Top slope (awaiting review)

Given a mod-$\ell$ Galois representation $\rho$, let $K$ be its splitting field. Then the local $\ell$-adic algebra $K\otimes\mathbb{\Q}_{\ell}$ is isomorphic to a product of $\ell$-adic fields. The top Artin slope of the representation is the largest Artin slope among these fields, which equals the top Artin slope for their compositum.

Note that a top slope of $0$ signifies that the field is unramified at $\ell$, and a top slope of $1$ indicates that it is ramified, but only tamely. Otherwise the top slope is the largest wild slope.