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Properties

Label 22.0.52722623986...7819.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,11^{40}\cdot 19^{11}$
Root discriminant $341.05$
Ramified primes $11, 19$
Class number $44082841$ (GRH)
Class group $[11, 4007531]$ (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![478967764799, -142892130293, 163349993499, -54759490940, 38104926805, -11812333698, 6065332130, -1494976967, 748155507, -198441551, 53555326, -6619506, 5317631, -1384724, 72600, 41206, 22033, -8184, 77, 319, 0, -11, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^21 + 319*x^19 + 77*x^18 - 8184*x^17 + 22033*x^16 + 41206*x^15 + 72600*x^14 - 1384724*x^13 + 5317631*x^12 - 6619506*x^11 + 53555326*x^10 - 198441551*x^9 + 748155507*x^8 - 1494976967*x^7 + 6065332130*x^6 - 11812333698*x^5 + 38104926805*x^4 - 54759490940*x^3 + 163349993499*x^2 - 142892130293*x + 478967764799)
 
gp: K = bnfinit(x^22 - 11*x^21 + 319*x^19 + 77*x^18 - 8184*x^17 + 22033*x^16 + 41206*x^15 + 72600*x^14 - 1384724*x^13 + 5317631*x^12 - 6619506*x^11 + 53555326*x^10 - 198441551*x^9 + 748155507*x^8 - 1494976967*x^7 + 6065332130*x^6 - 11812333698*x^5 + 38104926805*x^4 - 54759490940*x^3 + 163349993499*x^2 - 142892130293*x + 478967764799, 1)
 

Normalized defining polynomial

\( x^{22} - 11 x^{21} + 319 x^{19} + 77 x^{18} - 8184 x^{17} + 22033 x^{16} + 41206 x^{15} + 72600 x^{14} - 1384724 x^{13} + 5317631 x^{12} - 6619506 x^{11} + 53555326 x^{10} - 198441551 x^{9} + 748155507 x^{8} - 1494976967 x^{7} + 6065332130 x^{6} - 11812333698 x^{5} + 38104926805 x^{4} - 54759490940 x^{3} + 163349993499 x^{2} - 142892130293 x + 478967764799 \)

