Elliptic curves and triangles with three rational medians
type: Item
instance of: scholarly article
title: Elliptic curves and triangles with three rational medians
zbMATH Open document ID: 1271.11062
author: Andrej Dujella
author: Juan Carlos Peral
published in: Journal of Number Theory
publication date: 2013-04-15T00:00:00Z
review text: In [\textit{R. H. Buchholz}, J. Number Theory 97, No. 1, 113--131 (2002; Zbl 1037.11020)], it is proved that every rational sided triangle with three rational medians corresponds to a points on a one-parameter family of elliptic curves \(E[m]:y^2=x^3+9(1+m)^2x^2-576(m^2-1)^2m^2x\) and the rank of \(E[m]\) over \(\mathbb Q(m)\) is at least \(2\). In addition, Buchholz states that high rank curves with rank \(5,6,7\) emerge among \(E[m]~(m\in\mathbb Q)\). In this article, the authors prove that the rank of the elliptic curve \(E[m]\) over \(\mathbb Q(m)\) is exactly \(3\). By choosing \(m=(u^2-9)/(6u-73)\) or \(m=-(u^2-9)/4u\) they exhibit subfamilies of elliptic curves whose rank at least \(4\) over \(\mathbb Q(u)\). Further they prove the existence of infinitely many curves of rank at least \(5\) over \(\mathbb Q\) parameterized by an elliptic curve of positive rank and show examples of curves within those families having rank \(9\) and \(10\). These results are obtained from constructing independent points explicitly.
reviewed by: Noburo Ishii
Mathematics Subject Classification ID: 11G05
zbMATH DE Number: 6154108
zbMATH Keywords: elliptic curves
zbMATH Keywords: rank
zbMATH Keywords: triangles
zbMATH Keywords: medians
Wikidata QID: Q57595844
MaRDI profile type: Publication
full work available at URL: j.jnt.2012.12.003
OpenAlex ID: W2002270675
DOI: 10.1016/J.JNT.2012.12.003
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description: scientific article; zbMATH DE number 6154108