Curves of genus two with maps of every degree to a fixed elliptic curve

We show that up to isomorphism there are exactly twenty pairs $(C,E)$, where $C$ is a genus-$2$ curve over $\mathbf{C}$, where $E$ is an elliptic curve over $\mathbf{C}$, and where for every integer $n>1$ there is a map of degree $n$ from $C$ to $E$. We also show that for every genus-$2$ curve $C$, there is an integer $n$ with $1<n\le 59$ such that there is no minimal degree-$n$ map from $C$ to an elliptic curve.