↑

Properties

Label 22.0.11654522436...4819.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,419^{21}$
Root discriminant $318.44$
Ramified prime $419$
Class number $5829723$ (GRH)
Class group $[5829723]$ (GRH)
Galois group $C_{22}$ (as 22T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1515754471721, -2451856901550, 2553256282724, -1843489099333, 1062844895210, -463698132550, 156515778679, -33713397579, 1727107332, 2186797028, -992590046, 232972284, -21380891, -2060852, 1407458, -173956, 19486, -3051, 1885, -130, 10, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + 10*x^20 - 130*x^19 + 1885*x^18 - 3051*x^17 + 19486*x^16 - 173956*x^15 + 1407458*x^14 - 2060852*x^13 - 21380891*x^12 + 232972284*x^11 - 992590046*x^10 + 2186797028*x^9 + 1727107332*x^8 - 33713397579*x^7 + 156515778679*x^6 - 463698132550*x^5 + 1062844895210*x^4 - 1843489099333*x^3 + 2553256282724*x^2 - 2451856901550*x + 1515754471721)
 
gp: K = bnfinit(x^22 - x^21 + 10*x^20 - 130*x^19 + 1885*x^18 - 3051*x^17 + 19486*x^16 - 173956*x^15 + 1407458*x^14 - 2060852*x^13 - 21380891*x^12 + 232972284*x^11 - 992590046*x^10 + 2186797028*x^9 + 1727107332*x^8 - 33713397579*x^7 + 156515778679*x^6 - 463698132550*x^5 + 1062844895210*x^4 - 1843489099333*x^3 + 2553256282724*x^2 - 2451856901550*x + 1515754471721, 1)
 

Normalized defining polynomial

\( x^{22} - x^{21} + 10 x^{20} - 130 x^{19} + 1885 x^{18} - 3051 x^{17} + 19486 x^{16} - 173956 x^{15} + 1407458 x^{14} - 2060852 x^{13} - 21380891 x^{12} + 232972284 x^{11} - 992590046 x^{10} + 2186797028 x^{9} + 1727107332 x^{8} - 33713397579 x^{7} + 156515778679 x^{6} - 463698132550 x^{5} + 1062844895210 x^{4} - 1843489099333 x^{3} + 2553256282724 x^{2} - 2451856901550 x + 1515754471721 \)

