Let $A$ be an abelian surface given by the quotient of a product of two generic elliptic curves $E_1 \times E_2$ by the product $T_1 \times T_2$ of two translations by $2$-torsion points. Then $A$ admits two natural projections to $E_i':=E_i/T_i$, and the fibers are $E_j'$ with $i \neq j$.

Is the group of divisors up to numerical equivalence generated by $E_1',E_2'$? Do they intersect in $2$ points?