TITLE: Concrete bounds on the discriminant of a number field
QUESTION [5 upvotes]: There exists a well known concrete bound on the discriminant of a number field by Minkowski.
Are there any concrete (completely explicit) improvements of this bound? 
I know of a bound by Odlyzko, but it is only asymptotic, and thus not explicit/concrete enough. 
In particular, I do not know how to apply this bound to a number field of degree (say) $7$.
That is, I am looking for a bound on the discriminant as a function of the degree $n$ of the number field over $\mathbb{Q}$ (improving Minkowski's bound even if $n$ is not too large). 
I am most interested in the case of at most two real embeddings.

REPLY [8 votes]: Theorem 2.4 of the paper Local corrections of discriminant bounds and small degree extensions of quadratic base fields by Brueggeman and Doud gives two lower bounds for the discriminant of a number field of arbitrary signature. The formula is a bit complicated, but can easily be computed with a computer algebra system like SAGE. Note that the formula also makes use of Poitou's refinement of the Odlyzko bounds in that the bound incorporates knowledge about the existence of primes of small norm. (For instance, the bound implies that a degree 10 number field with 2 real embeddings has root discriminant at least 5.834, whereas a degree 10 number field with 2 real embeddings in which 2 splits completely has root discriminant at least 28.951.)