Ramification of the x-coordinate on the Klein quartic
= Ramification of the x-coordinate on the Klein quartic
The coordinate function $x$ has degree three on the Klein quartic. It has ramification index three at $(0,0)$, index two at the seven affine points satisfying $y^7=-3/8$ and $x=2y^3$, and index two at $[0:1:0]$. The total ramification is ten, so the <Riemann-Hurwitz formula> gives genus three.