This is one of the six possible cases for the image of the mod $p$ Galois Representation if $p$ is a non-surjective prime for an elliptic curve $E$.

The label **Ns** means that $G$ is contained in the normalizer of the split Cartan subgroup but not in the split Cartan subgroup itself.

The label **Ns.a.b** means that $G$ is contained in the subgroup of the normalizer of the split Cartan subgroug generated by
\[
\begin{pmatrix}a&0\\0&1/a\end{pmatrix}, \begin{pmatrix}0&b\\-r/b&0\end{pmatrix},
\]
where $r$ be the least positive integer that generates $\F_p^*$.

The label **Ns.a.b.c** means that $G$ is contained in the subgroup of the normalizer of the split Cartan subgroup generated by
\[
\begin{pmatrix}a&0\\0&1/a\end{pmatrix}, \begin{pmatrix}0&b\\-1/b&0\end{pmatrix}, \begin{pmatrix}0&c\\-r/c&0\end{pmatrix}
\]
where $r$ be the least positive integer that generates $\F_p^*$. Conjecturally, this case arises only when $E$ has CM.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by John Jones on 2018-06-19 18:51:13

**Referred to by:**

Not referenced anywhere at the moment.

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

- 2018-06-19 18:51:13 by John Jones (Reviewed)