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Properties

Label 22.0.43604905791...6031.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,23^{20}\cdot 31^{11}$
Root discriminant $96.30$
Ramified primes $23, 31$
Class number $3128667$ (GRH)
Class group $[3128667]$ (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![68117781041, -41797446694, 66136100859, -34852602875, 30359360950, -13884271574, 8681282523, -3469345651, 1716577617, -601120919, 246269894, -75415028, 26133577, -6940762, 2047836, -463428, 115721, -21547, 4455, -633, 103, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 9*x^21 + 103*x^20 - 633*x^19 + 4455*x^18 - 21547*x^17 + 115721*x^16 - 463428*x^15 + 2047836*x^14 - 6940762*x^13 + 26133577*x^12 - 75415028*x^11 + 246269894*x^10 - 601120919*x^9 + 1716577617*x^8 - 3469345651*x^7 + 8681282523*x^6 - 13884271574*x^5 + 30359360950*x^4 - 34852602875*x^3 + 66136100859*x^2 - 41797446694*x + 68117781041)
 
gp: K = bnfinit(x^22 - 9*x^21 + 103*x^20 - 633*x^19 + 4455*x^18 - 21547*x^17 + 115721*x^16 - 463428*x^15 + 2047836*x^14 - 6940762*x^13 + 26133577*x^12 - 75415028*x^11 + 246269894*x^10 - 601120919*x^9 + 1716577617*x^8 - 3469345651*x^7 + 8681282523*x^6 - 13884271574*x^5 + 30359360950*x^4 - 34852602875*x^3 + 66136100859*x^2 - 41797446694*x + 68117781041, 1)
 

Normalized defining polynomial

\( x^{22} - 9 x^{21} + 103 x^{20} - 633 x^{19} + 4455 x^{18} - 21547 x^{17} + 115721 x^{16} - 463428 x^{15} + 2047836 x^{14} - 6940762 x^{13} + 26133577 x^{12} - 75415028 x^{11} + 246269894 x^{10} - 601120919 x^{9} + 1716577617 x^{8} - 3469345651 x^{7} + 8681282523 x^{6} - 13884271574 x^{5} + 30359360950 x^{4} - 34852602875 x^{3} + 66136100859 x^{2} - 41797446694 x + 68117781041 \)

