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Properties

Label 22.0.89492336403...7943.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,7^{11}\cdot 11^{40}$
Root discriminant $207.01$
Ramified primes $7, 11$
Class number $173041$ (GRH)
Class group $[173041]$ (GRH)
Galois group $C_{22}$ (as 22T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5519069021, -1268438105, 860884596, -341803154, 515905885, -406562805, -87574498, 112398451, 88950411, -52267061, -18077708, 10421760, 3638591, -1913648, -351846, 222574, 18073, -16599, 77, 649, -33, -11, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^21 - 33*x^20 + 649*x^19 + 77*x^18 - 16599*x^17 + 18073*x^16 + 222574*x^15 - 351846*x^14 - 1913648*x^13 + 3638591*x^12 + 10421760*x^11 - 18077708*x^10 - 52267061*x^9 + 88950411*x^8 + 112398451*x^7 - 87574498*x^6 - 406562805*x^5 + 515905885*x^4 - 341803154*x^3 + 860884596*x^2 - 1268438105*x + 5519069021)
 
gp: K = bnfinit(x^22 - 11*x^21 - 33*x^20 + 649*x^19 + 77*x^18 - 16599*x^17 + 18073*x^16 + 222574*x^15 - 351846*x^14 - 1913648*x^13 + 3638591*x^12 + 10421760*x^11 - 18077708*x^10 - 52267061*x^9 + 88950411*x^8 + 112398451*x^7 - 87574498*x^6 - 406562805*x^5 + 515905885*x^4 - 341803154*x^3 + 860884596*x^2 - 1268438105*x + 5519069021, 1)
 

