want to emphasise that the level-lowering result might be hard to prove for small primes (which divide the order of elliptic points). For this reason, we include an unconditional proof (not relying on modularity) for $\ell=2$ and $3$ in Theorem \[thm:mainthm\] for the nine fields considered.

Assume otherwise that the residual image is absolutely reducible, i.e. it is either reducible over $\F_{\ell}$, or irreducible but reducible over $\F_{\ell^2}$. For the first possibility, we prove that either such curves do not exist, or otherwise there is an isogenous curve whose discriminant valuation is prime to $\ell$ (the case $\ell=2$ being hardest). When $\ell$ is odd, the second case is eliminated by a detailed study of the residual Galois representation. Lastly, for $\ell=2$ we prove that the discriminant valuation is at most $2$, where (for certain of the fields $K$ only) the exceptional curves with square discriminant arise as described above.

The article is organized as follows. The first section contains a brief description of Bianchi modular forms and modularity of elliptic curves. It also contains the conjectural level-lowering statement in the spirit of Ribet’s result, which we expect to hold in the setting of Bianchi modular forms. An important difference in 