Non-isogenous superelliptic Jacobians

Let ℓ be an odd prime. Let K be a field of characteristic zero with algebraic closure K_a. Let n, m ≥ 4 be integers that are not divisible by ℓ. Let f(x), h(x) ∈ Kx] 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 of roots of f and that Gal(h) acts doubly transitively on 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_h,ℓ) are not isogenous over K_a if the splitting fields of f and h are linearly disjoint over K and K contains a primitive ℓth root of unity.