  Given a pair of elliptic curves $E_1,E_2$ over a field $k$, we have a natural
map $\text{CH}^1(E_1)_0\otimes\text{CH}^1(E_2)_0\to\text{CH}^2(E_1\times E_2)$,
and a conjecture due to Beilinson predicts that the image of this map is finite
when $k$ is a number field. We construct a $2$-parameter family of elliptic
curves that can be used to produce examples of pairs $E_1,E_2$ where this image
is finite. The family is constructed to guarantee the existence of a rational
curve passing through a specified point in the Kummer surface of $E_1\times
E_2$.
