Completeness of number field data

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 complex places is $0$ or $1$. In other cases, the bounds have been extended by Jones and Roberts. The bounds for octics come from results of Diaz y Diaz and Battistoni and those for nonics are due to Battistoni.

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	[6,1]	$79{,}259{,}702$
8	[4,2]	$20{,}829{,}049$
8	[2,3]	$5{,}726{,}300$
8	[0,4]	$1{,}656{,}109$
9	[3,3]	$146{,}723{,}910$
9	[1,4]	$39{,}657{,}561$

For fields computed by Jones and Roberts, the 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 $\leq 47$ and conductor $\leq 1000$

Degree $8$ fields with Galois group $Q_8$ which are unramified outside of $\{2,3,5,7,11,13,17,19,23\}$

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	$512$
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, plus the work of others for the Galois group 17T7, 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 $n<20$.