Number of cusps of Gamma 1 of prime level ($|{\Gamma }_1(p)\backslash {\mathbb{P}}^1(\mathbb{Q})|$)

For an odd prime $p$, the group ${\Gamma }_1(p)$ has $p-1$ cusps. The general primitive-vector count givesThe exceptional group ${\Gamma }_1(2)={\Gamma }_0(2)$ has two cusps, represented by infinity and zero.