  We study the problem of deciding whether or not the image of an irreducible
representation of the braid group $\B_3$ of degree $\leq 5$ has finite image if
we are only given the eigenvalues of a generator. We provide a partial
algorithm that determines when the images are finite or infinite in all but
finitely many cases, and use these results to study examples coming from
quantum groups. Our technique uses two classification theorems and the
computational group theory package GAP.
