Normalizer of the split Cartan case (reviewed)

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.