  Let $K$ be a field of characteristic $p \neq 2$, and let $f(x)$ be a sextic
polynomial irreducible over $K$ with no repeated roots, whose Galois group is
isomorphic to $\A_5$. 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 $D$.
