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 $K=\Q$.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by David Roe on 2022-12-11 16:37:04

**Referred to by:**

**History:**(expand/hide all)

- 2022-12-11 16:37:04 by David Roe (Reviewed)
- 2022-03-23 15:46:24 by Andrew Sutherland (Reviewed)
- 2021-09-18 14:58:21 by Andrew Sutherland (Reviewed)
- 2021-09-18 09:29:05 by Andrew Sutherland
- 2021-07-17 14:33:04 by Andrew Sutherland (Reviewed)
- 2021-07-17 14:31:45 by Andrew Sutherland
- 2021-07-17 14:31:28 by Andrew Sutherland
- 2018-06-18 16:36:35 by John Jones (Reviewed)

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