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The database consists of fields from four sources:

  1. The PARI database from the Bordeaux PARI group [pari.math.u-bordeaux.fr/pub/numberfields/]
  2. Totally real fields of degrees from 6 to 10 computed by John Voight [arXiv:0802.0194, math.dartmouth.edu/~jvoight/nf-tables/index.html]
  3. Fields from John Jones-David Roberts database [arXiv:1404.0266, hobbes.la.asu.edu/NFDB] .
  4. Fields from Jürgen Klüners-Gunter Malle database [arXiv:math/0102232, galoisdb.math.upb.de/] .

Details of the fields contained in the database

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]\(1\)
2all\(2\cdot 10^6\)
3all\(3{,}375{,}000\)
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]\(14{,}071{,}004\)
7[3,2]\(5\cdot 10^6\)
7[1,3]\(5\cdot10^6\)
8[0,4]\(1{,}656{,}109\)

The Jones-Roberts database provides complete lists of fields satisfying a variety of conditions. 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 \(p\lt q \leq 5\) with \(n\leq 7\)
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.

Degree 7
\(t\) \(B\)
3\(26\)
5\(38\)
Degree 8
\(t\) \(B\)
3\(20\)
5\(50\)
15\(15\)
18\(15\)
22\(15\)
26\(15\)
29\(15\)
32\(15\)
34\(15\)
36\(15\)
39\(15\)
41\(15\)
45\(15\)
46\(15\)
Degree 9
\(t\) \(B\)
2\(20\)
5\(20\)
6\(20\)
7\(30\)
7\(30\)
8\(15\)
12\(15\)
13\(12\)
14\(18\)
15\(18\)
16\(12\)
17\(18\)
18\(12\)
19\(18\)
21\(15\)
23\(17\)
24\(12\)
25\(15\)
26\(15\)
29\(10\)
30\(10\)
31\(10\)

Selected fields from the Klüners-Malle 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.