Let H be a discrete cocompact subgroup of SL2(C). We conjecture that the quotient manifold X=SL2(C)/H contains infinitely many non-isogeneous elliptic curves and prove that this is indeed the case if Schanuel's conjecture holds. We also prove it in the special case where the intersection of H and SL2(R) is cocompact in SL2(R). Furthermore, we deduce some consequences for the geodesic length spectra of real hyperbolic 2- and 3-folds.