On Ito's observation on coefficients of the modular polynomial (Q1817245)

As a result of his extensive computation of the classical modular polynomial, \textit{H. Ito} [ibid. 71, 48-50 (1995; Zbl 0832.11017)] found interesting congruence properties among coefficients of the polynomial modulo the squares of a certain finite number of primes, and remarked that these primes are precisely the prime factors of the order of the simple group ``Monster''. In this note we give an explanation for his observation by employing the well-known fact that the set of primes in question is identical with the set of \(p\)'s with the proprty that all the \(j\)-invariants of supersingular elliptic curves in characteristic \(p\) belong to the prime field \(\mathbb{F}_p\).