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Heron Quadrilaterals via Elliptic Curves

Published

Author(s)

Farzali Izadi, Foad Khoshnam, Dustin Moody

Abstract

A Heron quadrilateral is a cyclic quadrilateral with rational area. In this work, we establish a correspondence between Heron quadrilaterals and a family of elliptic curves of the form y^2=x^3+\alpha x^2-n^2 x. This correspondence generalizes the notions of Goins and Maddox who established a similar connection between Heron triangles and elliptic curves. We further study this family of elliptic curves, looking at their torsion groups and ranks. We also explore their connection with congruent numbers, which are the \alpha=0 case. Congruent numbers are positive integers which are the area of a right triangle with rational side lengths.
Citation
Rocky Mountain Journal of Mathematics
Volume
7
Issue
4

Keywords

Heron quadrilaterals, cyclic quadrilaterals, congruent numbers, elliptic curves
Created August 5, 2017, Updated November 10, 2018