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:  \(-52722623986774764924550522590931307567802704396522287819=-\,11^{40}\cdot 19^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $341.05$
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:  $11, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2299=11^{2}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{2299}(1728,·)$, $\chi_{2299}(1,·)$, $\chi_{2299}(683,·)$, $\chi_{2299}(837,·)$, $\chi_{2299}(1673,·)$, $\chi_{2299}(1101,·)$, $\chi_{2299}(1937,·)$, $\chi_{2299}(210,·)$, $\chi_{2299}(1046,·)$, $\chi_{2299}(56,·)$, $\chi_{2299}(1882,·)$, $\chi_{2299}(474,·)$, $\chi_{2299}(1310,·)$, $\chi_{2299}(2146,·)$, $\chi_{2299}(419,·)$, $\chi_{2299}(1255,·)$, $\chi_{2299}(2091,·)$, $\chi_{2299}(1519,·)$, $\chi_{2299}(628,·)$, $\chi_{2299}(265,·)$, $\chi_{2299}(1464,·)$, $\chi_{2299}(892,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{10} + \frac{1}{9} a^{6} + \frac{2}{9} a^{2} + \frac{1}{9}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{11} + \frac{1}{9} a^{7} + \frac{2}{9} a^{3} + \frac{1}{9} a$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{10} + \frac{1}{9} a^{8} + \frac{1}{9} a^{6} + \frac{2}{9} a^{4} + \frac{1}{3} a^{2} + \frac{1}{9}$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{11} + \frac{1}{9} a^{9} + \frac{1}{9} a^{7} + \frac{2}{9} a^{5} + \frac{1}{3} a^{3} + \frac{1}{9} a$, $\frac{1}{81} a^{16} + \frac{4}{81} a^{15} - \frac{1}{81} a^{14} - \frac{1}{27} a^{13} - \frac{4}{81} a^{12} - \frac{1}{81} a^{11} - \frac{13}{81} a^{10} + \frac{13}{81} a^{9} + \frac{1}{9} a^{8} - \frac{8}{81} a^{7} - \frac{2}{81} a^{6} - \frac{1}{81} a^{5} + \frac{19}{81} a^{4} - \frac{4}{27} a^{3} + \frac{10}{81} a^{2} + \frac{1}{81} a - \frac{4}{81}$, $\frac{1}{81} a^{17} + \frac{1}{81} a^{15} + \frac{1}{81} a^{14} - \frac{1}{81} a^{13} - \frac{1}{27} a^{12} + \frac{1}{9} a^{11} + \frac{2}{81} a^{10} + \frac{2}{81} a^{9} + \frac{10}{81} a^{8} + \frac{4}{27} a^{7} - \frac{11}{81} a^{6} + \frac{32}{81} a^{5} - \frac{7}{81} a^{4} + \frac{40}{81} a^{3} + \frac{2}{27} a^{2} + \frac{28}{81} a + \frac{25}{81}$, $\frac{1}{243} a^{18} - \frac{1}{243} a^{17} - \frac{1}{243} a^{16} + \frac{1}{243} a^{15} - \frac{5}{243} a^{13} + \frac{2}{243} a^{12} + \frac{22}{243} a^{11} - \frac{37}{243} a^{10} + \frac{2}{27} a^{9} - \frac{16}{243} a^{8} + \frac{20}{243} a^{7} + \frac{2}{243} a^{6} - \frac{73}{243} a^{5} - \frac{11}{27} a^{4} - \frac{82}{243} a^{3} + \frac{20}{243} a^{2} - \frac{5}{243} a - \frac{62}{243}$, $\frac{1}{111051} a^{19} - \frac{25}{111051} a^{18} - \frac{469}{111051} a^{17} + \frac{547}{111051} a^{16} + \frac{1784}{37017} a^{15} + \frac{871}{111051} a^{14} - \frac{169}{111051} a^{13} + \frac{5977}{111051} a^{12} - \frac{18502}{111051} a^{11} + \frac{3166}{37017} a^{10} - \frac{1072}{111051} a^{9} + \frac{776}{111051} a^{8} + \frac{8585}{111051} a^{7} + \frac{15020}{111051} a^{6} + \frac{17818}{37017} a^{5} + \frac{42602}{111051} a^{4} - \frac{32758}{111051} a^{3} + \frac{28459}{111051} a^{2} - \frac{5717}{111051} a + \frac{2234}{37017}$, $\frac{1}{333153} a^{20} - \frac{1}{333153} a^{19} - \frac{68}{37017} a^{18} - \frac{523}{111051} a^{17} + \frac{1571}{333153} a^{16} + \frac{11870}{333153} a^{15} + \frac{9767}{333153} a^{14} + \frac{14717}{333153} a^{13} - \frac{9869}{333153} a^{12} - \frac{50213}{333153} a^{11} - \frac{13502}{333153} a^{10} - \frac{51001}{333153} a^{9} + \frac{41833}{333153} a^{8} + \frac{20437}{333153} a^{7} + \frac{21371}{333153} a^{6} - \frac{99428}{333153} a^{5} - \frac{64609}{333153} a^{4} + \frac{11879}{37017} a^{3} - \frac{136}{457} a^{2} + \frac{107134}{333153} a + \frac{6382}{333153}$, $\frac{1}{7387510153670582081822344375786563477777904025009280372112586722772356907045004539465031} a^{21} - \frac{8583670300514286904794020134836955956732264737271363465008178788001315274851006746}{7387510153670582081822344375786563477777904025009280372112586722772356907045004539465031} a^{20} - \frac{15678801858982710571971488099778075528658858626495727728168797831352367294308074123}{7387510153670582081822344375786563477777904025009280372112586722772356907045004539465031} a^{19} - \frac{3895781938253591706513465358565234412199529441501452114750114062539839635418802663881}{2462503384556860693940781458595521159259301341669760124037528907590785635681668179821677} a^{18} - \frac{27643187603043330174598812545722461459480244679572917036727905405229775074863492227722}{7387510153670582081822344375786563477777904025009280372112586722772356907045004539465031} a^{17} - \frac{1322977222184368085306374503232627491536353808656817856442943096950444823150958444295}{820834461518953564646927152865173719753100447223253374679176302530261878560556059940559} a^{16} - \frac{146521336398058110629879398840551641913451327014621642848573299613240491754558834501}{30401276352553835727663968624636063694559275823083458321450974167787476983724298516317} a^{15} + \frac{24035566989885872726661581341830660585039283398677545102822869066135646264887218682887}{2462503384556860693940781458595521159259301341669760124037528907590785635681668179821677} a^{14} - \frac{203120865315543730012126921805912934626659237404478560094931816643405524174958364578589}{7387510153670582081822344375786563477777904025009280372112586722772356907045004539465031} a^{13} - \frac{93337355557655921520818555431963472865565082743775494745125534353997301860486474111699}{2462503384556860693940781458595521159259301341669760124037528907590785635681668179821677} a^{12} - \frac{76264534712949439213754714926578176200011130920747041806432492231255568302755673243478}{2462503384556860693940781458595521159259301341669760124037528907590785635681668179821677} a^{11} - \frac{480407301113304460501491788575117187543947799174512713682677952792573491460466624527925}{7387510153670582081822344375786563477777904025009280372112586722772356907045004539465031} a^{10} - \frac{450698392495373123722090244340461672614492962591082094745676448003612155601711306956369}{7387510153670582081822344375786563477777904025009280372112586722772356907045004539465031} a^{9} + \frac{38033723353613454559574691665387616251351284399437591530336652999098134382549848072534}{273611487172984521548975717621724573251033482407751124893058767510087292853518686646853} a^{8} + \frac{298138907119457440648658667088964641208440852777210859032901144165114942473445620837177}{7387510153670582081822344375786563477777904025009280372112586722772356907045004539465031} a^{7} + \frac{569912162223153537450928647305922944992853318022116133205160200416413768349709807933970}{7387510153670582081822344375786563477777904025009280372112586722772356907045004539465031} a^{6} - \frac{237453057726254060185307642408374063407271395804636206666984756220062785968435984855785}{7387510153670582081822344375786563477777904025009280372112586722772356907045004539465031} a^{5} + \frac{2071616435857404691299649081337091708325862956118513106393343918475805983735429971388509}{7387510153670582081822344375786563477777904025009280372112586722772356907045004539465031} a^{4} - \frac{278343901870191702708785121060857669225224109501912369462929840235249918073402500025340}{820834461518953564646927152865173719753100447223253374679176302530261878560556059940559} a^{3} + \frac{649267734734019344371014219394636267116323242698375127537679635336115393098932109295698}{7387510153670582081822344375786563477777904025009280372112586722772356907045004539465031} a^{2} - \frac{778602006397768371565583712011143221853883670928141516791675076764668069467397325011578}{2462503384556860693940781458595521159259301341669760124037528907590785635681668179821677} a + \frac{2805890290782414933259450021022730023573687838550985497886327913100862791926655049433122}{7387510153670582081822344375786563477777904025009280372112586722772356907045004539465031}$

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

Class group and class number

$C_{11}\times C_{4007531}$, which has order $44082841$ (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

\(\Q(\sqrt{-19}) \), 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 $22$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ R $22$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ R ${\href{/LocalNumberField/23.11.0.1}{11} }^{2}$ $22$ $22$ $22$ $22$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/47.11.0.1}{11} }^{2}$ $22$ $22$

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
$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]$
19Data not computed