Number of cusps of Gamma 1 of prime level
= Number of cusps of Gamma 1 of prime level
{c}
{title2=$|\Gamma_1(p)\backslash\mathbb P^1(\mathbb Q)|$}
For an odd prime $p$, the group $\Gamma_1(p)$ has $p-1$ cusps. The general primitive-vector count gives
$$
\frac12\sum_{d\mid p}\varphi(d)\varphi(p/d)=p-1.
$$
The exceptional group $\Gamma_1(2)=\Gamma_0(2)$ has two cusps, represented by infinity and zero.