Generators of the congruence subgroup $\Gamma (5) \subset SL(2,\mathbb Z)$

Question

Recall that $SL(2,\mathbb Z)=\left\{A=\begin{pmatrix} a&b \\ c&d \end{pmatrix}: \det(A)=1; a,b,c,d \in \mathbb Z \right\} $ and $\Gamma(5)=\left\{A=\begin{pmatrix} a&b \\ c&d \end{pmatrix}: \det(A)=1; a\equiv d\equiv \pm1, b \equiv c\equiv0 \right\}$. What are the generators of $\Gamma(5)$? Does anyone have a reference for this?

Answer 1

This is a one-liner in Sage (try the online calculator): if you type

Gamma(5).generators()

then you get the response

[
[1 5]  [-24   5]  [-109   40]  [11 -5]  [-39  25]  [ 6 -5]  [-64 105]
[0 1], [ -5   1], [ -30   11], [20 -9], [-25  16], [ 5 -4], [-25  41],

[ 11 -20]  [-89 235]  [ 16 -45]  [ 21 -80]
[  5  -9], [-25  66], [  5 -14], [  5 -19]
]

(This is a slightly different $\Gamma(5)$ than yours, since Sage's definition of $\Gamma(5)$ requires that $a = d = 1 \bmod(5)$, not just $\pm 1$, but that's easily rectified by adding $-1$ to the generator list.)