Do there exist line bundles over the product of elliptic curves $E\times E$ which have a non-trivial chern class contained in $H^1(E)\otimes H^1(E)$?

Question

Given a projective elliptic curve $E = E'$ we can construct line bundles $\mathcal{O}(a,b)$ on $E\times E'$ pulled back from $\mathbb{P}^2\times\mathbb{P}^2$. Since the chern classes are pulled back from the two copies of $\mathbb{P}^2$, they live in the module $$ H^2(E)\otimes H^0(E') \oplus H^0(E)\otimes H^2(E') $$ Do there exist line bundles over $E\times E'$ whose chern classes are contained in $H^1(E)\otimes H^1(E')$? If so, what is example of such a line bundle?

Answer 1

If $E = E'$ then an example of such line bundle is the ideal of the diagonal $L = \mathcal{O}_{E \times E}(-\Delta)$.

More generally, if $E$ and $E'$ are isogenous, the ideal of the graph of an isogeny gives such an example.