  Quantum Teichmuller theory assigns invariants to three-manifolds via
projective representations of mapping class groups derived from the
representation of a noncommutative torus. Here, we focus on a representation of
the simplest non-commutative torus which remains fixed by all elements of the
mapping class group of the torus, $SL_2(\mathbb{Z})$. Also known as the modular
group. We use this representation to associate a matrix to each element of
$SL_2(\mathbb{Z})$; we then compute the trace and determinant of the associated
matrix.
