2+6)*x + (8*z2+8) over Finite Field in z2 of size 11^2
  To:   Elliptic Curve defined by y^2 = x^3 + (9*z2+3)*x + (3*z2+7) over Finite Field in z2 of size 11^2
sage: EllipticCurveHom_frobenius(E, 2)
Frobenius endomorphism of degree 121 = 11^2:
  From: Elliptic Curve defined by y^2 = x^3 + (2*z2+6)*x + (8*z2+8) over Finite Field in z2 of size 11^2
  To:   Elliptic Curve defined by y^2 = x^3 + (2*z2+6)*x + (8*z2+8) over Finite Field in z2 of size 11^2

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.hom_frobenius import EllipticCurveHom_frobenius
>>> E = EllipticCurve(j=GF(Integer(11)**Integer(2)).gen())
>>> EllipticCurveHom_frobenius(E)
Frobenius isogeny of degree 11:
  From: Elliptic Curve defined by y^2 = x^3 + (2*z2+6)*x + (8*z2+8) over Finite Field in z2 of size 11^2
  To:   Elliptic Curve defined by y^2 = x^3 + (9*z2+3)*x + (3*z2+7) over Finite Field in z2 of size 11^2
>>> EllipticCurveHom_frobenius(E, Integer(2))
Frobenius endomorphism of degree 121 = 11^2:
  From: Elliptic Curve defined by y^2 = x^3 + (2*z2+6)*x + (8*z2+8) over Finite Field in z2 of size 11^2
  To:   Elliptic Curve defined by y^2 = x^3 + (2*z2+6)*x + (8*z2+8) over Finite Field in z2 of size 11^2

Sage Live

from sage.schemes.elliptic_curves.hom_frobenius import EllipticCurveHom_frobenius
E = EllipticCurve(j=GF(11^2).gen())
EllipticCurveHom_frobenius(E)
EllipticCurveHom_frobenius(E, 2)

dual()[source]¶

Compute the dual of this Frobenius isogeny.

This method returns an EllipticCurveHom object.

EXAMPLES:

An ordinary example:

Sage

sage: from sage.schemes.elliptic_curves.hom_scalar import EllipticCurveHom_scalar
sage: from sage.schemes.elliptic_curves.hom_frobenius import EllipticCurveHom_frobenius
sage: E = EllipticCurve(GF(31), [0,1])
sage: f = EllipticCurveHom_frobenius(E)
sage: f.dual() * f == EllipticCurveHom_scalar(f.domain(), 31)
True
sage: f * f.dual() == EllipticCurveHom_scalar(f.codomain(), 31)
True

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.hom_scalar import EllipticCurveHom_scalar
>>> from sage.schemes.elliptic_curves.hom_frobenius import EllipticCurveHom_frobenius
>>> E = EllipticCurve(GF(Integer(31)), [Integer(0),Integer(1)])
>>> f = EllipticCurveHom_frobenius(E)
>>> f.dual() * f == EllipticCurveHom_scalar(f.domain(), Integer(31))
True
>>> f * f.dual() == EllipticCurveHom_scalar(f.codomain(), Integer(31))
True

Sage Live

from sage.schemes.elliptic_curves.hom_scalar import EllipticCurveHom_scalar
from sage.schemes.elliptic_curves.hom_frobenius import EllipticCurveHom_frobenius
E = EllipticCurve(GF(31), [0,1])
f = EllipticCurveHom_frobenius(E)
f.dual() * f == EllipticCurveHom_scalar(f.domain(), 31)
f * f.dual() == EllipticCurveHom_scalar(f.codomain(), 31)

A supersingular example:

Sage

sage: E = EllipticCurve(GF(31), [1,0])
sage: f = EllipticCurveHom_frobenius(E)
sage: f.dual() * f == EllipticCurveHom_scalar(f.domain(), 31)
True
sage: f * f.dual() == EllipticCurveHom_scalar(f.codomain(), 31)
True

Python

>>> from sage.all import *
>>> E = EllipticCurve(GF(Integer(31)), [Integer(1),Integer(0)])
>>> f = EllipticCurveHom_frobenius(E)
>>> f.dual() * f == EllipticCurveHom_scalar(f.domain(), Integer(31))
True
>>> f * f.dual() == EllipticCurveHom_scalar(f.codomain(), Integer(31))
True

Sage Live

E = EllipticCurve(GF(31), [1,0])
f = EllipticCurveHom_frobenius(E)
f.dual() * f == EllipticCurveHom_scalar(f.domain(), 31)
f * f.dual() == EllipticCurveHom_scalar(f.codomain(), 31)

ALGORITHM:

For supersingular curves, the dual of Frobenius is again purely inseparable, so we start out with a Frobenius isogeny of equal degree in the opposite direction.
For ordinary curves, we immediately reduce to the case of prime degree. The kernel of the dual is the unique subgroup of size \(p\), which we compute from the \(p\)-division polynomial.

In both cases, we then search for the correct post-isomorphism using find_post_isomorphism().

inseparable_degree()[source]¶

Return the inseparable degree of this Frobenius isogeny.

Since this class implements only purely inseparable isogenies, the inseparable degree equals the degree.

