Searching on Galois images (reviewed)

To search for elliptic curves with a particular $\ell$-adic image of Galois that is not maximal, enter the label (or list of labels) of a proper subgroup of $\GL_2(\Z_\ell)$. You can also enter the label of a subgroup of $\GL_2(\F_\ell)$, in which case only the mod-$\ell$ image will be constrained.

To search for elliptic curves with maximal $\ell$-adic Galois image, enter the desired prime or primes $\ell$ in the list of non-maximal primes and choose the "exclude" quantifier.

Unless you explicitly specify complex multiplication in your search criteria, the search will by limited to elliptic curves without complex multiplication.