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Rank of elliptic curves associated to the Brahmagupta quadrilaterals

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.