The PARI database contains complete lists of fields with the absolute discriminant $D$ less than given bounds, and was compiled from the work of several authors. Fields from Voight extend these lists in cases where the number of real places is \(0\) or \(1\). In other cases, the bounds have been extended by Jones and Roberts. The bound for octics come from results of Diaz y Diaz.
degree  signature(s)  absolute discriminant bound 

1  [1,0]  \(1\) 
2  all  \(2\cdot 10^6\) 
3  all  \(3{,}375{,}000\) 
4  [4,0]  \(10^7\) 
4  [2,1]  \(4\cdot 10^6\) 
4  [0,2]  \(4\cdot 10^6\) 
5  [5,0]  \(10^8\) 
5  [3,1]  \(1.2\cdot10^7\) 
5  [1,2]  \(1.2\cdot10^7\) 
6  [6,0]  \(481{,}890{,}304\) 
6  [4,1]  \(10^7\) 
6  [2,2]  \(10^7\) 
6  [0,3]  \(10^7\) 
7  [7,0]  \(214{,}942{,}297\) 
7  [5,1]  \(2\cdot 10^8\) 
7  [3,2]  \(2\cdot 10^8\) 
7  [1,3]  \(2\cdot10^8\) 
8  [0,4]  \(1{,}656{,}109\) 
The JonesRoberts database provides complete lists of fields satisfying a variety of conditions. Accordingly, number fields in the LMFDB are complete in the following cases. The degree of a field is given by \(n\).
Degree $2$ fields unramified outside \(\{2,3,5,7,11,13,17\}\) 
Degree $3$ fields unramified outside \(\{2,3,5,7,11,13,17,19\}\) 
Degree $4$ fields unramified outside \(\{2,3,5,7,11,13\}\) 
Degree $5$ fields unramified outside \(\{2,3,5,7\}\) 
Fields unramified outside \(\{2,3\}\) with \(n\leq 7\) 
Fields ramified at only one prime \(p\) with \(p<102\) with \(n\leq 7\) 
Fields ramified at only two primes \(p\lt q \lt 500\) with \(n\leq 4\) 
Fields ramified at only two primes \(p\lt q \leq 5\) with \(5\leq n\leq 7\) 
Fields ramified at only three primes \(p\lt q \lt r \lt 100\) with \(n\leq 4\) 
All abelian fields of degree $\leq 15$ and conductor $\leq 300$ 
For the remaining cases, the bound depends on the Galois group. Galois groups are given by \(t\)number. The bound \(B\) is for the root discriminant.



Selected fields from the KlünersMalle database provide examples of fields with many different Galois groups. As a result, the LMFDB contains at least one field for each Galois group (transitive subgroup of $S_n$ up to conjugation) which in degree $n<20$, with the exception of 17T7  no one knows of an example of such a field.
 Review status: reviewed
 Last edited by John Jones on 20190701 16:18:02
 20190701 18:48:50 by John Jones
 20190701 16:18:02 by John Jones (Reviewed)
 20190317 21:03:07 by John Jones (Reviewed)
 20190317 17:59:06 by John Jones
 20181019 18:57:55 by John Jones