Optimal curves differing by a 5-isogeny
type: Item
instance of: scholarly article
title: Optimal curves differing by a 5-isogeny
zbMATH Open document ID: 1331.11041
author: Dongho Byeon
author: Taekyung Kim
published in: Acta Arithmetica
publication date: 2014-10-13T00:00:00Z
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 14K02
zbMATH DE Number: 6354988
zbMATH Keywords: elliptic curves
zbMATH Keywords: optimal curves
zbMATH Keywords: isogeny
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4064/aa165-4-5
OpenAlex ID: W2069055256
cites work: Rational torsion on optimal curves and rank-one quadratic twists
DOI: 10.4064/AA165-4-5
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review text: For a positive integer \(N\), let \(X_1(N)\), \(X_0(N)\) denote the usual modular curves. Let \(\mathcal{C}\) denote an isogeny class of elliptic curves defined over \(\mathbb{Q}\) of conductor \(N\). For \(i=0, 1\), there is a unique curve \(E_i\in \mathcal{C}\) and a parametrization \(\phi_i:X_i(N)\rightarrow E_i\) such that for any \(E\in \mathcal{C}\) and a parametrization \(\phi'_i:X_i(N)\rightarrow E\), there is an isogeny \(\pi_i:E_i\rightarrow E\) such that \(\pi_i \circ\phi_i=\phi'_i\). The curve \(E_i\) is called the \(X_i(N)\)-optimal curve. Typically, the optimal curves \(E_0\) and \(E_1\) for a given isogeny class are the same, but not always. Based on numerical observations, \textit{W. A. Stein} and \textit{M. Watkins} [Lect. Notes Comput. Sci. 2369, 267--275 (2002; Zbl 1058.11036)] conjecture that \(E_0\) and \(E_1\) differ by a \(5\)-isogeny if and only if \(E_0=X_0(11)\) and \(E_1=X_1(11)\). In this paper, it is proved that this conjecture is true if \(N\) is square-free and not divisible by \(5\). Moreover, in the process of the proof of this result, the authors prove that a conjecture of \textit{T. Hadano} [Nagoya Math. J. 66, 99--108 (1977; Zbl 0343.14013)] on the classification of rational elliptic curves with a rational point of order 5, is not true.
reviewed by: Nuria Vila
description: scientific article; zbMATH DE number 6354988