Normalized defining polynomial

\( x^{22} - 11 x^{21} - 33 x^{20} + 649 x^{19} + 77 x^{18} - 16599 x^{17} + 18073 x^{16} + 222574 x^{15} - 351846 x^{14} - 1913648 x^{13} + 3638591 x^{12} + 10421760 x^{11} - 18077708 x^{10} - 52267061 x^{9} + 88950411 x^{8} + 112398451 x^{7} - 87574498 x^{6} - 406562805 x^{5} + 515905885 x^{4} - 341803154 x^{3} + 860884596 x^{2} - 1268438105 x + 5519069021 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 11]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-894923364032259692352641685868501220854496476917943=-\,7^{11}\cdot 11^{40}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $207.01$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(847=7\cdot 11^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{847}(1,·)$, $\chi_{847}(386,·)$, $\chi_{847}(771,·)$, $\chi_{847}(265,·)$, $\chi_{847}(650,·)$, $\chi_{847}(78,·)$, $\chi_{847}(463,·)$, $\chi_{847}(342,·)$, $\chi_{847}(727,·)$, $\chi_{847}(155,·)$, $\chi_{847}(540,·)$, $\chi_{847}(34,·)$, $\chi_{847}(419,·)$, $\chi_{847}(804,·)$, $\chi_{847}(232,·)$, $\chi_{847}(617,·)$, $\chi_{847}(111,·)$, $\chi_{847}(496,·)$, $\chi_{847}(309,·)$, $\chi_{847}(694,·)$, $\chi_{847}(188,·)$, $\chi_{847}(573,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{10} + \frac{1}{9} a^{6} + \frac{2}{9} a^{2} + \frac{1}{9}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{11} + \frac{1}{9} a^{7} + \frac{2}{9} a^{3} + \frac{1}{9} a$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{10} + \frac{1}{9} a^{8} + \frac{1}{9} a^{6} + \frac{2}{9} a^{4} + \frac{1}{3} a^{2} + \frac{1}{9}$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{11} + \frac{1}{9} a^{9} + \frac{1}{9} a^{7} + \frac{2}{9} a^{5} + \frac{1}{3} a^{3} + \frac{1}{9} a$, $\frac{1}{81} a^{16} + \frac{4}{81} a^{15} + \frac{2}{81} a^{14} + \frac{1}{27} a^{13} + \frac{2}{81} a^{12} + \frac{11}{81} a^{11} + \frac{11}{81} a^{10} + \frac{10}{81} a^{9} - \frac{1}{9} a^{8} - \frac{8}{81} a^{7} - \frac{11}{81} a^{6} + \frac{20}{81} a^{5} + \frac{25}{81} a^{4} + \frac{7}{27} a^{3} - \frac{20}{81} a^{2} - \frac{32}{81} a + \frac{17}{81}$, $\frac{1}{81} a^{17} + \frac{4}{81} a^{15} + \frac{4}{81} a^{14} - \frac{1}{81} a^{13} + \frac{1}{27} a^{12} - \frac{2}{27} a^{11} + \frac{11}{81} a^{10} - \frac{4}{81} a^{9} + \frac{10}{81} a^{8} - \frac{2}{27} a^{7} - \frac{8}{81} a^{6} + \frac{8}{81} a^{5} + \frac{20}{81} a^{4} + \frac{22}{81} a^{3} + \frac{7}{27} a^{2} + \frac{10}{81} a - \frac{32}{81}$, $\frac{1}{243} a^{18} - \frac{1}{243} a^{17} - \frac{1}{243} a^{16} - \frac{2}{243} a^{15} + \frac{1}{81} a^{14} + \frac{7}{243} a^{13} - \frac{10}{243} a^{12} - \frac{20}{243} a^{11} + \frac{38}{243} a^{10} - \frac{2}{27} a^{9} + \frac{20}{243} a^{8} + \frac{20}{243} a^{7} + \frac{17}{243} a^{6} + \frac{56}{243} a^{5} - \frac{2}{81} a^{4} + \frac{119}{243} a^{3} - \frac{55}{243} a^{2} + \frac{100}{243} a + \frac{82}{243}$, $\frac{1}{243} a^{19} + \frac{1}{243} a^{17} - \frac{2}{243} a^{15} + \frac{1}{243} a^{14} + \frac{1}{81} a^{13} + \frac{4}{81} a^{12} - \frac{7}{81} a^{11} + \frac{5}{243} a^{10} - \frac{7}{243} a^{9} + \frac{16}{243} a^{8} - \frac{32}{243} a^{7} + \frac{16}{243} a^{6} + \frac{80}{243} a^{5} - \frac{49}{243} a^{4} - \frac{50}{243} a^{3} + \frac{34}{81} a^{2} + \frac{89}{243} a + \frac{37}{243}$, $\frac{1}{333153} a^{20} - \frac{661}{333153} a^{19} - \frac{22}{12339} a^{18} + \frac{11}{4113} a^{17} - \frac{1351}{333153} a^{16} + \frac{5936}{333153} a^{15} + \frac{2762}{333153} a^{14} + \frac{224}{333153} a^{13} - \frac{968}{333153} a^{12} + \frac{21070}{333153} a^{11} - \frac{40127}{333153} a^{10} + \frac{11363}{333153} a^{9} + \frac{8203}{333153} a^{8} - \frac{31358}{333153} a^{7} + \frac{48305}{333153} a^{6} + \frac{115483}{333153} a^{5} - \frac{92614}{333153} a^{4} + \frac{35660}{111051} a^{3} - \frac{30712}{111051} a^{2} - \frac{155513}{333153} a - \frac{164918}{333153}$, $\frac{1}{1130412535762515617501575600775312760197613084001115126467157736181854404250147} a^{21} - \frac{740565514821052214394370527896124113170149513975659870571107498982595616}{1130412535762515617501575600775312760197613084001115126467157736181854404250147} a^{20} - \frac{1996597580934664376664555471943685171200751880130786881150001219123458016562}{1130412535762515617501575600775312760197613084001115126467157736181854404250147} a^{19} - \frac{318700110478179272698685719335541187290159860560454016464607933495194868504}{376804178587505205833858533591770920065871028000371708822385912060618134750049} a^{18} + \frac{6406740217452095504867200425778322473067536568147603880315520952415606986309}{1130412535762515617501575600775312760197613084001115126467157736181854404250147} a^{17} - \frac{318712206428920412952459083426861562847896743794591114776480163872810066688}{376804178587505205833858533591770920065871028000371708822385912060618134750049} a^{16} + \frac{3359787791623016504896375965638724170753890746318985585894480071207707114149}{125601392862501735277952844530590306688623676000123902940795304020206044916683} a^{15} - \frac{6119509672192055763603224552220860680857782955712140664128863087204156294099}{125601392862501735277952844530590306688623676000123902940795304020206044916683} a^{14} - \frac{10110242256252382797514758882502691788980178071809389853168443377640181351482}{1130412535762515617501575600775312760197613084001115126467157736181854404250147} a^{13} - \frac{7036091780627632665663738615247263445351522021055457650276390882698659063932}{376804178587505205833858533591770920065871028000371708822385912060618134750049} a^{12} - \frac{59536959328609504518465932260007401284808270127893699035862896375311403166877}{376804178587505205833858533591770920065871028000371708822385912060618134750049} a^{11} - \frac{64718016490303598480054649426194948958873670899382502475863156214085316910920}{1130412535762515617501575600775312760197613084001115126467157736181854404250147} a^{10} + \frac{115106664837458048137638604005874133477106937752303310649565897656451915517502}{1130412535762515617501575600775312760197613084001115126467157736181854404250147} a^{9} + \frac{44157115038411027549796763684110211810817890781453556468040235812438843368618}{376804178587505205833858533591770920065871028000371708822385912060618134750049} a^{8} - \frac{57795164752580034348719713526884872829543824875852777416495800465741096477806}{1130412535762515617501575600775312760197613084001115126467157736181854404250147} a^{7} - \frac{160174685824622642328434971621690429189150068562904628716331809621865453184930}{1130412535762515617501575600775312760197613084001115126467157736181854404250147} a^{6} - \frac{117019770405701373257724337952523233558671113982131852972857097355327569546452}{1130412535762515617501575600775312760197613084001115126467157736181854404250147} a^{5} - \frac{488717726364997732585495759183423255186619161180885468652479761105948722347745}{1130412535762515617501575600775312760197613084001115126467157736181854404250147} a^{4} - \frac{12762796452217428194815106417242083896864542509383001983992204630339832446555}{376804178587505205833858533591770920065871028000371708822385912060618134750049} a^{3} - \frac{467331954833779719961217281013933758504646734837452210691001730057803558555444}{1130412535762515617501575600775312760197613084001115126467157736181854404250147} a^{2} - \frac{174738974291594206062627845121500869090717078833252494341526162185731777169672}{376804178587505205833858533591770920065871028000371708822385912060618134750049} a + \frac{426895904337018846983083206851944578964715854122106428121184854677397701242960}{1130412535762515617501575600775312760197613084001115126467157736181854404250147}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{173041}$, which has order $173041$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $10$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 285114946276.13544 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 22
The 22 conjugacy class representatives for $C_{22}$
Character table for $C_{22}$ is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), 11.11.672749994932560009201.1

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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{11}$ $22$ R R $22$ $22$ $22$ ${\href{/LocalNumberField/23.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ $22$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
7Data not computed
$11$11.11.20.9$x^{11} - 11 x^{10} + 11$$11$$1$$20$$C_{11}$$[2]$
11.11.20.9$x^{11} - 11 x^{10} + 11$$11$$1$$20$$C_{11}$$[2]$