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:  \(-11654522436820240107274506932479072962338428405475204819=-\,419^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $318.44$
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:  $419$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(419\)
Dirichlet character group:    $\lbrace$$\chi_{419}(1,·)$, $\chi_{419}(69,·)$, $\chi_{419}(129,·)$, $\chi_{419}(267,·)$, $\chi_{419}(13,·)$, $\chi_{419}(334,·)$, $\chi_{419}(85,·)$, $\chi_{419}(406,·)$, $\chi_{419}(152,·)$, $\chi_{419}(348,·)$, $\chi_{419}(350,·)$, $\chi_{419}(418,·)$, $\chi_{419}(102,·)$, $\chi_{419}(360,·)$, $\chi_{419}(169,·)$, $\chi_{419}(71,·)$, $\chi_{419}(300,·)$, $\chi_{419}(119,·)$, $\chi_{419}(290,·)$, $\chi_{419}(250,·)$, $\chi_{419}(59,·)$, $\chi_{419}(317,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} - \frac{3}{7} a^{5} + \frac{2}{7} a^{4} + \frac{1}{7} a^{3} - \frac{3}{7} a^{2} + \frac{2}{7} a$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{3}$, $\frac{1}{49} a^{10} - \frac{1}{49} a^{9} + \frac{2}{49} a^{8} - \frac{1}{49} a^{7} + \frac{3}{49} a^{6} - \frac{23}{49} a^{5} + \frac{5}{49} a^{4} - \frac{24}{49} a^{3} - \frac{11}{49} a^{2}$, $\frac{1}{49} a^{11} + \frac{1}{49} a^{9} + \frac{1}{49} a^{8} + \frac{2}{49} a^{7} + \frac{1}{49} a^{6} + \frac{17}{49} a^{5} + \frac{23}{49} a^{4} - \frac{2}{7} a^{3} + \frac{24}{49} a^{2} - \frac{1}{7} a$, $\frac{1}{49} a^{12} + \frac{2}{49} a^{9} + \frac{2}{49} a^{7} - \frac{10}{49} a^{5} + \frac{2}{49} a^{4} - \frac{15}{49} a^{3} - \frac{3}{49} a^{2} + \frac{3}{7} a$, $\frac{1}{49} a^{13} + \frac{2}{49} a^{9} - \frac{2}{49} a^{8} + \frac{2}{49} a^{7} - \frac{2}{49} a^{6} + \frac{6}{49} a^{5} + \frac{3}{49} a^{4} + \frac{10}{49} a^{3} + \frac{1}{49} a^{2} - \frac{3}{7} a$, $\frac{1}{343} a^{14} - \frac{3}{343} a^{13} - \frac{1}{343} a^{12} - \frac{1}{343} a^{11} + \frac{3}{343} a^{10} + \frac{9}{343} a^{9} + \frac{16}{343} a^{8} - \frac{6}{343} a^{7} - \frac{1}{49} a^{6} - \frac{31}{343} a^{5} - \frac{12}{343} a^{4} - \frac{115}{343} a^{3} + \frac{3}{7} a$, $\frac{1}{343} a^{15} - \frac{3}{343} a^{13} + \frac{3}{343} a^{12} - \frac{3}{343} a^{10} - \frac{6}{343} a^{9} - \frac{2}{49} a^{8} + \frac{24}{343} a^{7} + \frac{18}{343} a^{6} - \frac{13}{49} a^{5} + \frac{73}{343} a^{4} + \frac{26}{343} a^{3} - \frac{11}{49} a^{2} + \frac{1}{7} a$, $\frac{1}{2401} a^{16} - \frac{3}{2401} a^{15} - \frac{2}{2401} a^{14} + \frac{9}{2401} a^{13} + \frac{4}{2401} a^{12} + \frac{17}{2401} a^{11} + \frac{6}{2401} a^{10} + \frac{111}{2401} a^{9} + \frac{103}{2401} a^{8} + \frac{10}{2401} a^{7} + \frac{65}{2401} a^{6} + \frac{41}{343} a^{5} - \frac{1017}{2401} a^{4} + \frac{1088}{2401} a^{3} + \frac{169}{343} a^{2} + \frac{18}{49} a - \frac{1}{7}$, $\frac{1}{2401} a^{17} + \frac{3}{2401} a^{15} + \frac{3}{2401} a^{14} - \frac{11}{2401} a^{13} + \frac{22}{2401} a^{12} + \frac{8}{2401} a^{11} - \frac{11}{2401} a^{10} - \frac{40}{2401} a^{9} - \frac{122}{2401} a^{8} - \frac{10}{2401} a^{7} + \frac{48}{2401} a^{6} - \frac{891}{2401} a^{5} - \frac{941}{2401} a^{4} - \frac{1020}{2401} a^{3} - \frac{53}{343} a^{2} - \frac{9}{49} a - \frac{3}{7}$, $\frac{1}{2401} a^{18} - \frac{2}{2401} a^{15} + \frac{2}{2401} a^{14} + \frac{16}{2401} a^{13} - \frac{4}{2401} a^{12} - \frac{20}{2401} a^{11} + \frac{5}{2401} a^{10} - \frac{23}{343} a^{9} + \frac{38}{2401} a^{8} - \frac{164}{2401} a^{7} + \frac{34}{2401} a^{6} + \frac{284}{2401} a^{5} + \frac{92}{2401} a^{4} - \frac{51}{2401} a^{3} - \frac{171}{343} a^{2} + \frac{2}{49} a + \frac{3}{7}$, $\frac{1}{789929} a^{19} - \frac{79}{789929} a^{18} - \frac{13}{112847} a^{17} + \frac{27}{789929} a^{16} - \frac{340}{789929} a^{15} + \frac{1123}{789929} a^{14} - \frac{2316}{789929} a^{13} + \frac{706}{789929} a^{12} - \frac{1324}{789929} a^{11} + \frac{290}{789929} a^{10} + \frac{43444}{789929} a^{9} + \frac{50963}{789929} a^{8} - \frac{12123}{789929} a^{7} + \frac{6952}{789929} a^{6} + \frac{18549}{112847} a^{5} - \frac{103641}{789929} a^{4} + \frac{529}{2303} a^{3} - \frac{6998}{16121} a^{2} + \frac{121}{329} a - \frac{2}{7}$, $\frac{1}{768600917} a^{20} - \frac{4}{15685733} a^{19} + \frac{76926}{768600917} a^{18} - \frac{90329}{768600917} a^{17} - \frac{148588}{768600917} a^{16} - \frac{2662}{5529503} a^{15} + \frac{152073}{109800131} a^{14} - \frac{5775342}{768600917} a^{13} + \frac{5184022}{768600917} a^{12} - \frac{3454590}{768600917} a^{11} + \frac{3619631}{768600917} a^{10} + \frac{1604637}{109800131} a^{9} + \frac{4909833}{109800131} a^{8} - \frac{51096217}{768600917} a^{7} + \frac{29721652}{768600917} a^{6} + \frac{225974904}{768600917} a^{5} + \frac{230069912}{768600917} a^{4} + \frac{464985}{2336173} a^{3} - \frac{5361774}{15685733} a^{2} + \frac{587383}{2240819} a - \frac{2182}{6811}$, $\frac{1}{386085007906316194949838525624095134817388215352571421869727495034654654369620113087747} a^{21} - \frac{13101947203477997336742854289439580148787449549791522331772438860290047264057}{386085007906316194949838525624095134817388215352571421869727495034654654369620113087747} a^{20} + \frac{2977557748684076873471964107121168547425093072200536516305155888316480266317916}{386085007906316194949838525624095134817388215352571421869727495034654654369620113087747} a^{19} + \frac{61745684005908904028491745082270729275606705643873427318990946927506689332051952989}{386085007906316194949838525624095134817388215352571421869727495034654654369620113087747} a^{18} + \frac{9943779236630606244179158376492382905459323285143186802226551212284369873579645507}{386085007906316194949838525624095134817388215352571421869727495034654654369620113087747} a^{17} - \frac{73362686285418921919145351639041135109694529052471510772863858923171642332847446151}{386085007906316194949838525624095134817388215352571421869727495034654654369620113087747} a^{16} + \frac{312468453003161547117140541577049744861711686272614544311058941633675580800357453561}{386085007906316194949838525624095134817388215352571421869727495034654654369620113087747} a^{15} - \frac{540155292642660597461333169294948142782932190699485247114211368015992089911163479}{1940125667870935653014263947859774546821046308304379004370489924797259569696583482853} a^{14} - \frac{234633791465101619316533918735476855664728310770555292960447854552501434722406447622}{386085007906316194949838525624095134817388215352571421869727495034654654369620113087747} a^{13} + \frac{234660033822477610373956629030158906270792057931751535639043604273122429830381130715}{386085007906316194949838525624095134817388215352571421869727495034654654369620113087747} a^{12} + \frac{53061864666225537277824871947758954049143478966966513772982952263900842546430887784}{7879285875639106019384459706614186424844657456174926976933214184380707232033063532403} a^{11} - \frac{1011954845403125611086052858980542000571395440604199311359043774384557227962385974232}{386085007906316194949838525624095134817388215352571421869727495034654654369620113087747} a^{10} - \frac{425650881401043398517403095092089122400864648774559996609725063527275932245598747061}{55155001129473742135691217946299304973912602193224488838532499290664950624231444726821} a^{9} + \frac{1925407677079952200186498066769569172918766142370758336298876228126807766355885331271}{386085007906316194949838525624095134817388215352571421869727495034654654369620113087747} a^{8} - \frac{79783564961038355361680023044998550641897822283507756307274073921802511410487685076}{386085007906316194949838525624095134817388215352571421869727495034654654369620113087747} a^{7} - \frac{17493946392022093870586014596150804048228382081253612268982690377233195954013388489763}{386085007906316194949838525624095134817388215352571421869727495034654654369620113087747} a^{6} - \frac{123462177359227805631922848929787459833576101695924570221770592722971757796654350315405}{386085007906316194949838525624095134817388215352571421869727495034654654369620113087747} a^{5} + \frac{189498160129470276981849596662192672912601506391968438597245336717272221639589121072210}{386085007906316194949838525624095134817388215352571421869727495034654654369620113087747} a^{4} + \frac{16931115233012762786805346385975610130224923673174867804701347077821730858490201685474}{55155001129473742135691217946299304973912602193224488838532499290664950624231444726821} a^{3} - \frac{430384651029905893886331385747095138244445081850303344520013105960798807408533350543}{7879285875639106019384459706614186424844657456174926976933214184380707232033063532403} a^{2} - \frac{238213807203829498631340771803582763486488792341232300465444930705696947987126993236}{1125612267948443717054922815230598060692093922310703853847602026340101033147580504629} a + \frac{328786846835370862175307843347911470607233034508833567389391972883696237148771498}{3421313884341774215972409772737380123684176055655634814126449928085413474612706701}$

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

Class group and class number

$C_{5829723}$, which has order $5829723$ (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:  \( 14055597684554.104 \) (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

\(\Q(\sqrt{-419}) \), 11.11.166778563814477267272573801.1

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 $22$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{22}$ $22$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/23.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{22}$ $22$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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
419Data not computed