Construction of an infinite family of elliptic curves of 2-Selmer rank 1 from Heron triangles

Abstract

Given any positive integer n, there exist triangles, called Heron triangles, with rational sides whose area is n. Assuming the finiteness of the Shafarevich–Tate group, we construct a family of infinitely many Heronian elliptic curves of rank 1 arising from Heron triangles of a certain type. We then explicitly produce a separate family of infinitely many Heronian elliptic curves with 2-Selmer rank lying between 1 and 3.