Frobenius polynomial of hyperelliptic curve does not account for base field

[kedlaya]

When one defines a hyperelliptic curve over a prime field and then changes the base ring to an extension field, the Frobenius polynomial should change accordingly. Currently it does not:

sage: P.<x> = PolynomialRing(GF(37))
sage: u = x^5 + 2*x + 1
sage: H = HyperellipticCurve(u)
sage: H.frobenius_polynomial()
x^4 - 2*x^3 + 48*x^2 - 74*x + 1369
sage: F.<a> = GF(37^2)
sage: HF = H.change_ring(F)
sage: HF.frobenius_polynomial()
x^4 - 2*x^3 + 48*x^2 - 74*x + 1369

CC: @jpflori

Component: number theory

Keywords: hyperelliptic curve, Frobenius polynomial, base change

Reviewer: Jean-Pierre Flori

Issue created by migration from https://trac.sagemath.org/ticket/15754

[jpflori]

comment:3

Fixed by #15148.

[jpflori]

Reviewer: Jean-Pierre Flori

[jpflori]

comment:4

Note that the second test in the ticket description would take a very long time (as it's done by computing naively the number of points on the curve).

We hsould though include an example showing that the base ring is taken into account on a manageable example.

[jpflori]

comment:6

In fact there are already a bunch of tests from #15148 showing that hypellfrob is only used on prime fields and is correclty done with the naive method in other cases and that it is very slow and so on.
So let's close this one as won't fix.