So if we think of $T:C\rightarrow C$ as a multivalued function and $T(P)=\sum_{i} [P_i]$ (formal sum) then are you saying that $(-T)(P)=\sum_{i} [-P_i]$?

Yes you are right! So this solves my question. Many thanks!

Hi @Will. I agree with everything you said except that I don't quite see why $\Delta\cdot\Gamma$ corresponds to $[(1-\sqrt{-D})^{-1}\mathcal{O}_K:\mathcal{O}_K]$

I thought a bit a bout my problem and now I realized that the way I set it up is probably not ideal. It is probably better to work with homology since then one can make pictures. The group $H_2(E,\mathbf{Z})$ has $3$ natural $\mathbf{Z}$-linearly independant elements, namely $E_1=E\times\{0\}$, $E_2=\{0\}\times E$ and $\Delta$ (the diagonal). Intuitively we should have $E_2\cdot \Gamma=1$ and $E_1\cdot\Gamma=D$. Though the intersection $\Delta\cdot \Gamma$ seems to be more complicated to compute.

Thanks, now I think I see how to prove it. The point is that multiplication on the left or the right by an invertible matrix with coefficients is Z preserves the determinant of the $k\times k$ minors.

no, I really meant what I wrote.

Thanks for the data. Yes my questions are closely related to asking for a "canonical Grossencharacter" of a quadratic CM field, but from what you say it seems that such an object does not really exist.

So the point is as @Cesnavicius wrote, the support of the conductor won't be bounded.