Hyperelliptic jacobians with real multiplication

Let K be a field of characteristic p≠2, and let f(x) be a sextic polynomial irreducible over K with no repeated roots, whose Galois group is isomorphic to A5. If the jacobian J(C) of the hyperelliptic curve C:y2=f(x) admits real multiplication over the ground field from an order of a real quadratic field D, then either its endomorphism algebra is isomorphic to D, or p>0 and J(C) is a supersingular abelian variety. The supersingular outcome cannot occur when p splits in D.