Some observations on the arithmetic of Eisenstein series for the modular group SL(2, \Bbb Z)

For even integers k\geqq4k\geqq4, let j_k(X)\varphi_k(X) be the separable rational polynomial that encodes the j-invariants of non-elliptic zeroes of the Eisenstein series E_k for the modular group SL(2,Bbb Z)(2,{Bbb Z}). We prove Kummer-type congruence properties for the j_k\varphi_k and, based on extensive calculations, make observations about the Galois group, the discriminant, and the distribution of zeroes of j_k(X)\varphi_k(X).