  We describe a collection of computer scripts written in PARI/GP to compute,
for reflection groups determined by finite-volume polyhedra in $\mathbb{H}^3$,
the commensurability invariants known as the invariant trace field and
invariant quaternion algebra. Our scripts also allow one to determine
arithmeticity of such groups and the isomorphism class of the invariant
quaternion algebra by analyzing its ramification.
  We present many computed examples of these invariants. This is enough to show
that most of the groups that we consider are pairwise incommensurable. For
pairs of groups with identical invariants, not all is lost: when both groups
are arithmetic, having identical invariants guarantees commensurability. We
discover many ``unexpected'' commensurable pairs this way. We also present a
non-arithmetic pair with identical invariants for which we cannot determine
commensurability.
