Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 137 / 3 / c / Solution

The genus formula for a modular curve iswhere $\mu =[PS{L}_2(\mathbb{Z}):\overline{\Gamma }]$, $e_2,e_3$ count elliptic orbits of orders two and three, and $c$ is the number of cusps.

For ${\Gamma }_0(3)$,There are two cusps, represented by infinity and zero. The congruence $x^2+1\equiv 0(\mathrm{mod}3)$ has no solution, so there is no elliptic orbit of order two. The congruence $x^2+x+1\equiv 0(\mathrm{mod}3)$ has the single solution $x=1$, giving one elliptic orbit of order three. Thereforewhich is the Modular curve X0 3 calculation.