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compute duals of Îlu isogenies correctly in all cases #41538

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compute duals of Îlu isogenies correctly in all cases #41538

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62 changes: 60 additions & 2 deletions src/sage/schemes/elliptic_curves/hom_velusqrt.py
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Original file line number Diff line number Diff line change
Expand Up @@ -1072,6 +1072,39 @@ def kernel_polynomial(self):
return R(h).monic()

@cached_method
def as_EllipticCurveIsogeny(self):
r"""
Return the mathematically identical isogeny represented as a
:class:`EllipticCurveIsogeny` object.

.. NOTE::

The result is computed by :class:`EllipticCurveIsogeny`,
hence it obviously does not benefit from the square-root
Vélu speedup.

EXAMPLES::

sage: E = EllipticCurve(GF(101^2), [1, 1, 1, 1, 1])
sage: K = E.cardinality() // 11 * E.gens()[0]
sage: phi = E.isogeny(K, algorithm='velusqrt'); phi
Elliptic-curve isogeny (using square-root Vélu) of degree 11:
From: Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 + x + 1 over Finite Field in z2 of size 101^2
To: Elliptic Curve defined by y^2 = x^3 + 39*x + 40 over Finite Field in z2 of size 101^2
sage: psi = phi.as_EllipticCurveIsogeny(); psi
Isogeny of degree 11
from Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 + x + 1 over Finite Field in z2 of size 101^2
to Elliptic Curve defined by y^2 = x^3 + 39*x + 40 over Finite Field in z2 of size 101^2
sage: phi == psi
True
"""
ker = self.kernel_polynomial()
phi = self.domain().isogeny(ker, degree=self.degree(), codomain=self.codomain(), check=False)
from sage.schemes.elliptic_curves.hom import find_post_isomorphism
iso = find_post_isomorphism(self, phi)
return iso * phi

# not explicitly cached here since .as_EllipticCurveIsogeny() and EllipticCurveIsogeny.dual() already cache their results
def dual(self):
r"""
Return the dual of this square-root Vélu
Expand All @@ -1082,6 +1115,10 @@ def dual(self):
The dual is computed by :class:`EllipticCurveIsogeny`,
hence it does not benefit from the square-root Vélu speedup.

ALGORITHM: In the separable case, similar to :meth:`EllipticCurveIsogeny.dual`.
In the inseparable case, converts to an :class:`EllipticCurveIsogeny` using
:meth:`as_EllipticCurveIsogeny`, then runs :meth:`EllipticCurveIsogeny.dual`.

EXAMPLES::

sage: E = EllipticCurve(GF(101^2), [1, 1, 1, 1, 1])
Expand All @@ -1091,13 +1128,34 @@ def dual(self):
From: Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 + x + 1 over Finite Field in z2 of size 101^2
To: Elliptic Curve defined by y^2 = x^3 + 39*x + 40 over Finite Field in z2 of size 101^2
sage: phi.dual()
Isogeny of degree 11 from Elliptic Curve defined by y^2 = x^3 + 39*x + 40 over Finite Field in z2 of size 101^2 to Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 + x + 1 over Finite Field in z2 of size 101^2
Isogeny of degree 11
from Elliptic Curve defined by y^2 = x^3 + 39*x + 40 over Finite Field in z2 of size 101^2
to Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 + x + 1 over Finite Field in z2 of size 101^2
sage: phi.dual() * phi == phi.domain().scalar_multiplication(11)
True
sage: phi * phi.dual() == phi.codomain().scalar_multiplication(11)
True

Inseparable duals are computed correctly::

sage: # needs sage.rings.finite_rings
sage: z2 = GF(71^2).gen()
sage: E = EllipticCurve(j=57*z2+51)
sage: E.isogeny(3*E.lift_x(0), algorithm='velusqrt').dual()
Composite morphism of degree 71 = 71*1^2:
From: Elliptic Curve defined by y^2 = x^3 + (8*z2+70)*x + (3*z2+49) 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), algorithm='velusqrt').dual()
Composite morphism of degree 213 = 71*3:
From: Elliptic Curve defined by y^2 = x^3 + (50*z2+61)*x + (22*z2+25) 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
"""
# FIXME: This code fails if the degree is divisible by the characteristic.
if self.base_ring().characteristic().divides(self.degree()):
# The dual is inseparable.
#TODO: This is a lazy workaround; it could be optimized more.
return self.as_EllipticCurveIsogeny().dual()

# The dual is separable.
F = self._raw_domain.base_ring()
from sage.schemes.elliptic_curves.weierstrass_morphism import WeierstrassIsomorphism
isom = ~WeierstrassIsomorphism(self._raw_domain, (~F(self._degree), 0, 0, 0))
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