Global number field 22.0.15329357105602142437812938855438783298481123683.1

• Label: 22.0.15329357105...3683.1
• Degree: $22$
• Signature: $[0, 11]$
• Discriminant: $-\,3^{11}\cdot 89^{21}$
• Root discriminant: $125.70$
• Ramified primes: $3, 89$
• Class number: $51658$ (GRH)
• Class group: $[51658]$ (GRH)
• Galois group: $C_{22}$ (as 22T1)

Normalized defining polynomial

\( x^{22} - x^{21} + 47 x^{20} - 52 x^{19} + 792 x^{18} - 1785 x^{17} + 6826 x^{16} - 27519 x^{15} + 65725 x^{14} - 239894 x^{13} + 428795 x^{12} - 1132774 x^{11} + 3194187 x^{10} - 2353532 x^{9} + 12889600 x^{8} - 10825379 x^{7} + 17897782 x^{6} - 53427788 x^{5} + 50690086 x^{4} - 31652698 x^{3} + 43810715 x^{2} - 2705016 x + 8609149 \)

Invariants

Degree:		$22$
Signature:		$[0, 11]$
Discriminant:		\(-15329357105602142437812938855438783298481123683=-\,3^{11}\cdot 89^{21}\)
Root discriminant:		$125.70$
Ramified primes:		$3, 89$
This field is Galois and abelian over $\Q$.
Conductor:		\(267=3\cdot 89\)
Dirichlet character group:		$\lbrace$$\chi_{267}(256,·)$, $\chi_{267}(1,·)$, $\chi_{267}(67,·)$, $\chi_{267}(4,·)$, $\chi_{267}(263,·)$, $\chi_{267}(64,·)$, $\chi_{267}(266,·)$, $\chi_{267}(11,·)$, $\chi_{267}(16,·)$, $\chi_{267}(146,·)$, $\chi_{267}(203,·)$, $\chi_{267}(217,·)$, $\chi_{267}(91,·)$, $\chi_{267}(223,·)$, $\chi_{267}(97,·)$, $\chi_{267}(170,·)$, $\chi_{267}(44,·)$, $\chi_{267}(176,·)$, $\chi_{267}(200,·)$, $\chi_{267}(50,·)$, $\chi_{267}(121,·)$, $\chi_{267}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{101} a^{20} - \frac{11}{101} a^{19} - \frac{27}{101} a^{18} + \frac{20}{101} a^{17} + \frac{5}{101} a^{16} + \frac{40}{101} a^{15} - \frac{49}{101} a^{14} - \frac{49}{101} a^{13} - \frac{14}{101} a^{12} + \frac{47}{101} a^{11} + \frac{35}{101} a^{10} + \frac{33}{101} a^{9} - \frac{42}{101} a^{8} - \frac{26}{101} a^{7} - \frac{11}{101} a^{6} + \frac{49}{101} a^{5} - \frac{5}{101} a^{4} + \frac{23}{101} a^{3} - \frac{13}{101} a^{2} + \frac{34}{101} a - \frac{23}{101}$, $\frac{1}{1733915904584460205979701102743525951855149611913460601229251202203388170363590573105401303} a^{21} + \frac{2262998991165935744674468921578330889398020917061649375158172857306191709022735068351026}{1733915904584460205979701102743525951855149611913460601229251202203388170363590573105401303} a^{20} - \frac{794890469433170335860140988798399160893899685289122826883654807500223099167953210052640787}{1733915904584460205979701102743525951855149611913460601229251202203388170363590573105401303} a^{19} + \frac{333497372431053603343668125465326618236475558285676228576040850527141061924727178455501165}{1733915904584460205979701102743525951855149611913460601229251202203388170363590573105401303} a^{18} - \frac{211531682133671640262497901312660011321492778167691316443458178173657770960684109763204731}{1733915904584460205979701102743525951855149611913460601229251202203388170363590573105401303} a^{17} + \frac{749783726867902384751003760697894952904333953316086657856024323445632796850330912392091996}{1733915904584460205979701102743525951855149611913460601229251202203388170363590573105401303} a^{16} + \frac{465786878325139563486255260317330823949234331342704450363992106845621018755514176108622350}{1733915904584460205979701102743525951855149611913460601229251202203388170363590573105401303} a^{15} - \frac{582382304645668058906877339664220582740909467513362571431948795339287938894720883358023255}{1733915904584460205979701102743525951855149611913460601229251202203388170363590573105401303} a^{14} - \frac{845274026981708466981641794930067122794388129045894491005046247896957554181649338358891551}{1733915904584460205979701102743525951855149611913460601229251202203388170363590573105401303} a^{13} - \frac{747674330140762323639978979105894329885362392468317821265248194832997055670502426130124762}{1733915904584460205979701102743525951855149611913460601229251202203388170363590573105401303} a^{12} - \frac{413398247128336539617351228542283299677310742431708778977050962471407107126925196624269538}{1733915904584460205979701102743525951855149611913460601229251202203388170363590573105401303} a^{11} + \frac{63880539308850299905682356606852895657663145153257962624487714776253370016704578471299310}{1733915904584460205979701102743525951855149611913460601229251202203388170363590573105401303} a^{10} + \frac{250430984985184782176019154839465155448882657065565799102394117275071251329080795004086989}{1733915904584460205979701102743525951855149611913460601229251202203388170363590573105401303} a^{9} + \frac{739681684677979141463510146071181109336602877612047552111254244818140089744744573405660766}{1733915904584460205979701102743525951855149611913460601229251202203388170363590573105401303} a^{8} + \frac{451523240824477464850167442731422934008471970036828808292356775546303190292497711732426236}{1733915904584460205979701102743525951855149611913460601229251202203388170363590573105401303} a^{7} + \frac{151187583499538480623353474160506591678742794273460460280244652103586219052471760389444095}{1733915904584460205979701102743525951855149611913460601229251202203388170363590573105401303} a^{6} + \frac{456302249579771926300957917545989164215242149276255339988391123720301119591018700353682999}{1733915904584460205979701102743525951855149611913460601229251202203388170363590573105401303} a^{5} - \frac{481940102547779279716951540191507347766505902504300551641169629500425139674598432224831134}{1733915904584460205979701102743525951855149611913460601229251202203388170363590573105401303} a^{4} - \frac{198102512093516976507598249350756303416122663877126598176542100941938645603947009679045717}{1733915904584460205979701102743525951855149611913460601229251202203388170363590573105401303} a^{3} + \frac{833437615420422522792221799287280143378971461070474543316588394045247879525580929472294656}{1733915904584460205979701102743525951855149611913460601229251202203388170363590573105401303} a^{2} + \frac{579502798419730880374580853860941548554847245170048428109318601199655288660206781459605566}{1733915904584460205979701102743525951855149611913460601229251202203388170363590573105401303} a - \frac{613503759900798124714922165751186662909914184055159068003966147872889527785340721403589132}{1733915904584460205979701102743525951855149611913460601229251202203388170363590573105401303}$

Class group and class number

$C_{51658}$, which has order $51658$ (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:		\( 866679281.3791491 \) (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

\(\Q(\sqrt{-267}) \), 11.11.31181719929966183601.1

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	$22$	R	$22$	$22$	$22$	$22$	$22$	$22$	${\href{/LocalNumberField/23.11.0.1}{11} }^{2}$	${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$	$22$	${\href{/LocalNumberField/37.2.0.1}{2} }^{11}$	${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$	$22$	$22$	$22$	${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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
89	Data not computed