= Elliptic and cusp ramification for principal level
For $N\geq2$, 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 $N\geq3$; level one is the identity cover.
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