Construction of an infinite family of elliptic curves of 2-selmer rank 1 from heron triangles

Abstract

Given any positive integer n, it is well known that there always exist triangles with rational sides a, b and c such that the area of the triangle is n. Assuming finiteness of the Shafarevich-Tate group, we first construct a family of infinitely many Heronian elliptic curves of rank exactly 1 from Heron triangles of a certain type. We also explicitly produce a separate family of infinitely many Heronian elliptic curves with 2-Selmer rank lying between 1 and 3.