Elliptic curves having the same 6-torsion of a given elliptic curve
type: Item
instance of: scholarly article
title: Elliptic curves having the same 6-torsion of a given elliptic curve
zbMATH Open document ID: 0968.11025
author: Yanyan Li
published in: Journal of Number Theory
publication date: 2001-09-14T00:00:00Z
review text: Let \(K\) be a field of characteristic different from \(2\) and \(3\), and let \(E\) be an elliptic curve defined over \(K\). Let \(E[6]\) be its group of \(6\)-torsion points. A question raised by Mazur is: Is there a finite or an infinite number of elliptic curves \(F\), defined over \(K\), such that \(F[6]\) is isomorphic to \(E[6]\) as a \(\text{Gal}(\overline{K}/K)\)-module? This moduli-problem is parametrized by the elliptic curve \(\widetilde{E}\) of equation \(y^2 = x^3 + \Delta_E\), whereas \(\Delta_E\) is the discriminant of a Weierstrass equation of \(E\). Hence the answer to the problem stated depends on the \(K\)-rank of \(\widetilde{E}\).
Mathematics Subject Classification ID: 11G05
zbMATH DE Number: 1389293
zbMATH Keywords: torsion of elliptic curves
reviewed by: Franck Leprévost
MaRDI profile type: Publication
full work available at URL: jnth.1999.2410
OpenAlex ID: W1981952705
cites work: Rational isogenies of prime degree. (With an appendix by D. Goldfeld)
cites work: Galois properties of points of finite order of elliptic curves
cites work: Q3710637
cites work: Q3871017
DOI: 10.1006/JNTH.1999.2410
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description: scientific article; zbMATH DE number 1389293