The database consists of fields from four sources:

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

### Details of the fields contained in the database

- 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] \(1\) 2 all \(10^6\) 3 [3,0] \(2\cdot10^6\) 3 [1,1] \(10^6\) 4 all \(10^6\) 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] \(1{,}656{,}109\) 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 \(14\) 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 \(10\) 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.