  Let $A$ be an abelian variety over a finite field $k$ with $|k|=q=p^m$. Let
$\pi\in \text{End}_k(A)$ denote the Frobenius and let $v=\frac{q}{\pi}$ denote
Verschiebung. Suppose the Weil $q$-polynomial of $A$ is irreducible. When
$\text{End}_k(A)=\mathbb{Z}[\pi,v]$, we construct a matrix which describes the
action of $\pi$ on the prime-to-$p$-torsion points of $A$. We employ this
matrix in an algorithm that detects when $p$ is an obstruction to the
monogeneity of division fields of certain abelian varieties.
