A family of congruent number elliptic curves of rank three

Abstract

Recent progress in the theory of Heron triangles and their elliptic curves led to new families of congruent number elliptic curves with rank at least two. Based on these results, we derive a parametric family of congruent number elliptic curves with rank at least three. It turns out that this family is isomorphic to a family which was recently discovered by the third-named author, however the new approach is simpler, more flexible and gives new insight. In particular, it provides in addition three formulae for congruent numbers.