Given $q = e^{2\pi i \tau}$ and the Eisenstein series $E_{2k}(\tau)$, i.e.,

$$E_2(\tau) = 1-24\sum_{n=1}^\infty \frac{n q^n}{1-q^n}$$

$$E_4(\tau) = 1+240\sum_{n=1}^\infty \frac{n^3 q^n}{1-q^n}$$

and so on. Define the function,

$$F_{2k}(\tau) = \frac{E_{2k}(\tau)}{\left(E_2(\tau)-\frac{3}{\pi\; \Im(\tau)}\right)^k}$$

for $k \geq 2$, where $\tau = \frac{1+\sqrt{-d}}{2}$, $\Im(\tau)$ is the imaginary part of $\tau$, and $d$ has class number $h(-d) = m$. For example, we have,

$$F_4\left(\tfrac{1+\sqrt{-163}}{2}\right) = \frac{5\cdot23\cdot29\cdot163}{2^2\cdot3\cdot181^2}$$

$$F_6\left(\tfrac{1+\sqrt{-163}}{2}\right) = \frac{7\cdot11\cdot19\cdot127\cdot163^2}{2^9\cdot181^3}$$

$$F_8(\tau) = F_4^2(\tau)$$

and so on.

**Question**: In general, is it true that for $k \geq 2$ the function $F_{2k}$, ** like the j-function, is an algebraic number of degree m = h(-d)**? (I've tested it with

*d*with higher class numbers, and it seems to be true.)