When the Jacobian of a hyperelliptic curve 
$$y^2=x(x-1)(x-a)(x-b)(x-c)$$
is a product of two elliptic curves? 
(This is a sort of reverse to
http://mathoverflow.net/questions/35060/when-is-a-product-of-elliptic-curves-isogenous-to-the-jacobian-of-a-hyperellipti). Obviously, it means some algebraic relations between $a,b,c$; the question is, which ones.

P.S. This must be a classic, but I have some trouble figuring it out or finding it in the literature. There is an old example due to Jacobi, but is it all?