Generation of Hauptmoduln of \(\Gamma_{1}(N)\) by Weierstrass units and application to class fields (Q651299)

Let \(X_1(N)\) be the modular curve associated with the modular group \(\Gamma_1(N)\) for a positive integer \(N\). The authors construct a generator (Hauptmodul) of the function field of \(X_1(N)\) in the case that its genus is \(0\), thus, \(1\leq N\leq 10\) and \(N=12\) by using Weierstrass units. Let \(\alpha\) be an imaginary quadratic point on the complex upper plane. Then the value of this generator at \(\alpha\) is an algebraic integer and generates a ray class field over \(\mathbb Q(\alpha)\) with conductor dividing \(N\).