ELLIPTIC CURVES ARISING FROM BRAHMAGUPTA QUADRILATERALS

A Brahmagupta quadrilateral is a cyclic quadrilateral whose sides, diagonals and area are all integer values. In this article, we characterise the notions of Brahmagupta, introduced by K. R. S. Sastry ['Brahmagupta quadrilaterals', Forum Geom. 2 (2002), 167-173], by means of elliptic curves. Motivated by these characterisations, we use Brahmagupta quadrilaterals to construct infinite families of elliptic curves with torsion group Z/2Z x Z/2Z having ranks (at least) four, five and six. Furthermore, by specialising we give examples from these families of specific curves with rank nine.