Maximal $\ell$-adic Galois representation (reviewed)

Let $E$ be an elliptic curve over a number field $K$, let $\ell$ be prime, and let \[ \rho_{E,\ell}\colon \Gal(\overline{K}/K)\to \Aut(E[\ell^\infty]) \simeq \GL_2(\Z_\ell) \] be the $\ell$-adic Galois representation associated to $E$.

If $E$ does not have potential complex multiplication, then $\rho_{E,\ell}$ is maximal if its image contains $\SL_2(\Z_\ell)$.

In general, let $\mathcal{O}$ be the geometric endomorphism ring of $E$. Then $E[\ell^\infty]$ is an $\mathcal{O}$-module, and we view $\Aut_\mathcal{O}(E[\ell^\infty])$ as a subgroup of $\Aut(E[\ell^\infty]) \simeq \GL_2(\Z_\ell)$ that contains the image of $\rho_{E,\ell}$ whenever $K$ contains $\mathcal O$. We say that $\rho_{E,\ell}$ is maximal if its image contains $\SL_2(\Z_\ell) \cap \Aut_{\mathcal{O}}(E[\ell^\infty])$, in which case we call $\ell$ a maximal prime for $E$.