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Let $\tau_0$ be CM point. The values of $E_4,E_6$ at this point is algebraic multiple of the elliptic curve period power. The same is true about the modular completion of the second Eisenstein series - $$E_2^{*}(\tau_0) = E_2(\tau_0) - \dfrac{3}{\pi Im(\tau_0)} = \alpha \Omega^2, \ \alpha \in \overline {\mathbb Q}.$$ There are just 13 points such that $E_4,E_6$ are not algebraic, but indeed rational multiples of the period power. Is it true for the $E_2^*$ too?

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