Global number field 22.0.333299588378851543700346160352728825135889034361065589007483.1

• Label: 22.0.33329958837...7483.1
• Degree: $22$
• Signature: $[0, 11]$
• Discriminant: $-\,683^{21}$
• Root discriminant: $507.67$
• Ramified prime: $683$
• Class number: $1047240485$ (GRH)
• Class group: $[1047240485]$ (GRH)
• Galois group: $C_{22}$ (as 22T1)

Normalized defining polynomial

\( x^{22} - x^{21} + 16 x^{20} - 336 x^{19} - 300 x^{18} - 8892 x^{17} - 82988 x^{16} + 568691 x^{15} + 488956 x^{14} + 27309015 x^{13} + 83250417 x^{12} - 666527265 x^{11} + 3371076691 x^{10} - 45350489518 x^{9} + 323098105759 x^{8} - 1640795352234 x^{7} + 8669584750392 x^{6} - 33274597103613 x^{5} + 77339144364048 x^{4} - 119648150339229 x^{3} + 126007273791531 x^{2} - 75637782101187 x + 39680501725269 \)

Invariants

Degree:		$22$
Signature:		$[0, 11]$
Discriminant:		\(-333299588378851543700346160352728825135889034361065589007483=-\,683^{21}\)
Root discriminant:		$507.67$
Ramified primes:		$683$
This field is Galois and abelian over $\Q$.
Conductor:		\(683\)
Dirichlet character group:		$\lbrace$$\chi_{683}(128,·)$, $\chi_{683}(1,·)$, $\chi_{683}(2,·)$, $\chi_{683}(427,·)$, $\chi_{683}(4,·)$, $\chi_{683}(342,·)$, $\chi_{683}(64,·)$, $\chi_{683}(8,·)$, $\chi_{683}(651,·)$, $\chi_{683}(256,·)$, $\chi_{683}(16,·)$, $\chi_{683}(341,·)$, $\chi_{683}(512,·)$, $\chi_{683}(667,·)$, $\chi_{683}(32,·)$, $\chi_{683}(555,·)$, $\chi_{683}(675,·)$, $\chi_{683}(679,·)$, $\chi_{683}(681,·)$, $\chi_{683}(682,·)$, $\chi_{683}(171,·)$, $\chi_{683}(619,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{9} a^{6} + \frac{1}{9} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{27} a^{7} + \frac{1}{27} a^{6} + \frac{4}{27} a^{5} + \frac{1}{27} a^{4} + \frac{1}{27} a^{3} - \frac{2}{27} a^{2} - \frac{2}{9} a$, $\frac{1}{27} a^{8} - \frac{1}{9} a^{5} - \frac{1}{9} a^{4} - \frac{1}{9} a^{3} + \frac{2}{27} a^{2} + \frac{2}{9} a$, $\frac{1}{81} a^{9} + \frac{1}{27} a^{6} - \frac{4}{27} a^{5} + \frac{1}{27} a^{4} - \frac{7}{81} a^{3} - \frac{2}{27} a^{2} + \frac{2}{9} a$, $\frac{1}{243} a^{10} + \frac{1}{243} a^{9} - \frac{1}{81} a^{8} + \frac{2}{81} a^{6} + \frac{5}{81} a^{5} + \frac{20}{243} a^{4} + \frac{20}{243} a^{3} - \frac{8}{81} a^{2} - \frac{4}{27} a$, $\frac{1}{243} a^{11} - \frac{1}{243} a^{9} + \frac{1}{81} a^{8} - \frac{1}{81} a^{7} + \frac{1}{27} a^{6} + \frac{14}{243} a^{5} + \frac{7}{243} a^{3} - \frac{4}{81} a^{2} - \frac{2}{27} a$, $\frac{1}{243} a^{12} + \frac{1}{243} a^{9} + \frac{1}{81} a^{8} + \frac{2}{243} a^{6} - \frac{4}{81} a^{5} - \frac{2}{27} a^{4} - \frac{7}{243} a^{3} + \frac{4}{81} a^{2} + \frac{2}{27} a$, $\frac{1}{729} a^{13} + \frac{1}{729} a^{12} - \frac{1}{729} a^{11} - \frac{1}{729} a^{10} - \frac{1}{243} a^{9} - \frac{1}{243} a^{8} + \frac{5}{729} a^{7} + \frac{5}{729} a^{6} - \frac{56}{729} a^{5} - \frac{2}{729} a^{4} - \frac{2}{27} a^{3} + \frac{4}{27} a$, $\frac{1}{6561} a^{14} - \frac{1}{6561} a^{13} - \frac{1}{2187} a^{12} + \frac{10}{6561} a^{11} - \frac{1}{6561} a^{10} - \frac{2}{2187} a^{9} + \frac{11}{6561} a^{8} + \frac{103}{6561} a^{7} + \frac{104}{2187} a^{6} - \frac{601}{6561} a^{5} - \frac{401}{6561} a^{4} - \frac{119}{729} a^{3} + \frac{37}{81} a^{2} + \frac{58}{243} a - \frac{4}{9}$, $\frac{1}{6561} a^{15} - \frac{4}{6561} a^{13} + \frac{7}{6561} a^{12} + \frac{1}{729} a^{11} - \frac{7}{6561} a^{10} + \frac{5}{6561} a^{9} + \frac{38}{2187} a^{8} - \frac{71}{6561} a^{7} - \frac{46}{6561} a^{6} - \frac{253}{2187} a^{5} + \frac{958}{6561} a^{4} - \frac{83}{729} a^{3} + \frac{70}{243} a^{2} + \frac{58}{243} a - \frac{4}{9}$, $\frac{1}{19683} a^{16} + \frac{1}{19683} a^{15} + \frac{1}{19683} a^{14} - \frac{11}{19683} a^{13} - \frac{8}{19683} a^{12} - \frac{20}{19683} a^{11} + \frac{2}{19683} a^{10} + \frac{116}{19683} a^{9} - \frac{118}{19683} a^{8} + \frac{353}{19683} a^{7} + \frac{224}{19683} a^{6} - \frac{1249}{19683} a^{5} + \frac{947}{6561} a^{4} - \frac{22}{243} a^{3} + \frac{296}{729} a^{2} + \frac{32}{243} a + \frac{4}{9}$, $\frac{1}{177147} a^{17} - \frac{4}{177147} a^{16} + \frac{5}{177147} a^{15} + \frac{11}{177147} a^{14} - \frac{16}{177147} a^{13} + \frac{326}{177147} a^{12} - \frac{11}{59049} a^{11} + \frac{259}{177147} a^{10} - \frac{5}{177147} a^{9} - \frac{1865}{177147} a^{8} + \frac{2059}{177147} a^{7} + \frac{2644}{177147} a^{6} + \frac{16889}{177147} a^{5} - \frac{8089}{59049} a^{4} - \frac{580}{6561} a^{3} + \frac{1982}{6561} a^{2} + \frac{689}{2187} a + \frac{40}{81}$, $\frac{1}{35606547} a^{18} + \frac{77}{35606547} a^{17} + \frac{626}{35606547} a^{16} + \frac{2252}{35606547} a^{15} + \frac{1010}{35606547} a^{14} - \frac{25}{35606547} a^{13} - \frac{18461}{11868849} a^{12} - \frac{4871}{35606547} a^{11} + \frac{19381}{35606547} a^{10} + \frac{83131}{35606547} a^{9} - \frac{194825}{35606547} a^{8} - \frac{25922}{35606547} a^{7} + \frac{1617719}{35606547} a^{6} - \frac{1948462}{11868849} a^{5} - \frac{166840}{1318761} a^{4} - \frac{21238}{1318761} a^{3} + \frac{136886}{439587} a^{2} - \frac{2036}{5427} a - \frac{2}{9}$, $\frac{1}{106819641} a^{19} + \frac{1}{106819641} a^{18} - \frac{368}{35606547} a^{16} - \frac{970}{35606547} a^{15} + \frac{2612}{35606547} a^{14} + \frac{4003}{106819641} a^{13} - \frac{170327}{106819641} a^{12} - \frac{48830}{35606547} a^{11} + \frac{29510}{35606547} a^{10} - \frac{142703}{35606547} a^{9} + \frac{154918}{35606547} a^{8} - \frac{641249}{106819641} a^{7} - \frac{3004622}{106819641} a^{6} + \frac{988676}{35606547} a^{5} - \frac{1496708}{11868849} a^{4} + \frac{401068}{3956283} a^{3} + \frac{214460}{1318761} a^{2} - \frac{86276}{439587} a - \frac{31}{243}$, $\frac{1}{320458923} a^{20} - \frac{1}{320458923} a^{19} + \frac{1}{320458923} a^{18} + \frac{37}{35606547} a^{17} - \frac{1216}{106819641} a^{16} - \frac{2489}{35606547} a^{15} + \frac{20908}{320458923} a^{14} + \frac{127916}{320458923} a^{13} - \frac{262631}{320458923} a^{12} + \frac{114460}{106819641} a^{11} + \frac{128002}{106819641} a^{10} - \frac{108389}{35606547} a^{9} - \frac{3639215}{320458923} a^{8} + \frac{1285661}{320458923} a^{7} - \frac{859445}{320458923} a^{6} - \frac{1067917}{106819641} a^{5} + \frac{5013193}{35606547} a^{4} - \frac{639746}{11868849} a^{3} + \frac{1577939}{3956283} a^{2} + \frac{148939}{1318761} a + \frac{221}{729}$, $\frac{1}{19162386736636176100299339183092171064668184924947129504249322215233541010251947269124656623829} a^{21} - \frac{20914930649745056106010705078182078600470263875520844911613411672207007095341044401317}{19162386736636176100299339183092171064668184924947129504249322215233541010251947269124656623829} a^{20} - \frac{58867053491530555344532747017141084812565336034603815176660421486324766952563205220908}{19162386736636176100299339183092171064668184924947129504249322215233541010251947269124656623829} a^{19} + \frac{88085396371585485628621307358523091323234447229183040223713166497686012222120073214582}{6387462245545392033433113061030723688222728308315709834749774071744513670083982423041552207943} a^{18} + \frac{3437058532167453142853494727186377727454914862714895581680100146105550672195762218879639}{6387462245545392033433113061030723688222728308315709834749774071744513670083982423041552207943} a^{17} + \frac{7552543871270522970801489757414004659054394456902731836756972350395166117458524927021377}{709718027282821337048123673447858187580303145368412203861086007971612630009331380337950245327} a^{16} + \frac{449497997399157149568141256252476292782321431102593577352931303686983355918474633148216099}{19162386736636176100299339183092171064668184924947129504249322215233541010251947269124656623829} a^{15} - \frac{952267402668237045365213014306564