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