Let $X\_0$ be a smooth projective geometrically connected curve over $\mathbf{F}\_p$ and let $X = X\_0 \otimes\_{\mathbf{F}\_p} \overline{\mathbf{F}\_p}$. 

Assume that the genus $g\geq 2$.

Let $f:X\to X$ be an endomorphism. Then $f$ is (EDIT) bijective if (EDIT) $f$ is non-constant.

How can one determine the action of $f$ on $H^1(X\_{et},\mathbf{Q}\_\ell)$ computationally?

Explicitly:

Consider the curve $X$ given by the equation $x^{100}+y^{100} = z^{100}$. Let $f:X\to X$ be defined as $(x:y:z)\mapsto (\zeta\_{100} x: y: \zeta\_{100} z)$, where $\zeta\_{100}^{100} = 1$.

It is easy to determine the number of fixed points with the trace formula, but I need more. I need a $(2g\times 2g)$-matrix over $\overline{\mathbf{Q}\_\ell}$ giving the action of $f$ on $H^1$.

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