  Pick a random matrix $\gamma$ in $\Gamma={\rm SL}(2,\mathbb{Z})$. Denote by
$\mathcal{O}_K$ the Dedekind ring generated by its eigenvalues, and let
$\Delta_K$, $\Delta_\gamma$ and $\Delta = {\rm Tr}(\gamma)^2-4$ be the
respective discriminant of the rings $\mathcal{O}_K$, the multiplier ring
$M(2,\mathbb{Z})\cap \mathbb{Q}[\gamma]$ and $\mathbb{Z}[\gamma]$. We show that
their ratios admit probability limit distributions. In particular, 42% of the
elements of $\Gamma$ have a fundamental discriminant, and $\mathbb{Z}[\gamma]$
is a ring of integers with probability 32%.
