Companion forms and explicit computation of PGL2 number fields with very little ramification

Abstract: In previous works, we described algorithms to compute the number field cut out by the mod ell representation attached to a modular form of level N=1. In this article, we explain how these algorithms can be generalised to forms of higher level N. As an application, we compute the Galois representations attached to a few forms which are supersingular or admit a companion mod ell with ell=13 (and soon ell=41), and we obtain previously unknown number fields of degree ell+1 whose Galois closure has Galois group PGL(2,ell) and a root discriminant that is so small that it beats records for such number fields. Finally, we give a formula to predict the discriminant of the fields obtained by this method, and we use it to find other interesting examples, which are unfortunately out of our computational reach.