  Let $E$ be an elliptic curve defined over $\mathbf{Q}$ without complex
multiplication. For each prime $\ell$, there is a representation
$\rho_{E,\ell}\colon \text{Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) \to
\text{GL}_2(\mathbf{F}_{\ell})$ that describes the Galois action on the
$\ell$-torsion points of $E$. Building on recent work of Rouse--Zureick-Brown
and Zywina, we find models for composite level modular curves whose rational
points classify elliptic curves over $\mathbf{Q}$ with simultaneously
non-surjective, composite image of Galois. We also provably determine the
rational points on almost all of these curves. Finally, we give an application
of our results to the study of entanglement fields.
