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Elliptic curves with $2$-torsion contained in the $3$-torsion field

Authors: Julio Brau and Nathan Jones
Journal: Proc. Amer. Math. Soc. 144 (2016), 925-936
MSC (2010): Primary 11G05
Published electronically: July 8, 2015
MathSciNet review: 3447646
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Abstract: There is a modular curve $X’(6)$ of level $6$ defined over $\mathbb {Q}$ whose $\mathbb {Q}$-rational points correspond to $j$-invariants of elliptic curves $E$ over $\mathbb {Q}$ that satisfy $\mathbb {Q}(E[2]) \subseteq \mathbb {Q}(E[3])$. In this note we characterize the $j$-invariants of elliptic curves with this property by exhibiting an explicit model of $X’(6)$. Our motivation is two-fold: on the one hand, $X’(6)$ belongs to the list of modular curves which parametrize non-Serre curves (and is not well known), and on the other hand, $X’(6)(\mathbb {Q})$ gives an infinite family of examples of elliptic curves with non-abelian “entanglement fields”, which is relevant to the systematic study of correction factors of various conjectural constants for elliptic curves over $\mathbb {Q}$.

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Additional Information

Julio Brau
Affiliation: Faculty of Mathematics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, United Kingdom

Nathan Jones
Affiliation: Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 322 Science and Engineering Offices (M/C 249), 851 S. Morgan Street, Chicago, Illinois 60607-7045
MR Author ID: 842244

Received by editor(s): June 8, 2014
Received by editor(s) in revised form: February 4, 2015
Published electronically: July 8, 2015
Communicated by: Romyar T. Sharifi
Article copyright: © Copyright 2015 American Mathematical Society