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Let $E$ be an elliptic curve defined over $\mathbf{Q}$ without complexmultiplication. 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-Brownand Zywina, we find models for composite level modular curves whose rationalpoints classify elliptic curves over $\mathbf{Q}$ with simultaneouslynon-surjective, composite image of Galois. We also provably determine therational points on almost all of these curves. Finally, we give an applicationof our results to the study of entanglement fields.

Composite images of Galois for elliptic curves over $\mathbf{Q}$ & Entanglement fields