Rational Equivalences on Products of Elliptic Curves in a Family

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 $E_1\times E_2$.