Consider the elliptic curve $E$ defined by $y^2z= x^3 +xz^2$ over an algebraic closure $\overline{\mathbf{Q}}$ of $\mathbf{Q}$. Consider the endomorphism $f:E\to E$ given by $(x:y:z)\mapsto (-x:iy:z)$. Note that $f$ has precisely two fixed points: $(0:0:1)$ and $(0:1:0)$.
How do I compute the trace of $f$ on the (etale) of $E$? More precisely, there are three cohomology groups: $H^0$, $H^1$ and $H^2$.
If I'm right, $H^0$ and $H^2$ are $1$-dimensional. So in each case the action is given by the multiplication of some number. What are these numbers? (Note that $f$ is an automorphism. Maybe this helps?)
The $H^1$ is two-dimensional. How do I find the action of $f$ on $H^1$. (It's given by a two by two matrix.)
My motivation for this question is the trace formula. I want to see examples! I plan on working out more examples with CM elliptic curves. Then maybe some higher genus curves and finally some higher-dimensional varieties.