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 (p³,q³,r³,s³) 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 u⁶+v⁶+p⁶+q⁶ = 2(r⁶+s⁶), uv = pq, which, by previous work, has infinitely many integer solutions.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm6556-12-2015,
author = {Farzali Izadi and Foad Khoshnam and Arman Shamsi Zargar},
title = {Rank of elliptic curves associated to Brahmagupta quadrilaterals},
journal = {Colloquium Mathematicae},
volume = {144},
year = {2016},
pages = {187-192},
zbl = {06574981},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm6556-12-2015}
}
Farzali Izadi; Foad Khoshnam; Arman Shamsi Zargar. Rank of elliptic curves associated to Brahmagupta quadrilaterals. Colloquium Mathematicae, Tome 144 (2016) pp. 187-192. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm6556-12-2015/