  The question of computing the reductions modulo $p$ of two-dimensional
crystalline $p$-adic Galois representations has been studied extensively, and
partial progress has been made for representations that have small weights,
very small slopes, or very large slopes. It was conjectured by Breuil, Buzzard,
and Emerton that these reductions are irreducible if they have even weight and
non-integer slope. We prove some instances of this conjecture for slopes up to
$\frac{p-1}{2}$.
