When is the Jacobian of a hyperelliptic curve $$y^2=x(x-1)(x-a)(x-b)(x-c)$$ a product of two elliptic curves? (This is a sort of reverse to When is a product of elliptic curves isogenous to the Jacobian of a hyperelliptic curve?). Obviously, it implies some algebraic relations between $a,b,c$; the question is: which ones?

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

EDIT. As pointed out by abx, the set of parameters $(a,b,c)$ for which the Jacobian is isogenous to a product is actually dense (e.g. in $\mathbb{C}^3$ with complex topology). For this reason, the question in this form does not have a reasonable answer. (In hindsight, it is obvious, but I missed it.)

isogenousto a product of two elliptic curves; (b) the Jacobian of $C$ isisomorphicto a product of two elliptic curves [but the polarization on $J_C$ is not the product polarization on the product of elliptic curves]. Which do you mean? $\endgroup$ – Dan Petersen Oct 2 '16 at 7:09