On the $j$-invariant and the Legendre Representation of Elliptic Curves with Complex Multiplication

Abstract

By introducing a class of meromorphic functions with certain ramification structures on ${\mathbb{CP}}^1$, a new method for the determination of the Legendre representation of elliptic curves with complex multiplication is introduced. These functions reduce the desired representation to the solution of an explicitly given system of polynomial equations and makes no use of the knowledge of Hilbert class polynomial on which standard computations depend. As a byproduct, an algorithm for computing $j(k\tau )$ in terms of $j(\tau )$ is obtained and implemented for $k=2,3$. Solving the system of equations provides a method for computing certain special values of the modular function $j:\mathbb{H}\to \mathbb{C}$.