Rank of elliptic curves associated to Brahmagupta quadrilaterals
Volume 143 / 2016
Colloquium Mathematicum 143 (2016), 187-192
MSC: Primary 11G05; Secondary 14H52.
DOI: 10.4064/cm6556-12-2015
Published online: 21 December 2015
Abstract
We construct a family of elliptic curves with six parameters, arising from a system of Diophantine equations, whose rank is at least five. To do so, we use the Brahmagupta formula for the area of cyclic quadrilaterals $(p^3,q^3,r^3,s^3)$ not necessarily representing genuine geometric objects. It turns out that, as parameters of the curves, the integers $p,q,r,s$ along with the extra integers $u,v$ satisfy $u^6+v^6+p^6+q^6=2(r^6+s^6)$, $uv=pq$, which, by previous work, has infinitely many integer solutions.
