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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.

    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]\(6\cdot10^5\)

  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.