Let $E/\mathbf{C}$ be an elliptic curve with CM by the maximal order of $\mathbf{Q}(\sqrt{-D})$ where $D$ is positive and square-free integer. Consider the endomorphism $[\sqrt{-D}]:E\rightarrow E$. Let $\Gamma\subseteq E\times E$ be the graph of $[\sqrt{-D}]$. Then $\Gamma$ is isomorphic (as a complex manifold) to $E$, in particular it is a 2-cycle on $E\times E$. Now let $\eta$ be the Poincare dual of $\Gamma$: so if $\iota:\Gamma\hookrightarrow E\times E$ denotes the inclusion then for all $\omega\in A^2(E\times E)$, we have $\int_{\Gamma}\iota^*\eta=\int_{E\times E}\omega\wedge \eta$. Let $a,b\in H^1(E,\mathbf{Z})$ be the standard oriented basis: if one thinks of the torus as a rectangle with the usual side idtentifications, then $a$ corresponds to the horizontal (left to right) edge, $b$ to the vertical (bottom to top) edge and $a\cdot b=1$ (so $b\cdot a=-1$). From the Kuenneth formula we have a priviledged isomorphism $H^*(E\times E)\simeq\oplus_{i=1}^4 H^{i}(E)\otimes H^{4-i}(E)$. Via this isomorphism, there exists real numbers $c_i$ ($i=1\ldots 4$) such that $$ \eta=c_1\cdot(a\otimes a)+c_2\cdot(a\otimes b)+c_3\cdot(b\otimes a)+c_4\cdot(b\otimes b). $$ I would expect in general the $c_i$'s to depend on $D$.

Q: Is there a method to compute these numbers in a systematic way, so that we can clearly see their dependence on $D$?