The background question is: Let $C$ be a curve of genus $2$ over a field $k$. When is there a degree $2$ morphism from $C$ to an elliptic curve (and therefore an isogeny from the Jacobian $\mbox{Jac}(C)$ to a product of elliptic curves)?

On the homepage for their paper, Bruin and Doerksen list some equations for the absolute Igusa invariants of $C$ and degree 2, 3 and 4 of the morphism. If $\mbox{Jac}(C)$ is split, then the invariants satisfy the respective polynomial equation. On the other hand, their method suggests the converse may not hold.

Does anyone know a curve whose invariants satisfy the equation, but whose Jacobian is not (2,2)-split?