Given two smooth elliptic curves $C_1$ and $C_2$ over $\mathbb{C}$. Assume they are not isogenous. I'm interested in the structure of $Pic(A)$ and $Pic^{0}(A)$ for $A:=C_1 \times C_2$.

Reading Birkenhake/Lange - Complex Abelian Varieties, i think this has to do with correspondences of curves. Since an elliptic curve is its own Jacobian and the two curves are not isogenous, we have $Hom(C_1,C_2)=0$. So the space of correspondences $Corr(C_1,C_2)$ is trivial, i.e. every line bundle $L$ on $A$ is of the form $L=q^{\*}M\otimes p^{\*}N$, where q and p are the projections on the factors and $M$ and $N$ are line bundles on the factors. This implies $Pic(A)=Pic(C_1)\times Pic(C_2)$.

Does this impliy $Pic^{0}(A)=Pic^{0}(C_1)\times Pic^{0}(C_2)$? That is, is the Picard variety of $A$ isomorphic to $A$ in this case?