Borel 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 $E$.

Any label starting with B means that the group $G$ is contained in the Borel subgroup in $\GL(2,\F_p)$, i.e. a conjugate of the upper triangular group in $\GL(2,\F_p)$, and not contained in the subgroup of diagonal matrices.

The label B means that the group $G$ is the full Borel subgroup in $\GL(2,\F_p)$.

The label B.a.b means that the group $G$ is contained in the subgroup of the upper triangular group generated by the matrices \[ \begin{pmatrix}a&0\\0&1/a\end{pmatrix}, \begin{pmatrix}b&0\\0&r/b\end{pmatrix}, \begin{pmatrix}1&1\\0&1\end{pmatrix}. \]
where $r$ be the least positive integer that generates $\F_p^*$.