Given two non-isogenous elliptic curves $E_1$ and $E_2$ over $\mathbb{C}$. Set $A:=E_1 \times E_2$. Given a nontrivial sheaf of quaternion algebras $D$ over $A$, what is the dimension of the vector space $H^1(A,D)$?

If one thinks of $D$ as an element in the Brauer group $Br(A)$, then it is $2$-torsion, hence belongs to $Br(A)[2]$. Since the curves are non-isogenous there is an isomorphism $Pic(E_1)[2] \otimes Pic(E_2)[2] \to Br(A)[2]$. So there should be a connection between such quaternions and $2$-torsion line bundles on the curves, but i cannot find an explicit description for this isomorphism. If there is one, i thought one could use the Künneth formula to compute $H^1(A,D)$ in terms of the cohomology of the line bundles on the curves.

For now i could only work out the bound $d=dim(H^1(A,D)) \geq 2$: using Hirzebruch-Riemann-Roch and simplifying terms one gets $d=c_2(D)+2$. After a result of M.Lieblich one has $c_2(D)\geq 0$.

Does anyone see/have an explicit description of the isomorphism mentioned above? Is the idea using Künneth a promising approach to this problem at all? Or does anyone have another approach? Are there some calculations regarding this in the literature (i couldn't find one)?

Another question in this context is: what is the image of such an algebra under the map $Br(A) \rightarrow Br(\mathbb{C}(A))$. This should be nontrivial $\mathbb{C}(A)$-quaternions, since the map "looking at the genric point $\eta$" is injective, i.e. $D_{\eta}$ is generated by elements $i,j$ with $i^2=a, j^2=b and ij=-ji$. But what are a resp. b? I think they should have something to do with functions h such that 2*Y=div(h), where Y defines one of the line bundles. Is this true?