EXAMPLES:

Sage

sage: from sage.schemes.elliptic_curves.hom_frobenius import EllipticCurveHom_frobenius
sage: E = EllipticCurve(GF(11), [1,1])
sage: pi = EllipticCurveHom_frobenius(E, 4)
sage: pi.inseparable_degree()
14641
sage: pi.inseparable_degree() == pi.degree()
True

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.hom_frobenius import EllipticCurveHom_frobenius
>>> E = EllipticCurve(GF(Integer(11)), [Integer(1),Integer(1)])
>>> pi = EllipticCurveHom_frobenius(E, Integer(4))
>>> pi.inseparable_degree()
14641
>>> pi.inseparable_degree() == pi.degree()
True

Sage Live

from sage.schemes.elliptic_curves.hom_frobenius import EllipticCurveHom_frobenius
E = EllipticCurve(GF(11), [1,1])
pi = EllipticCurveHom_frobenius(E, 4)
pi.inseparable_degree()
pi.inseparable_degree() == pi.degree()

kernel_polynomial()[source]¶

Return the kernel polynomial of this Frobenius isogeny as a polynomial in \(x\). This method always returns \(1\).

EXAMPLES:

Sage

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

Python

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

Sage Live

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

rational_maps()[source]¶

Return the explicit rational maps defining this Frobenius isogeny as (sparse) bivariate rational maps in \(x\) and \(y\).

EXAMPLES:

Sage

sage: from sage.schemes.elliptic_curves.hom_frobenius import EllipticCurveHom_frobenius
sage: E = EllipticCurve(GF(11), [1,1])
sage: pi = EllipticCurveHom_frobenius(E, 4)
sage: pi.rational_maps()
(x^14641, y^14641)

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.hom_frobenius import EllipticCurveHom_frobenius
>>> E = EllipticCurve(GF(Integer(11)), [Integer(1),Integer(1)])
>>> pi = EllipticCurveHom_frobenius(E, Integer(4))
>>> pi.rational_maps()
(x^14641, y^14641)

Sage Live

from sage.schemes.elliptic_curves.hom_frobenius import EllipticCurveHom_frobenius
E = EllipticCurve(GF(11), [1,1])
pi = EllipticCurveHom_frobenius(E, 4)
pi.rational_maps()

scaling_factor()[source]¶

Return the Weierstrass scaling factor associated to this Frobenius morphism.

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

EXAMPLES:

Sage

sage: from sage.schemes.elliptic_curves.hom_frobenius import EllipticCurveHom_frobenius
sage: E = EllipticCurve(GF(11), [1,1])
sage: pi = EllipticCurveHom_frobenius(E)
sage: pi.formal()
t^11 + O(t^33)
sage: pi.scaling_factor()
0
sage: pi = EllipticCurveHom_frobenius(E, 3)
sage: pi.formal()
t^1331 + O(t^1353)
sage: pi.scaling_factor()
0
sage: pi = EllipticCurveHom_frobenius(E, 0)
sage: pi == E.scalar_multiplication(1)
True
sage: pi.scaling_factor()
1

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.hom_frobenius import EllipticCurveHom_frobenius
>>> E = EllipticCurve(GF(Integer(11)), [Integer(1),Integer(1)])
>>> pi = EllipticCurveHom_frobenius(E)
>>> pi.formal()
t^11 + O(t^33)
>>> pi.scaling_factor()
0
>>> pi = EllipticCurveHom_frobenius(E, Integer(3))
>>> pi.formal()
t^1331 + O(t^1353)
>>> pi.scaling_factor()
0
>>> pi = EllipticCurveHom_frobenius(E, Integer(0))
>>> pi == E.scalar_multiplication(Integer(1))
True
>>> pi.scaling_factor()
1

Sage Live

from sage.schemes.elliptic_curves.hom_frobenius import EllipticCurveHom_frobenius
E = EllipticCurve(GF(11), [1,1])
pi = EllipticCurveHom_frobenius(E)
pi.formal()
pi.scaling_factor()
pi = EllipticCurveHom_frobenius(E, 3)
pi.formal()
pi.scaling_factor()
pi = EllipticCurveHom_frobenius(E, 0)
pi == E.scalar_multiplication(1)
pi.scaling_factor()

The scaling factor lives in the base ring:

Sage

sage: pi.scaling_factor().parent()
Finite Field of size 11

Python

>>> from sage.all import *
>>> pi.scaling_factor().parent()
Finite Field of size 11

Sage Live

pi.scaling_factor().parent()

ALGORITHM: Inseparable isogenies of degree \(>1\) have scaling factor \(0\).

x_rational_map()[source]¶

Return the \(x\)-coordinate rational map of this Frobenius isogeny as a (sparse) univariate rational map in \(x\).

EXAMPLES:

Sage

sage: from sage.schemes.elliptic_curves.hom_frobenius import EllipticCurveHom_frobenius
sage: E = EllipticCurve(GF(11), [1,1])
sage: pi = EllipticCurveHom_frobenius(E, 4)
sage: pi.x_rational_map()
x^14641

Python

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.hom_frobenius import EllipticCurveHom_frobenius
>>> E = EllipticCurve(GF(Integer(11)), [Integer(1),Integer(1)])
>>> pi = EllipticCurveHom_frobenius(E, Integer(4))
>>> pi.x_rational_map()
x^14641

Sage Live

from sage.schemes.elliptic_curves.hom_frobenius import EllipticCurveHom_frobenius
E = EllipticCurve(GF(11), [1,1])
pi = EllipticCurveHom_frobenius(E, 4)
pi.x_rational_map()