This looks like a bug to me:

sage: F=GF(3)
sage: R.<X,Y,Z>=F[]
sage: C=Curve(X^8+Y^8-Z^8)
sage: C.count_points(1) # correct count over GF(3^1)
[4]
sage: C.count_points(8) # should give counts over GF(3^n) for n=1..8 but it
crashes

TypeError: F (=[X^8 + Y^8 - Z^8]) must be a list or tuple of polynomials of
the coordinate ring of A  (=Projective Space of dimension 2 over Finite
Field in z2 of size 3^2)

-- note that the error message looks suspicious.

Also the more naive

sage: F=GF(3^8)
sage: R.<X,Y,Z>=F[]
sage: C=Curve(X^8+Y^8-Z^8)
sage: C.count_points(1)
[4696]

works but is so slow it must be doing something very basic such as testing
every point in the plane (of which there are 3^16+3^8+1...)

John


On 13 April 2018 at 11:55, David Joyner <wdjoy...@gmail.com> wrote:

> Hi:
>
> The question below is posted for Gary McGuire, who is not a subscriber
> to this list:
>
> "I would like to know the number of rational points on the (projective)
> curve
> x^8+y^8=z^8
> over the field of order 3^{18}.
>
> My question is, can Sage do this calculation, and how?"
>
> - David
>
> PS: About 3 years ago, a related question was posted:
> https://groups.google.com/forum/#!topic/sage-support/s59iDjhu2zU
> For some reason, the method described there is no longer implemented.
>
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