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

Brau, Julio; Jones, Nathan

doi:10.1090/proc/12786

Elliptic curves with $2$-torsion contained in the $3$-torsion field
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by Julio Brau and Nathan Jones PDF

Proc. Amer. Math. Soc. 144 (2016), 925-936 Request permission

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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