On the coefficients of certain family of modular equations

The n-th modular equation for the elliptic modular function j(z) has large coefficients even for small n, and those coefficients grow rapidly as n → ∞. The growth of these coefficients was first obtained by Cohen ([5]). And, recently Cais and Conrad ([1], §7) considered this problem for the Hauptmodul j₅(z) of the principal congruence group (5). They found that the ratio of logarithmic heights of n-th modular equations for j(z) and j₅(z) converges to 60 as n → ∞, and observed that 60 is the group index [(1):(5)]. In this paper we prove that their observation is true for Hauptmoduln of somewhat general Fuchsian groups of the first kind with genus zero.