The database consists of fields from four sources:

1. The PARI database from the Bordeaux PARI group
2. Additional totally real fields of degrees from 6 to 10 computed by John Voight.
3. Additional fields from John Jones-David Roberts database.
4. Additional fields from Jürgen Klüners-Gunter Malle database.

### Details of the fields contained in the database

1. PARI database: the database is complete for the absolute discriminant $|D|$ less than the given bounds. The bounds for octics come from results of Diaz y Diaz.

degree signature(s) absolute discriminant bound
1[1,0]$1$
2all$10^6$
3[3,0]$2\cdot10^6$
3[1,1]$10^6$
4all$10^6$
5[5,0]$2\cdot10^7$
5[3,1]$10^6$
5[1,2]$10^6$
6[6,0]$10^7$
6[4,1]$10^6$
6[2,2]$4\cdot10^5$
6[0,3]$2\cdot10^5$
7[7,0]$15\cdot10^7$
7[5,1]$12\cdot10^6$
7[3,2]$18\cdot10^5$
7[1,3]$6\cdot10^5$
8[0,4]$1{,}656{,}109$

2. The Voight database is included and is complete for totally real fields with root discriminant less than or equal to the given bound.

degree root discriminant bound
6$20.5$
7$15.5$
8$17$
9$15$
10$14$

3. The Jones-Roberts database provides complete lists of fields satisfying a variety of conditions. The degree of a field is given by $n$.

 Degree $3$ fields unramified outside $\{2,3,5,7,11,13,17,19,23\}$ 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$10$
4. 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.