Top slope (awaiting review)

Given a mod-$\ell$ Galois representation $\rho$, let $K$ be the fixed field of kernel $\ker(\rho)$. Then the local $\ell$-adic algebra is isomorphic to a product of $\ell$-adic fields. The top slope of the representation is the largest slope among these fields, which equals the top slope for the compositum of these fields.

Note, and 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.