Rank of elliptic curves associated to the Brahmagupta quadrilaterals

TL;DR

This paper constructs a family of elliptic curves with rank at least five using Brahmagupta's formula and solutions to a specific Diophantine system, revealing new insights into the rank properties of these curves.

Contribution

It introduces a novel method to generate elliptic curves of high rank based on Brahmagupta quadrilaterals and associated Diophantine equations.

Findings

Constructed a family of elliptic curves with rank ≥ 5.

Linked the solutions of a Diophantine system to rational points on an elliptic curve.

Demonstrated the existence of high-rank elliptic curves related to Brahmagupta quadrilaterals.

Abstract

In this paper, we construct a family of elliptic curves of rank at least five. To do so, we use the Brahmagupta formula for the area of cyclic quadrilaterals $(p^3,q^3,r^3,q^3)$ not necessarily standing for the genuine sides of quadrilaterals. It turns out that, as parameters of the curves, the integers $p,q,r,s$, along with the extra integers $u$, $v$ satisfy $u^6+v^6+p^6+q^6=2(r^6+s^6)$, $uv=pq$. However, we utilize a subset of the solutions of the above system via the rational points of a specific elliptic curve of positive rank lying on the system.