Let $G_{\mathbb{Q}}$ be the absolute Galois group of the rationals. I want to know how many isomorphism classes of 2-dimensional, irreducible, pure Galois representations are there over a finite field $\mathbb{F}_p$, that are unramified outside a fixed finite set of primes $S$.
Is there any literature on the classification of such representations? For example, are there finitely many? Do they all come from residual representations associated to Galois representations of modular forms of a given weight/conductor?
I am particularly interested in understanding the isomorphism classes of the semisimplifications of the residual representations associated to Tate modules of Elliptic curves with nice integral models.
Thanks in advance!
