Global number field 22.0.304011857053427966889939263171547.1

• Label: 22.0.30401185705...1547.1
• Degree: $22$
• Signature: $[0, 11]$
• Discriminant: $-\,3^{11}\cdot 23^{20}$
• Root discriminant: $29.96$
• Ramified primes: $3, 23$
• Class number: $23$ (GRH)
• Class group: $[23]$ (GRH)
• Galois group: $C_{22}$ (as 22T1)

Normalized defining polynomial

\( x^{22} - x^{21} + 11 x^{20} - 8 x^{19} + 73 x^{18} - 46 x^{17} + 301 x^{16} - 145 x^{15} + 883 x^{14} - 355 x^{13} + 1776 x^{12} - 498 x^{11} + 2527 x^{10} - 574 x^{9} + 2324 x^{8} - 251 x^{7} + 1358 x^{6} - 161 x^{5} + 400 x^{4} + 20 x^{3} + 51 x^{2} - 6 x + 1 \)

Invariants

Degree:		$22$
Signature:		$[0, 11]$
Discriminant:		\(-304011857053427966889939263171547=-\,3^{11}\cdot 23^{20}\)
Root discriminant:		$29.96$
Ramified primes:		$3, 23$
This field is Galois and abelian over $\Q$.
Conductor:		\(69=3\cdot 23\)
Dirichlet character group:		$\lbrace$$\chi_{69}(64,·)$, $\chi_{69}(1,·)$, $\chi_{69}(2,·)$, $\chi_{69}(4,·)$, $\chi_{69}(8,·)$, $\chi_{69}(13,·)$, $\chi_{69}(16,·)$, $\chi_{69}(25,·)$, $\chi_{69}(26,·)$, $\chi_{69}(29,·)$, $\chi_{69}(31,·)$, $\chi_{69}(32,·)$, $\chi_{69}(35,·)$, $\chi_{69}(41,·)$, $\chi_{69}(47,·)$, $\chi_{69}(49,·)$, $\chi_{69}(50,·)$, $\chi_{69}(52,·)$, $\chi_{69}(55,·)$, $\chi_{69}(58,·)$, $\chi_{69}(59,·)$, $\chi_{69}(62,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{2399572293754950878352937} a^{21} + \frac{308744703297147194070311}{2399572293754950878352937} a^{20} + \frac{88659209219352668960482}{2399572293754950878352937} a^{19} - \frac{907822485756664737749554}{2399572293754950878352937} a^{18} - \frac{1042019138452476636956430}{2399572293754950878352937} a^{17} - \frac{577190319491765609678848}{2399572293754950878352937} a^{16} + \frac{463114657921309389036085}{2399572293754950878352937} a^{15} + \frac{86466375033073064313788}{2399572293754950878352937} a^{14} - \frac{486489714829787901341730}{2399572293754950878352937} a^{13} - \frac{450837216637162129150073}{2399572293754950878352937} a^{12} - \frac{320414944655069441114285}{2399572293754950878352937} a^{11} + \frac{983567855953000123034087}{2399572293754950878352937} a^{10} - \frac{743678039035852251665094}{2399572293754950878352937} a^{9} - \frac{1090171357346250082163579}{2399572293754950878352937} a^{8} - \frac{132012807335142334236934}{2399572293754950878352937} a^{7} - \frac{674991336467845892683651}{2399572293754950878352937} a^{6} - \frac{218471332042608447330226}{2399572293754950878352937} a^{5} - \frac{486944050520569331229800}{2399572293754950878352937} a^{4} + \frac{582045276855027803441066}{2399572293754950878352937} a^{3} - \frac{821889983352727133765399}{2399572293754950878352937} a^{2} - \frac{899290321817951869787853}{2399572293754950878352937} a + \frac{1130012207994179206961748}{2399572293754950878352937}$

Class group and class number

$C_{23}$, which has order $23$ (assuming GRH)

Unit group

Rank:		$10$
Torsion generator:		\( \frac{365951618498234120673324}{2399572293754950878352937} a^{21} - \frac{345158176742881962674269}{2399572293754950878352937} a^{20} + \frac{3982551148800568962307725}{2399572293754950878352937} a^{19} - \frac{2672705178337699431702962}{2399572293754950878352937} a^{18} + \frac{26307336342927927421700759}{2399572293754950878352937} a^{17} - \frac{15101698170156634653261844}{2399572293754950878352937} a^{16} + \frac{107617446263662484381462434}{2399572293754950878352937} a^{15} - \frac{45547391916985823252087632}{2399572293754950878352937} a^{14} + \frac{313725721647580011080943047}{2399572293754950878352937} a^{13} - \frac{107449102908277895677240354}{2399572293754950878352937} a^{12} + \frac{624208609956033132913341787}{2399572293754950878352937} a^{11} - \frac{134857719815979527039251119}{2399572293754950878352937} a^{10} + \frac{878677052495348964966074511}{2399572293754950878352937} a^{9} - \frac{141651134653593414510577387}{2399572293754950878352937} a^{8} + \frac{790014705504716141562122246}{2399572293754950878352937} a^{7} - \frac{23488836215950872022333895}{2399572293754950878352937} a^{6} + \frac{450312759268360585572496091}{2399572293754950878352937} a^{5} - \frac{18986755263211477402816046}{2399572293754950878352937} a^{4} + \frac{121749280575651133363463060}{2399572293754950878352937} a^{3} + \frac{25197200282875729630436597}{2399572293754950878352937} a^{2} + \frac{14245208898295452180533159}{2399572293754950878352937} a + \frac{757739841033766827607698}{2399572293754950878352937} \) (order $6$)
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

\(\Q(\sqrt{-3}) \), \(\Q(\zeta_{23})^+\)

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	$22$	${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$	$22$	${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$	$22$	${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$	R	$22$	${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$	${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$	$22$	${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$	${\href{/LocalNumberField/47.2.0.1}{2} }^{11}$	$22$	$22$

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