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 | References | Similar Articles | Additional Information

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

MR Author ID:
842244

Email:
ncjones@uic.edu

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