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 that satisfy Q(E[2]) ⊆ 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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