$\infty$ is a regular cusp for the group $\Gamma_1(5)$

Question

How can we show that $\infty$ is a regular cusp for the group $\Gamma_1(5)$?

Answer 1

The stabilizer of $\infty$ is : $$ \Gamma_1(N)_{\infty} :=\left\{\begin{pmatrix} a & b \\ c & d \end{pmatrix} \ | \ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \infty=\infty \right\}=\left\{\begin{pmatrix} a & b \\ c & d \end{pmatrix} | \frac{a}{c}=\infty \right\}=\left\{ \begin{pmatrix} a & b \\ 0 & d \end{pmatrix}\right\} .$$