Lower bounds for GL₂ (F_) number fields

Let Fₙ (X;G) denote the set of number fields of degree n with absolute discriminant no larger than X and Galois group G. This set is known to be finite for any finite permutation group G and X 1. In this paper, we give a lower bound for the cases G=GL₂ (F_), \; PGL₂ (F_) for primes 13. We also provide a method to compute lower bounds for any permutation representations of these groups.