I recently read the statement "up to conjugacy there are 4 nontrivial finite subgroups of $SL_2(\mathbb{Z})$." They are generated by

$\left(\begin{array}{cc} -1&0 \\\ 0&-1\end{array}\right), \left(\begin{array}{cc} -1&-1 \\\ 1&0\end{array}\right), \left(\begin{array}{cc} 0&-1 \\\ 1&0\end{array}\right), \left(\begin{array}{cc} 0&-1 \\\ 1&1\end{array}\right) $ and are isomorphic to Z_2, Z_3, Z_4, and Z_6, respectively. Does someone know a reference for this statement? (Or, is it easy to see?) My attempt at a google search turned up this statement, but I wasn't able to find a reference.

I recently read the statement "up to conjugacy there are 4 nontrivial finite subgroups of ${\rm SL}_2(\mathbb{Z})$." They are generated by

$$\left(\begin{array}{cc} -1&0 \\\ 0&-1\end{array}\right), \left(\begin{array}{cc} -1&-1 \\\ 1&0\end{array}\right), \left(\begin{array}{cc} 0&-1 \\\ 1&0\end{array}\right), \left(\begin{array}{cc} 0&-1 \\\ 1&1\end{array}\right) $$ and are isomorphic to $\mathbb{Z}_2$, $\mathbb{Z}_3$, $\mathbb{Z}_4$, and $\mathbb{Z}_6$, respectively. Does someone know a reference for this statement? (Or, is it easy to see?) My attempt at a Google search turned up this statement, but I wasn't able to find a reference.