  Zarhin has extensively studied restrictions placed on the endomorphism
algebras of Jacobians $J$ for which the Galois group associated to their
2-torsion is insoluble and 'large' (relative to the dimension of $J$). In this
paper we examine what happens when this Galois group merely contains an element
of 'large' prime order. In doing so we obtain a partial converse to a result by
Guralnick and Kedlaya on the endomorphism field.
