If so, do people expect certain invariants (regulator, # of complex embeddings, etc) to fully 'discriminate' between number fields?



Are there two non-isomorphic number fields with the same degree, class number and discriminant?
John Jones has computed [tables of number fields of low degree with prescribed ramification](http://math.asu.edu/~jj/numberfields/). Though the tables just list the defining polynomials and the set of ramified primes, and not any other invariants, it's not hard to search them to find, e.g., that the three quartic fields obtained by adjoining a root of $x^4 - 6$, $x^4 - 24$, and $x^4 - 12x^2 - 16x + 12$ respectively all have degree $4$, class number $1$, and discriminant $-2^{11} \cdot 3^3$. On the other hand these three fields are non-isomorphic (e.g. the regulators distinguish them, the splitting fields distinguish them...).


