Number Fields
Users may be interested in the
paper associated with this database.
When citing the database, please do so via this paper.
Input notes
- Warning: the most common error is to treat an
entry blank as a bound. All entry blanks describe sets as described
below. So, entering "101" for "|D|" means that D
must equal 101 (so not many fields will be found).
- rd(K) is the root discriminant of the field K,
i.e., |D|^{1/n}.
- grd(K) is the Galois root discriminant of the field K,
i.e., root discriminant of the Galois closure of K.
- Galois groups are entered by T-numbers.
- For ramifying primes, c_{i} is the exponent of
p_{i} in the discriminant of the field.
- r_{1} and r_{2} are the number of real
and pairs of complex embeddings for the field
- p_{min} refers to the smallest ramifying prime
- p_{max} refers to the largest ramifying prime
- The check box "Only listed primes can ramify" only has meaning if some of
the prime entry blanks p_{i} are filled in, and each such entry
blank is a specific prime. (In general, such a blank could contain "2..11" meaning
that the condition has to be satisfied by any prime between 2 and 11. A query with
an entry of this type will ignore the "Only listed primes can ramify".)
Entry blanks for a variable x can be filled in with
- single values, such as "4", meaning that x = 4.
- ranges, as in "3..7", meaning 3 ≤ x ≤ 7. You can
use I to represent infinity, so "3..I" means x≥ 3.
- combinations of the above separated by commas. So,
"2, 3, 6..11" means x = 2 or x = 3 or 6 ≤ x ≤ 11.
- All values must be non-negative. Including negative values will
likely lead to unexpected results.
- Entry blanks will not do arithmetic for you, so "10^5" will not
produce a meaningful result.
Output notes
- all class number computations are done assuming GRH. No other
computations here are made under the assumption of GRH.
Sources
Most fields here were computed by J. Jones and D. Roberts. In addition,
there are fields computed by
E. Driver,
R. Wallington,
J. Voight,
J.
Klüners and G. Malle, Noam Elkies,
as
well as those
from the Bordeaux
PARI group.
Previous tables
The tables from which the database grew are
still accessible here.