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}
.