On the rank of the elliptic curves with a rational point of order 6
type: Item
instance of: scholarly article
title: On the rank of the elliptic curves with a rational point of order 6
zbMATH Open document ID: 1116.11038
author: Shoichi Kihara
published in: Proceedings of the Japan Academy. Series A
publication date: 2007-06-06T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.pja/1160485501
review text: The author is interested in the existence of elliptic curves over \(\mathbb Q\) of high rank, with a rational point of order \(6\). O. Lecacheux has previously obtained an elliptic curve over \(\mathbb Q(t)\) whose rank is at least \(3\) with a rational point of order \(6\). In his paper, the author constructs an infinite family of elliptic curves over \(\mathbb Q\), of rank at least \(4\) over \(\mathbb Q\), with a rational point of order \(6\). This family is parametrized by the rational points of the elliptic curve given by the equation \(y^2=x(x+4)(x+137)\), whose rank over \(\mathbb Q\) is equal to \(1\). This latest assertion can be justified by verifying that the derivative at \(1\) of its \(L\) function is not zero: it is equal about to \(7,8034\).
reviewed by: Alain Kraus
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 14G05
zbMATH DE Number: 5161644
zbMATH Keywords: Elliptic curves
zbMATH Keywords: rank
describes a project that uses: mwrank
MaRDI profile type: Publication
OpenAlex ID: W4253712682
cites work: Q2914641
cites work: Rank of elliptic curves with nontrivial torsion group
cites work: Rational isogenies of prime degree. (With an appendix by D. Goldfeld)
cites work: Q3710637
DOI: 10.3792/PJAA.82.81
description: scientific article; zbMATH DE number 5161644