I don't know anything yet about Complex Multiplication (on elliptic Curves). Do you know good introductory books on the Matter? I'm especially looking for a proof that the following terms are algebraic integers whenever $\tau$ is a quadratic irrationality: $$\frac{E_4(\tau)}{\eta^8(\tau)},\quad\frac{E_6(\tau)}{\eta^{12}(\tau)}\quad\text{and}\quad\sqrt{D}\cdot\frac{E_2(\tau)-\frac{3}{\pi\Im(\tau)}}{\eta^4(\tau)}$$ Here $\eta$ denotes the Dedekind $\eta$-Function, $E_k$ are the Eisenstein series of weight $k$, and $D=B^2-4AC$ where $A\tau^2+B\tau+C=0$
Edit: As @Mathmo123 commented (Thank you!!!), the first expression is a root of the polynomial $P(X)=X^3-1728J(\tau)$ and the second is a root of $Q(X)=X^2-1728+1728J(\tau)$, where $1728J(\tau)\in\mathbb Z$. Thus it only remains to prove that the third expression is an algebraic integer.
Edit: I verified numerically that the sixth power of the third expression is integral for all Discriminants with class number $1$, but I don't see how I can prove it. Unfortunately, I couldn't find anything in the books mentioned by @gandalf61 or by @Mathmo123 ... -> but this is another topic, moved to another question here.