A note on factorisation patterns of division polynomials of elliptic curves over finite fields

Let $E$ be an elliptic curve defined over a finite field $\mathbf{F}_{q}$, $q = p^{d}$, $p > 3$, and a prime number $\ell > 3$ such that $q \equiv 1 \pmod{\ell}$ and $\ell \mid \# E(\mathbf{F}_{q})$. In this paper we study the possible factorisation patterns over $\mathbf{F}_{q}[x]$ of the $\ell^{k}$-division polynomials associated to $E$ with $k \geq 2$, extending the work of Verdure [6] for $k=1$.