Non-isogenous elliptic curves and hyperelliptic jacobians

Let K be a field of characteristic different from 2, K̄ its algebraic closure. Let n ≥ 3 be an odd prime such that 2 is a primitive root modulo n. Let f(x) and h(x) be degree n polynomials with coefficients in K and without repeated roots. Let us consider genus (n − 1)/2 hyperelliptic curves C_f : y² = f(x) and C_h : y² = h(x), and their jacobians J(C_f) and J(C_h), which are (n − 1)/2-dimensional abelian varieties defined over K. Suppose that one of the polynomials is irreducible and the other reducible. We prove that if J(C_f) and J(C_h) are isogenous over K̄ then both jacobians are abelian varieties of CM type with multiplication by the field of nth roots of 1. We also discuss the case when both polynomials are irreducible while their splitting fields are linearly disjoint. In particular, we prove that if char(K) = 0, the Galois group of one of the polynomials is doubly transitive and the Galois group of the other is a cyclic group of order n, then J(C_f) and J(C_h) are not isogenous over K̄.