Global number field 22.22.6992272712228843238468603052945581.1: \(\Q(\zeta_{69})^+\)

• Label: 22.22.6992272712...5581.1
• Degree: $22$
• Signature: $[22, 0]$
• Discriminant: $3^{11}\cdot 23^{21}$
• Root discriminant: $34.55$
• Ramified primes: $3, 23$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $C_{22}$ (as 22T1)

Normalized defining polynomial

\( x^{22} - x^{21} - 22 x^{20} + 22 x^{19} + 208 x^{18} - 208 x^{17} - 1103 x^{16} + 1103 x^{15} + 3589 x^{14} - 3589 x^{13} - 7359 x^{12} + 7359 x^{11} + 9385 x^{10} - 9385 x^{9} - 7060 x^{8} + 7060 x^{7} + 2807 x^{6} - 2807 x^{5} - 482 x^{4} + 482 x^{3} + 24 x^{2} - 24 x + 1 \)

Invariants

Degree:		$22$
Signature:		$[22, 0]$
Discriminant:		\(6992272712228843238468603052945581=3^{11}\cdot 23^{21}\)
Root discriminant:		$34.55$
Ramified primes:		$3, 23$
This field is Galois and abelian over $\Q$.
Conductor:		\(69=3\cdot 23\)
Dirichlet character group:		$\lbrace$$\chi_{69}(64,·)$, $\chi_{69}(1,·)$, $\chi_{69}(4,·)$, $\chi_{69}(5,·)$, $\chi_{69}(68,·)$, $\chi_{69}(65,·)$, $\chi_{69}(11,·)$, $\chi_{69}(13,·)$, $\chi_{69}(14,·)$, $\chi_{69}(16,·)$, $\chi_{69}(17,·)$, $\chi_{69}(20,·)$, $\chi_{69}(25,·)$, $\chi_{69}(31,·)$, $\chi_{69}(38,·)$, $\chi_{69}(44,·)$, $\chi_{69}(49,·)$, $\chi_{69}(52,·)$, $\chi_{69}(53,·)$, $\chi_{69}(55,·)$, $\chi_{69}(56,·)$, $\chi_{69}(58,·)$$\rbrace$
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:		$21$
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:		\( 5503144493.95 \) (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

\(\Q(\sqrt{69}) \), \(\Q(\zeta_{23})^+\)

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	$22$	R	${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$	$22$	${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$	${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$	${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$	$22$	R	$22$	${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$	$22$	$22$	$22$	${\href{/LocalNumberField/47.2.0.1}{2} }^{11}$	${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$	$22$

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
3	Data not computed
23	Data not computed