Skip to main content Accessibility help
×
Home
Hostname: page-component-888d5979f-lgdn2 Total loading time: 0.374 Render date: 2021-10-28T05:22:53.214Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

ELLIPTIC CURVES ARISING FROM BRAHMAGUPTA QUADRILATERALS

Part of: Curves

Published online by Cambridge University Press:  10 April 2014

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
Rights & Permissions[Opens in a new window]

Abstract

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

MSC classification

Type
Research Article
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

References

Aguirre, J., Dujella, A. and Peral, J. C., ‘On the rank of elliptic curves coming from rational Diophantine triples’, Rocky Mountain J. Math. 42 (2012), 17591776.CrossRefGoogle Scholar
Alsina, C. and Nelsen, R. B., ‘On the diagonals of a cyclic quadrilateral’, Forum Geom. 7 (2007), 147149.Google Scholar
Buchholz, R. H. and Macdougall, J. A., ‘Heron quadrilaterals with sides in arithmetic or geometric progression’, Bull. Aust. Math. Soc. 59 (1999), 263269.CrossRefGoogle Scholar
Daia, L., ‘On a conjecture’, Gaz. Mat. 89 (1984), 276279 (in Romanian).Google Scholar
Dickson, L. E., History of the Theory of Numbers II (Chelsea, New York, 1971).Google Scholar
Dujella, A., ‘Diophantine triples and construction of high-rank elliptic curves over ℚ with three nontrivial 2-torsion points’, Rocky Mountain J. Math. 30 (2000), 157164.CrossRefGoogle Scholar
Dujella, A., ‘Irregular Diophantine m-tuples and elliptic curves of high rank’, Proc. Japan Acad. Ser. A Math. Sci. 76 (2000), 6667.CrossRefGoogle Scholar
Dujella, A. and Peral, J. C., ‘Elliptic curves coming from Heron triangles’, Rocky Mountain J. Math., to appear.Google Scholar
Elkies, N. D., ‘Three lectures on elliptic surfaces and curves of high rank’, Preprint, 2007, arXiv:0709.2908.Google Scholar
Gupta, R. C., ‘Parameśvara’s rule for the circumradius of a cyclic quadrilateral’, Historia Math. 4 (1977), 6774.CrossRefGoogle Scholar
Ismailescu, D. and Vojdany, A., ‘Class preserving dissections of convex quadrilatrals’, Forum Geom. 9 (2009), 195211.Google Scholar
Izadi, F., Khoshnam, F. and Nabardi, K., ‘A new family of elliptic curves with positive ranks arising from the Heron triangles’, Preprint, 2010, arXiv:1012.5835.Google Scholar
Mestre, J. -F., ‘Construction de courbes elliptiques sur ℚ de rang ≥ 12’, C. R. Acad. Sci. Paris Ser. I 295 (1982), 643644.Google Scholar
Nagao, K., ‘An example of elliptic curve over ℚ with rank ≥ 20’, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), 291293.CrossRefGoogle Scholar
SAGE software, Version 4.3.5, http://sagemath.org.Google Scholar
Sastry, K. R. S., ‘Brahmagupta quadrilaterals’, Forum Geom. 2 (2002), 167173.Google Scholar
Silverman, J. H., Advanced Topics in the Arithmetic of Elliptic Curves (Springer, New York, 1994).CrossRefGoogle Scholar
You have Access
5
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

ELLIPTIC CURVES ARISING FROM BRAHMAGUPTA QUADRILATERALS
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

ELLIPTIC CURVES ARISING FROM BRAHMAGUPTA QUADRILATERALS
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

ELLIPTIC CURVES ARISING FROM BRAHMAGUPTA QUADRILATERALS
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *