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 the intersection of the Humbert surfaces $H_{n^2}$, for $n$ ranging from 2 to 1811, is empty.