Hyperelliptic jacobians with real multiplication

Let $K$ be a field of characteristic $p\mathrm{\ne }2$, and let $f(x)$ be a sextic polynomial irreducible over $K$ with no repeated roots, whose Galois group is isomorphic to ${\text{A}}_5$. If the jacobian $J(C)$ of the hyperelliptic curve $C:y^2=f(x)$ 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 $D$.