Elliptic curves over real quadratic fields with everywhere good reduction and a non-trivial 3-division point

Abstract

This paper concerns elliptic curves defined over real quadratic fields with everywhere good reduction and a torsion point of order 3 over the same field. We characterize all such elliptic curves by using Tateʼs algorithm. A corollary is that there are infinitely many such elliptic curves. We also calculate all such elliptic curves over real quadratic fields with discriminants < 10 000 up to Galois conjugation.