We consider elliptic curves
E
/
Q
E / \mathbb {Q}
for which the image of the adelic Galois representation
ρ
E
\rho _E
is as large as possible given a constraint on the image modulo 2. For such curves, we give a characterization in terms of their
ℓ
\ell
-adic images, compute all examples of conductor at most 500,000, precisely describe the image of
ρ
E
\rho _E
, and offer an application to the cyclicity problem. In this way, we generalize some foundational results on Serre curves.