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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.

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