#11980: hyperelliptic curves: count_points and frobenius_polynomial do not work
in
all cases, not zeta_function available
-------------------------------+--------------------------------------------
Reporter: dkrenn | Owner: cremona
Type: enhancement | Status: new
Priority: major | Milestone: sage-4.7.2
Component: elliptic curves | Keywords:
Work_issues: | Upstream: N/A
Reviewer: | Author:
Merged: | Dependencies:
[http://trac.sagemath.org/sage_trac/ticket/11930 11930]
-------------------------------+--------------------------------------------
The following is not implemented (but can be done for hyperelliptic
curves):
{{{
sage: P.<x> = PolynomialRing(GF(9,'a'))
sage: H = HyperellipticCurve(x^5+x^2+1)
sage: H.count_points(5)
Traceback (most recent call last)
...
NotImplementedError: Point counting only implemented for schemes over
prime fields
}}}
Also, having a `count_points` for all cases, the Frobenius polynomial can
be calculated. At the moment, this is calculated via `frobenius_matrix`,
which is not always available:
{{{
sage: R.<t> = PolynomialRing(GF(8, 'a'))
sage: H = HyperellipticCurve(t^5 + t + 2, t + 1)
sage: H.frobenius_polynomial()
Traceback (most recent call last):
...
NotImplementedError: only implemented for curves y^2 = f(x)
}}}
Further, the zeta function can be calculated easily when the Frobenius
polynomial is known.
Patch follows.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/11980>
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