Modularity of a Ramanujan-Selberg continued fraction

We study a Ramanujan-Selberg continued fraction S(τ) by employing the modular function theory. We first find modular equations of S(τ) of level n for every positive integer n by using affine models of modular curves. This is an extension of Baruah-Saikia's results for level n=3, 5 and 7. We further show that the ray class field modulo 4 over an imaginary quadratic field K is obtained by the value of S²(τ), and we prove the integrality of 1/S(τ) to find its class polynomial for K with τ∈K∩H, where H is the complex upper half plane.