Number Fields

The number field database is closed indefinitely. Most, if not all of the fields here can be found in the LMFDB.

Users may be interested in the paper associated with this database. When citing the database, please do so via this paper.


Degree		r₁		r₂
\|D\|		rd(K)		grd(K)
Galois T-num.				h
\|{ram. primes}\|		p_min		p_max

p₁		c₁
p₂		c₂
p₃		c₃
p₄		c₄
p₅		c₅
	Only listed primes can ramify


Sort order:	1		2		3


Max fields shown per page:					The number field database is closed indefinitely

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.
r₁ and r₂ 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.

all class number computations are done assuming GRH. No other computations here are made under the assumption of GRH.

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.

The tables from which the database grew are still accessible here.