It is known conditionally on GRH that, for all $\alpha<44.7632 \dots$, there are are only finitely many number fields $K$ satisfying $\Delta\_K ^{ 1/[K:\mathbb Q]}< \alpha$.


For which $\alpha$ can we make this effective by giving an upper bound on $[K:\mathbb Q]$ for all fields satisfying $\Delta\_K ^{ 1/[K:\mathbb Q]}< \alpha$, assuming GRH and maybe some other conjecture?



Effective bounds on root discriminant assuming GRH?
In fact, the answer is contained in one of the references in the survey Lucia mentioned. The relevant table is [here](http://www.dtc.umn.edu/~odlyzko/unpublished/discr.bound.table1).


