The title says it all. Singular moduli of the j-function satisfy polynomials, but as the class number grows, these polynomial coefficients become very large. Weber functions are modular (not over the full modular group), and their values also satisfy polynomials. But the Weber polynomials tend to have much smaller coefficents. Why?



Why Are Weber Polynomial Coefficients Smaller than Hilbert Polynomial Coefficients?
Simply because they satisfy an equation of the form $P(f)-fj$ for some polynomial $P$. This immediately implies that the height of $f(z)$ will be around $1/deg(P)$ of the height of $j(z)$, or more precisely, asymptotic to it as the discriminant of $z$ goes to infinity.


See A. Enge and F. Morain's "Comparing invariants for class ﬁelds of imaginary quadratic
ﬁelds", ANTS-V 2002.


