Let E be an elliptic curve defined over Q. Associated to E, there is an adelic Galois representation. In this talk, I will discuss a joint work with Jacob Mayle where we consider those elliptic curves defined over Q whose adelic Galois image is as large as possible given a constraint on image modulo 2. For such curves, we give a characterization in terms of their l-adic images, compute all examples of conductor at most 500,000, precisely describe the image of their adelic Galois representation and offer an application to cyclicity problem. In this way, we generalize some foundational results on Serre curves.