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]$$2\cdot 10^8$$
7[3,2]$$2\cdot 10^8$$
7[1,3]$$2\cdot10^8$$
8[0,4]$$1{,}656{,}109$$

The Jones-Roberts 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.

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.