Computation of the cover of Shimura curves $X_0(2) \to X(1)$ for the cyclic cubic field of discriminant 13^2

TL;DR

This paper computes the canonical model of a specific Shimura curve cover related to a cyclic cubic field of discriminant 13^2 and identifies rational CM points, advancing explicit understanding of these algebraic structures.

Contribution

It provides the explicit computation of the canonical model of the Shimura curve cover for the specified cubic field, including rational CM points, which was not previously available.

Findings

Explicit equations for the Shimura curve cover are obtained.

Coordinates of rational CM points on X(1) are listed.

The results extend previous theoretical descriptions with concrete data.

Abstract

We compute the canonical model of the cover of Shimura curves $X_0(2) \to X(1)$ for the cubic field of discriminant 13^2 described at the end of Elkies' paper "Shimura curves for level 3 subgroups of the (2,3,7) triangle group". Last, we list the coordinates of some rational CM points on X(1).