It is a well-known result that the modular function $1728J(\tau) := \frac{1728E_4(\tau)^3}{E_4(\tau)^3-E_6(\tau)^2}$ has integral values if $\tau$ has class number 1 - for example at $\tau_{163}:=\frac{1+i\cdot\sqrt{163}}{2}$ you get $1728J(\tau_{163})=-640320^3$.

Now define the quasi-modular function $s_2(\tau):=\frac{E_4(\tau)}{E_6(\tau)}\cdot\left(E_2(\tau)-\frac{3}{\pi Im(\tau)}\right)$.

Then I have looked at all 13 class 1 discriminants and have verified numerically that $$M(\tau):=s_2(\tau)\cdot(1728J(\tau)-1728)=\frac{1728E_4(\tau)E_6(\tau)}{E_4(\tau)^3-E_6(\tau)^2}\left(E_2(\tau)-\frac{3}{\pi Im(\tau)}\right)$$ also has integral values for all these $\tau_N$.

For example $M(\tau_{163})=-2^{13}\cdot3^5\cdot5\cdot7\cdot11\cdot19\cdot23\cdot29\cdot127\cdot181$.

**My Question:**
How can I prove that $M(\tau)\in\mathbb Z$ for all $\tau$ with class number 1?
Or where can I find a proof for it?

*Edit:* After the answer of @Zavosh (thank you!!!) it remains to prove this question. Who can help?