show · ec.q.galois_rep_image_nsp_cartan all knowls · up · search:

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 $E$.

The label Cn means that $G$ is contained in the nonsplit Cartan subgroup which is, up to conjugation, the unique maximal abelian subgroups of $\GL(2,\F_p)$ isomorphic to $\F_{p^2}^*$.

For $p=2$ is the unique index 2 subgroup of $\GL(2,\F_2)$ and for $p$ odd consists of all matrices \[ \begin{pmatrix}x&ry\\y&x\end{pmatrix}, \] where $r$ be the least positive integer that generates $\F_p^*$.

Knowl status:
  • Review status: reviewed
  • Last edited by John Jones on 2018-06-19 18:54:21
Referred to by:

Not referenced anywhere at the moment.

History: (expand/hide all)