Image of mod p Galois Representation (reviewed)

Let $p$ be a prime and let $E$ be an elliptic curve defined over $\Q$. Assume that $p$ is a non-surjective prime for $E$.

Subgroups $G$ of $\GL(2,\F_p)$ that can arise as the image of the mod-$p$ Galois representation \[ \rho_{E,p}\colon {\Gal}(\overline{\Q}/\Q)\to \GL(2,\F_p) \] attached to $E$ are identified using the labels introduced by Sutherland in [arXiv:1504.07618, MR:3482279]. These labels have the form \[ \mathrm{\bf{S.a.b.c,}} \] where S is one B, Cs, Cn, Ns, Nn, or S4, and a, b, c are (optional) positive integers.

There are six cases: Borel B, split Cartan Cs, normalizer of the split Cartan Ns, nonsplit Cartan Cn, normalizer of the nonsplit Cartan Nn, exceptional S4.