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Conjugacy classes of $\mathrm{SL}_2(\mathbb{Z})$

One can proceed as follows for $SL_2(\mathbb{Z})$. First, the trace is a conjugacy invariant. For trace $0$ there are two conjugacy classes represented by $\pmatrix{0 & 1 \\ -1 & 0}$ and $\pmatrix{0 …
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