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

Abstract

There is a modular curve X ′ ( 6 ) X’(6) of level 6 6 defined over Q \mathbb {Q} whose Q \mathbb {Q} -rational points correspond to j j -invariants of elliptic curves E E over Q \mathbb {Q} that satisfy Q ( E [ 2 ] ) ⊆ Q ( E [ 3 ] ) \mathbb {Q}(E[2]) \subseteq \mathbb {Q}(E[3]) . In this note we characterize the j j -invariants of elliptic curves with this property by exhibiting an explicit model of X ′ ( 6 ) X’(6) . Our motivation is two-fold: on the one hand, X ′ ( 6 ) 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 ) 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 Q \mathbb {Q} .