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

Abstract : There is a modular curve X'(6) of level 6 defined over Q whose Q-rational points correspond to j-invariants of elliptic curves E over Q for which Q(E[2]) is a subfield of 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)(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 Q.
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Submitted on : Friday, January 30, 2015 - 8:36:02 PM
Last modification on : Friday, August 2, 2019 - 2:23:29 AM

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  • HAL Id : hal-01111744, version 1
  • ARXIV : 1406.1262

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Julio Brau, Nathan Jones. Elliptic curves with 2-torsion contained in the 3-torsion field. Proceedings of the American Mathematical Society, American Mathematical Society, 2016, 144, pp.925-936. ⟨hal-01111744⟩

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