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ELLIPTIC CURVES ARISING FROM BRAHMAGUPTA QUADRILATERALS

Part of: Curves

Published online by Cambridge University Press:  10 April 2014

FARZALI IZADI ,
FOAD KHOSHNAM ,
DUSTIN MOODY  and
ARMAN SHAMSI ZARGAR
FARZALI IZADI
Affiliation:
Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz 53751-71379, Iran email farzali.izadi@azaruniv.edu
FOAD KHOSHNAM
Affiliation:
Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz 53751-71379, Iran email khoshnam@azaruniv.edu
DUSTIN MOODY*
Affiliation:
Computer Security Division, National Institute of Standards and Technology, 100 Bureau Drive, Gaithersburg, MD 20899-8930, USA email dustin.moody@nist.gov
ARMAN SHAMSI  ZARGAR
Affiliation:
Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz 53751-71379, Iran email shzargar.arman@azaruniv.edu
*
✉dustin.moody@nist.gov
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Abstract

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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 $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ having ranks (at least) four, five and six. Furthermore, by specialising we give examples from these families of specific curves with rank nine.

Keywords

Brahmagupta quadrilateralelliptic curveHeron trianglerank

MSC classification

Secondary: 14H52: Elliptic curves
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 90 , Issue 1 , August 2014 , pp. 47 - 56
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

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