Non-isogenous superelliptic Jacobians

Lei ℓ be an odd prime. Let K be a field of characteristic zero with algebraic closure K_a. Let n, in ≥ 4 be integers that are not divisible by ℓ Let f(x), h(x) ∈ K [x] be irreducible separable polynomials of degree n and m respectively. Suppose that the Galois group Gal(f) of f acts doubly transitively on the set R_f of roots of f and that Gal(/z) acts doubly transitively on R_h as well. Let J(C _f,ℓ) and J(C_{h, ℓ}) be the Jacobians of the superelliptic curves C_f,ℓ: y_ℓ = f(x) and C _h,ℓ: y^ℓ = h(x) respectively. We prove that J(C_f,ℓ) and J(C_hℓ) are not isogenous over K _a if the splitting fields of / and h are linearly disjoint over K and K contains a primitive ℓth root of unity.