Global number field 22.0.7198079267989980836471065337135104.1

• Label: 22.0.71980792679...5104.1
• Degree: $22$
• Signature: $[0, 11]$
• Discriminant: $-\,2^{22}\cdot 23^{20}$
• Root discriminant: $34.59$
• Ramified primes: $2, 23$
• Class number: $67$ (GRH)
• Class group: $[67]$ (GRH)
• Galois group: $C_{22}$ (as 22T1)

Normalized defining polynomial

\( x^{22} + 21 x^{20} + 190 x^{18} + 969 x^{16} + 3060 x^{14} + 6188 x^{12} + 8008 x^{10} + 6435 x^{8} + 3003 x^{6} + 715 x^{4} + 66 x^{2} + 1 \)

Invariants

Degree:		$22$
Signature:		$[0, 11]$
Discriminant:		\(-7198079267989980836471065337135104=-\,2^{22}\cdot 23^{20}\)
Root discriminant:		$34.59$
Ramified primes:		$2, 23$
This field is Galois and abelian over $\Q$.
Conductor:		\(92=2^{2}\cdot 23\)
Dirichlet character group:		$\lbrace$$\chi_{92}(1,·)$, $\chi_{92}(3,·)$, $\chi_{92}(71,·)$, $\chi_{92}(9,·)$, $\chi_{92}(75,·)$, $\chi_{92}(13,·)$, $\chi_{92}(77,·)$, $\chi_{92}(81,·)$, $\chi_{92}(85,·)$, $\chi_{92}(73,·)$, $\chi_{92}(87,·)$, $\chi_{92}(25,·)$, $\chi_{92}(27,·)$, $\chi_{92}(29,·)$, $\chi_{92}(31,·)$, $\chi_{92}(35,·)$, $\chi_{92}(39,·)$, $\chi_{92}(41,·)$, $\chi_{92}(47,·)$, $\chi_{92}(49,·)$, $\chi_{92}(55,·)$, $\chi_{92}(59,·)$$\rbrace$
This is 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

$C_{67}$, which has order $67$ (assuming GRH)

Unit group

Rank:		$10$
Torsion generator:		\( a^{21} + 20 a^{19} + 171 a^{17} + 816 a^{15} + 2380 a^{13} + 4368 a^{11} + 5005 a^{9} + 3432 a^{7} + 1287 a^{5} + 220 a^{3} + 11 a \) (order $4$)
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:		\( 1038656.82438 \) (assuming GRH)

Galois group

$C_{22}$ (as 22T1):

A cyclic group of order 22

The 22 conjugacy class representatives for $C_{22}$

Character table for $C_{22}$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\zeta_{23})^+\)

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

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	R	$22$	${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$	$22$	$22$	${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$	${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$	$22$	R	${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$	$22$	${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$	${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$	$22$	${\href{/LocalNumberField/47.2.0.1}{2} }^{11}$	${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$	$22$

In the table, R denotes a ramified prime. 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
2	Data not computed
23	Data not computed