Superelliptic Jacobians and central simple representations

Let f ( x ) f(x) be a polynomial of degree at least 5 5 with complex coefficients and without repeated roots. Suppose that all the coefficients of f ( x ) f(x) lie in a subfield K K of C \mathbb {C} such that: K K contains a primitive p-th root of unity; f ( x ) f(x) is irreducible over K K ; the Galois group Gal ⁡ ( f ) \operatorname {Gal}(f) of f ( x ) f(x) acts doubly transitively on the set of roots of f ( x ) f(x) ; the index of every maximal subgroup of Gal ⁡ ( f ) \operatorname {Gal}(f) does not divide deg ⁡ ( f ) − 1 \deg (f)-1 . Then the endomorphism ring of the Jacobian of the superel