Bug in elliptic curve cardinality for j=0 in char. 3

[JohnCremona]

Dustin Moody reported

    While working on some things, I found a bug in SAGE:

 sage:k.<a>=GF(3^5)

 sage:E=EllipticCurve(k,[-1,-1])

 sage:E.trace_of_frobenius()
 0

 This isn't correct.  It should be -27.  I also discovered you can get
around it.

 sage:E.cardinality_exhaustive()
 271

 sage:E.trace_of_frobenius()
 -27

 Somehow, doing .cardinality_exhaustive() fixes the bug.

Component: number theory

Issue created by migration from https://trac.sagemath.org/ticket/2849

[JohnCremona]

Attachment: trac2849.patch.gz

[JohnCremona]

comment:1

Here's the patch: A case where q=3 (mod 4) only worked for p>3 and was being used for p=3, odd degree. Should be a trivial review.

Note that I am in the middle of implementing vastly better support for the cases j=0 and j=1728, which are not so straightforward in characterisitcs 2 and 3 but I am getting there!

[aghitza]

apply after trac2849.patch

[aghitza]

comment:2

Attachment: trac2849_doctest.patch.gz

Looks fine and it fixes the issue. I've added a mini-patch that puts in a doctest demonstrating the fixed status.

Apply both patches.

[sagetrac-mabshoff]

comment:3

Merged trac2849.patch and trac2849_doctest.patch in Sage 3.0.alpha3

[JohnCremona]

comment:4

Replying to @aghitza:

Looks fine and it fixes the issue. I've added a mini-patch that puts in a doctest demonstrating the fixed status.

Thanks, Alex -- I should have done that but only remembered after uploading the patch.

Apply both patches.