I want to count the number of points on a algebraic curve $C:y^2=x^5-x+1$ over $\mathbb{F}_{3^n} (n=2,3,4,...)$ by calculators (Pari/GP, Sage, Magma,...).
Can you give me a command that solves the above problem (without relying as much as possible on the fact that $C$ is a hyperelliptic curve) using any calculator you can use?
(Maybe it's just my lack of research, though they do a very good job with this problem for special curves like elliptic curves, but seem to be weak for general $n$ variable plane curves)