Relations between the Eisenstein series and the hypergeometric series

Question

It is known that $$E_4(\tau) = {}_{2}F_{1}\left(\frac{1}{12}, \frac{5}{12}; 1; \frac{1}{J(\tau)}\right)^4$$ and $$E_6(\tau) = {}_{2}F_{1}\left(\frac{1}{12}, \frac{7}{12}; 1; \frac{1}{1 - J(\tau)}\right)^6,$$ where $E_{2k}(\tau)$ are the (normalized) Eisenstein series of weight $2k$ and $J(\tau)$ is Klein's invariant. My question is: Are these the only ones? Is there a corresponding identity for $E_2(\tau)$, for example?