  Let $p$ be a prime and $a$ a quadratic non-residue $\bmod p$. Then the set of
integral solutions of the diophantine equation $x_0^2 - ax_1^2 -px_2^2 +
apx_3^2=1$ form a cocompact discrete subgroup $\Gamma_{p,a}\subset
SL(2,\mathbb{R})$ and is commensurable with the group of units of an order in a
quaternion algebra over $\mathbb{Q}$. The problem addressed in this paper is an
estimate for the traces of a set of generators for $\Gamma_{p,a}$. Empirical
results summarized in several tables show that the trace has significant and
irregular fluctuations which is reminiscent of the behavior of the size of a
generator for the solutions of Pell's equation. The geometry and arithmetic of
the group of units of an order in a quaternion algebra play a key role in the
development of the code for the purpose of this paper.
