Principal congruence modular curve genus ($g(X(N))=1+\frac{N^2(N-6)}{24}{\prod }_{p\mid N}(1-p^{-2})$)

For $N\ge 3$, the principal congruence subgroup has projective index $\mu =\frac{1}{2}{N}^3{\prod }_{p\mid N}(1-p^{-2})$, no effective elliptic stabilizers, and all cusp widths equal to $N$. Hence the number of cusps is $\mu /N$, and the genus formula for a modular curve gives the displayed formula. The exceptional levels one and two both have genus zero; the full genus-zero list is $N=1,2,3,4,5$. Level six has genus one and every larger level has positive genus.

For $N\ge 2$, the effective principal congruence subgroup is torsion-free. Its modular curve cover of $X(1)$ has analytic ramification indices $2,3,N$ above the two elliptic orbits and the modular cusp. If $\mu$ is its projective index, the numbers of points above these three values are $\mu /2,\mu /3,\mu /N$. Applying the Riemann-Hurwitz formula gives $2g-2=\mu (1/6-1/N)$. Here $\mu =6$ for $N=2$ and $\mu =N^3{\prod }_{p\mid N}(1-p^{-2})/2$ for $N\ge 3$; level one is the identity cover.