Image of mod-$\ell$ Galois representation (reviewed)

Let $\ell$ be a prime and let $E$ be an elliptic curve defined over a number field $K$.

Subgroups $G$ of $\GL(2,\F_\ell)$ that can arise as the image of the mod-$\ell$ Galois representation \[ \bar\rho_{E,\ell}\colon {\Gal}(\overline{K}/K)\to \GL(2,\F_\ell) \] attached to $E$ that do not contain $\SL(2,\F_\ell)$ are identified using the labels introduced by Sutherland in [arXiv:1504.07618, MR:3482279]. For groups with surjective determinant map (necessarily the case when $K=\Q$), these labels have the form \[ \texttt{LS.a.b.c}, \] where $\texttt{L}$ is the prime $\ell$, $\texttt{S}$ is one of G, B, Cs, Cn, Ns, Nn, A4, S4, A5, and $\texttt{a}$, $\texttt{b}$, $\texttt{c}$ are optional positive integers. When the determinant map is not surjective the label has "$\texttt{[d]}$", where $d$ is the index of the determinant image in $\F_\ell^\times$.

When $\bar\rho_{E,\ell}$ does not contain $\SL(2,\F_\ell)$ the possibilities for $\texttt{S}$ are: Borel B, split Cartan Cs, normalizer of the split Cartan Ns, nonsplit Cartan Cn, normalizer of the nonsplit Cartan Nn, exceptional A4, S4, A5. The cases A4 and A5 cannot occur when $K=\Q$.