Problem: Let $X,Y,Z$ be homogeneous coordinates in $\mathbb P^2\mathbb C$ and
$$C=\{[X,Y,Z] \in \mathbb P^2\mathbb C \mid X^4 + Y^4 + Z^4=0\}.$$
Let $i:C \to C$ be the map $i([X,Y,Z])=[-X,Y,Z]$. Calculate the genus of $C/i$.
Thoughts: $C$ is smooth of genus 3. $i^2=Id$. Morover $i$ is a ramified covering of index $2$ with four special points at $[0,a_i,1]$ with $a_i^4=-1$. I tried to study $f: C \to \mathbb P\mathbb C$ such that $f=\frac Y Z$ because it maps into the quotient as a map $f_i: C/i \to \mathbb P\mathbb C$. Since $f$ has 4 simple poles and 4 simple zeros given in pairs, $f_i$ has 2 simple poles and 2 simple zeros. We also find 4 ramification points at $[0,a_i,1]$ which remain the same. Thus $g=5$ by Hurwitz Formula.
This approach could be completely wrong anyway! The genus I found is even greater than the one of $C$ which is also weird!
Thanks!