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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
DOI: https://doi.org/10.1090/proc/12786
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
Email: jb711@cam.ac.uk

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
Email: ncjones@uic.edu

DOI: https://doi.org/10.1090/proc/12786
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