TITLE: Are there two non-isomorphic number fields with the same degree, class number and discriminant?
QUESTION [27 upvotes]: If so, do people expect certain invariants (regulator, # of complex embeddings, etc) to fully 'discriminate' between number fields?

REPLY [5 votes]: I recently heard a fascinating talk by Drew Sutherland on work he's done on this sort of question. Rather than trying to summarize his talk, here is a link to the slides for his talk. He also indicated that he'll be posting an article soon on the ArXiv, but it's not quite ready yet.