Endomorphism rings of certain Jacobians in finite characteristic

We prove that, under certain additional assumptions, the endomorphism ring of the Jacobian of a curve y^l = f(x) contains a maximal commutative subring isomorphic to the ring of algebraic integers of the lth cyclotomic field. Here l is an odd prime dividing the degree n of the polynomial f and different from the characteristic of the algebraically closed ground field; moreover, n ≥ 9. The additional assumptions stipulate that all coefficients of f lie in some subfield K over which its (the polynomial's) Galois group coincides with either the full symmetric group S_n or with the alternating group A_n.