A new family of elliptic curves with positive ranks arising from the Heron triangles

Abstract

The aim of this paper is to introduce a new family of elliptic curves with positive ranks. These elliptic curves have been constructed with certain rational numbers, namely a, b, and c as sides of Heron triangles having rational areas $k$. It turned out that the torsion groups of this family are of the form $\frac{\mathbb{Z}}{2\mathbb{Z}}\times \frac{\mathbb{Z}}{2\mathbb{Z}}$ and also the rank is positive.