808760727636009859210148313097876958643289728519990867546}{19162386736636176100299339183092171064668184924947129504249322215233541010251947269124656623829} a^{14} - \frac{12982395511950111603808168020859724008656383346600384993870656952030244810553672893720687610}{19162386736636176100299339183092171064668184924947129504249322215233541010251947269124656623829} a^{13} - \frac{4415941187744385247827787203791203861838397252883355828439453167822159422674903986344428574}{6387462245545392033433113061030723688222728308315709834749774071744513670083982423041552207943} a^{12} - \frac{4898618530342195227338761955506518368624932043832408365046009661849290413671013546514493934}{6387462245545392033433113061030723688222728308315709834749774071744513670083982423041552207943} a^{11} + \frac{157116683903095054653398107402600283421646099962880627466777972223807514548393398785412501}{2129154081848464011144371020343574562740909436105236611583258023914837890027994141013850735981} a^{10} - \frac{41863155010986397089698562092815418937477221634277029821553553657121531420225869259926013600}{19162386736636176100299339183092171064668184924947129504249322215233541010251947269124656623829} a^{9} + \frac{263530508040486664236704091392253770320282066436507823722789510454863009414457808480496598356}{19162386736636176100299339183092171064668184924947129504249322215233541010251947269124656623829} a^{8} - \frac{288112559871381706172067076103857262992557589842151483425162635128508037811383364345367851217}{19162386736636176100299339183092171064668184924947129504249322215233541010251947269124656623829} a^{7} - \frac{101388794367919443253751791625797390362802318244688939772621210568769191583478254066355869215}{2129154081848464011144371020343574562740909436105236611583258023914837890027994141013850735981} a^{6} + \frac{102799502801647138325818436849914858001419331833802462215082706056650617657511119675975883585}{709718027282821337048123673447858187580303145368412203861086007971612630009331380337950245327} a^{5} - \frac{101267647001098776104571322830463215623244176869041268003155682517570474021007406872080006869}{709718027282821337048123673447858187580303145368412203861086007971612630009331380337950245327} a^{4} - \frac{1279972275491774193258134106756312040580176571626281635157559781403389208582029805613201139}{78857558586980148560902630383095354175589238374268022651231778663512514445481264481994471703} a^{3} + \frac{3207244137439585472553107815509190411792309372436794775910504193087752491250996146582253599}{26285852862326716186967543461031784725196412791422674217077259554504171481827088160664823901} a^{2} - \frac{285358813144777175944845016896049737096767638718775779814062346582626788584778821971657684}{710428455738559896945068742190048235816119264633045249110196204175788418427759139477427673} a - \frac{1479406376602837916387766842611131964418977227534159426244350420523485767823620821742485}{14530598597195531336079349619144159604862583079835640805460066088725357369721994560898189}$

Class group and class number

$C_{1047240485}$, which has order $1047240485$ (assuming GRH)

Unit group

Rank:		$10$
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:		\( 2331588007058459.0 \) (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

\(\Q(\sqrt{-683}) \), 11.11.22090575837180674640752471449.1

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	$22$	${\href{/LocalNumberField/3.1.0.1}{1} }^{22}$	$22$	$22$	$22$	$22$	$22$	${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$	$22$	${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$	$22$	${\href{/LocalNumberField/37.2.0.1}{2} }^{11}$	$22$	$22$	$22$	${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$	${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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
683	Data not computed