Non-cyclic primes (reviewed)

We say that an isogeny class of abelian varieties over a finite field $\F_q$ is non-cyclic at $\ell$ if the $\ell$-primary part of the group of points $A(\F_q)$ is non-cyclic for some abelian variety $A$ in the isogeny class. If an isogeny class is not cyclic then there exists at least one prime $\ell$ so that it is non-cyclic at $\ell$.