Global number field 22.4.951194438369517944038992707.1

• Label: 22.4.95119443836...2707.1
• Degree: $22$
• Signature: $[4, 9]$
• Discriminant: $-\,139\cdot 1583^{2}\cdot 2731^{2}\cdot 6217^{2}\cdot 9473$
• Root discriminant: $16.84$
• Ramified primes: $139, 1583, 2731, 6217, 9473$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: 22T53

Normalized defining polynomial

\( x^{22} - 11 x^{21} + 66 x^{20} - 275 x^{19} + 879 x^{18} - 2268 x^{17} + 4871 x^{16} - 8878 x^{15} + 13903 x^{14} - 18847 x^{13} + 22192 x^{12} - 22691 x^{11} + 20056 x^{10} - 15173 x^{9} + 9644 x^{8} - 4974 x^{7} + 1932 x^{6} - 448 x^{5} - 27 x^{4} + 71 x^{3} - 24 x^{2} + x + 1 \)

Invariants

Degree:		$22$
Signature:		$[4, 9]$
Discriminant:		\(-951194438369517944038992707=-\,139\cdot 1583^{2}\cdot 2731^{2}\cdot 6217^{2}\cdot 9473\)
Root discriminant:		$16.84$
Ramified primes:		$139, 1583, 2731, 6217, 9473$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

Unit group

Rank:		$12$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
Regulator:		\( 43108.9874663 \) (assuming GRH)

Galois group

22T53:

A non-solvable group of order 81749606400

The 752 conjugacy class representatives for t22n53 are not computed

Character table for t22n53 is not computed

Intermediate fields

11.3.26877166541.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 22 sibling:	data not computed
Degree 44 siblings:	data not computed

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }^{2}$	${\href{/LocalNumberField/3.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$	${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$	$22$	${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$	$22$	${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }^{2}$	$18{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$	${\href{/LocalNumberField/29.14.0.1}{14} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$	${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$	${\href{/LocalNumberField/37.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$	$20{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
139	Data not computed
1583	Data not computed
2731	Data not computed
6217	Data not computed
9473	Data not computed