Irreducibility of the zero polynomials of Eisenstein series

Let $E_k$ be the normalized Eisenstein series of weight $k$ on ${\mathrm{S}\mathrm{L}}_2(\mathbb{Z})$. Let ${\phi }_k$ be the polynomial that encodes the $j$-invariants of non-elliptic zeros of $E_k$. In 2001, Gekeler observed that the polynomials ${\phi }_k$ seem to be irreducible (and verified this claim for $k\le 446$). We show that ${\phi }_k$ is irreducible for infinitely many $k$.