Opened 3 years ago

Last modified 2 years ago

#26487 closed defect

isogenies_prime_degree() does not work well for degree = characteristic — at Initial Version

Reported by: jdemeyer Owned by:
Priority: major Milestone: sage-8.5
Component: elliptic curves Keywords:
Cc: Merged in:
Authors: Reviewers:
Report Upstream: N/A Work issues:
Branch: Commit:
Dependencies: Stopgaps:

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Description

Depending on how ambitious one is, this is either a documentation bug or a missing feature. It seems that isogenies_prime_degree only finds separable isogenies, so it never finds the Frobenius. It does find the Verschiebung for ordinary elliptic curves (where it's found multiple times somehow) but not for supersingular elliptic curves.

Ordinary:

sage: E = EllipticCurve(GF(5), [1,1])
sage: E.trace_of_frobenius()
-3
sage: L = E.isogenies_prime_degree(5); L
[Isogeny of degree 5 from Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field of size 5 to Elliptic Curve defined by y^2 = x^3 + x + 4 over Finite Field of size 5,
 Isogeny of degree 5 from Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field of size 5 to Elliptic Curve defined by y^2 = x^3 + x + 4 over Finite Field of size 5,
 Isogeny of degree 5 from Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field of size 5 to Elliptic Curve defined by y^2 = x^3 + x + 4 over Finite Field of size 5,
 Isogeny of degree 5 from Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field of size 5 to Elliptic Curve defined by y^2 = x^3 + x + 4 over Finite Field of size 5,
 Isogeny of degree 5 from Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field of size 5 to Elliptic Curve defined by y^2 = x^3 + x + 4 over Finite Field of size 5]
sage: L[0] == L[1]
True

Supersingular:

sage: E = EllipticCurve(GF(5), [0,1])
sage: E.trace_of_frobenius()
0
sage: E.isogenies_prime_degree(5)
[]

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