Maximality of the 72-fold height reduction for class polynomials from modular functions

Prove that the asymptotic height reduction factor of 72 achieved by Weber modular functions for class polynomials, relative to those formed from the j-invariant, is maximal among all modular functions covering the j-line; equivalently, determine that no modular function yields class polynomials with logarithmic coefficient heights reduced by a factor strictly greater than 72 compared to the j-invariant case.

Background

Yui and Zagier showed that Weber functions produce class polynomials with remarkably small coefficients, asymptotically 72 times smaller in height than those built from the j-invariant. Subsequent work by Gee and by Enge–Morain generalized the framework for class invariants and analyzed height reductions for various eta-quotients and generalized Weber functions.

Despite these advances, no other modular function is currently known to match the 72-fold height reduction achieved by the classical Weber functions. The authors explicitly note that the factor 72 is conjecturally maximal, highlighting a central unresolved question about the theoretical limit of height reduction attainable via modular functions.