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:  \(-43604905791095263021253936224725999514116031=-\,23^{20}\cdot 31^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $96.30$
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:  $23, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(713=23\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{713}(1,·)$, $\chi_{713}(588,·)$, $\chi_{713}(650,·)$, $\chi_{713}(652,·)$, $\chi_{713}(464,·)$, $\chi_{713}(216,·)$, $\chi_{713}(466,·)$, $\chi_{713}(404,·)$, $\chi_{713}(278,·)$, $\chi_{713}(280,·)$, $\chi_{713}(154,·)$, $\chi_{713}(156,·)$, $\chi_{713}(94,·)$, $\chi_{713}(32,·)$, $\chi_{713}(123,·)$, $\chi_{713}(340,·)$, $\chi_{713}(683,·)$, $\chi_{713}(495,·)$, $\chi_{713}(371,·)$, $\chi_{713}(311,·)$, $\chi_{713}(185,·)$, $\chi_{713}(187,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{47} a^{17} - \frac{23}{47} a^{16} - \frac{9}{47} a^{15} + \frac{19}{47} a^{14} + \frac{8}{47} a^{13} + \frac{12}{47} a^{12} + \frac{14}{47} a^{11} + \frac{2}{47} a^{10} - \frac{6}{47} a^{9} - \frac{20}{47} a^{8} + \frac{23}{47} a^{7} - \frac{20}{47} a^{6} + \frac{13}{47} a^{5} - \frac{13}{47} a^{4} + \frac{3}{47} a^{3} + \frac{4}{47} a^{2} + \frac{21}{47} a + \frac{18}{47}$, $\frac{1}{47} a^{18} - \frac{21}{47} a^{16} + \frac{22}{47} a^{14} + \frac{8}{47} a^{13} + \frac{8}{47} a^{12} - \frac{5}{47} a^{11} - \frac{7}{47} a^{10} - \frac{17}{47} a^{9} - \frac{14}{47} a^{8} - \frac{8}{47} a^{7} + \frac{23}{47} a^{6} + \frac{4}{47} a^{5} - \frac{14}{47} a^{4} - \frac{21}{47} a^{3} + \frac{19}{47} a^{2} - \frac{16}{47} a - \frac{9}{47}$, $\frac{1}{47} a^{19} - \frac{13}{47} a^{16} + \frac{21}{47} a^{15} - \frac{16}{47} a^{14} - \frac{12}{47} a^{13} + \frac{12}{47} a^{12} + \frac{5}{47} a^{11} - \frac{22}{47} a^{10} + \frac{1}{47} a^{9} - \frac{5}{47} a^{8} - \frac{11}{47} a^{7} + \frac{7}{47} a^{6} - \frac{23}{47} a^{5} - \frac{12}{47} a^{4} - \frac{12}{47} a^{3} + \frac{21}{47} a^{2} + \frac{9}{47} a + \frac{2}{47}$, $\frac{1}{47} a^{20} + \frac{4}{47} a^{16} + \frac{8}{47} a^{15} + \frac{22}{47} a^{13} + \frac{20}{47} a^{12} + \frac{19}{47} a^{11} - \frac{20}{47} a^{10} + \frac{11}{47} a^{9} + \frac{11}{47} a^{8} - \frac{23}{47} a^{7} - \frac{1}{47} a^{6} + \frac{16}{47} a^{5} + \frac{7}{47} a^{4} + \frac{13}{47} a^{3} + \frac{14}{47} a^{2} - \frac{7}{47} a - \frac{1}{47}$, $\frac{1}{1556442115134851948898199241611841913085058448358302573221612453547341056746011} a^{21} + \frac{7516889370645302330534152177049021439682490565704238519108444154669174365034}{1556442115134851948898199241611841913085058448358302573221612453547341056746011} a^{20} + \frac{210016551140209516645437650802949509886323061038168444104658569876983107543}{1556442115134851948898199241611841913085058448358302573221612453547341056746011} a^{19} + \frac{4323227326338993424606013978918254158422579355707314022582538276529168728387}{1556442115134851948898199241611841913085058448358302573221612453547341056746011} a^{18} - \frac{10739616269218480902525193028959409309083359251090550634669855666216212429280}{1556442115134851948898199241611841913085058448358302573221612453547341056746011} a^{17} + \frac{257459695631682311796386729289160543473183913016123125857908232844738181903583}{1556442115134851948898199241611841913085058448358302573221612453547341056746011} a^{16} - \frac{268922592878468190544835550727224991888694856506429008614758565689258161820144}{1556442115134851948898199241611841913085058448358302573221612453547341056746011} a^{15} + \frac{251366665581091180576286697445527284094704548230998544042874331605702744811607}{1556442115134851948898199241611841913085058448358302573221612453547341056746011} a^{14} - \frac{173541905865268448067242603615287779716761762102130257399690290495719783309932}{1556442115134851948898199241611841913085058448358302573221612453547341056746011} a^{13} - \frac{586872893693892760175258049595440644302085508746554142125996977788673217004623}{1556442115134851948898199241611841913085058448358302573221612453547341056746011} a^{12} - \frac{80979512904310513842631369684750247824964848851815727132243282721689975737802}{1556442115134851948898199241611841913085058448358302573221612453547341056746011} a^{11} - \frac{314163998815408027418279442230912812890808399296416717394806315040384284400345}{1556442115134851948898199241611841913085058448358302573221612453547341056746011} a^{10} - \frac{241843409847998440824224409186925723397081904926481532605602567001934459675917}{1556442115134851948898199241611841913085058448358302573221612453547341056746011} a^{9} + \frac{374222462293599007389754553878338766042024925411266972214170482208287734261233}{1556442115134851948898199241611841913085058448358302573221612453547341056746011} a^{8} + \frac{552910552891988231784051813591616148256693123696432299183553734518265293294292}{1556442115134851948898199241611841913085058448358302573221612453547341056746011} a^{7} - \frac{222200988019830042442500978850123333416054347354683463095073385685269790493247}{1556442115134851948898199241611841913085058448358302573221612453547341056746011} a^{6} - \frac{92197485293670427731534173134214493837124397404233016668997368702280155671544}{1556442115134851948898199241611841913085058448358302573221612453547341056746011} a^{5} - \frac{706943111025705839917526609246800012083155860589437801213537275380276871954803}{1556442115134851948898199241611841913085058448358302573221612453547341056746011} a^{4} + \frac{566840111058401689327088161923448550573439952578859936536971773000862972168484}{1556442115134851948898199241611841913085058448358302573221612453547341056746011} a^{3} - \frac{333191049128292409025171540809016035633731925116962755227246174339148665994439}{1556442115134851948898199241611841913085058448358302573221612453547341056746011} a^{2} + \frac{212146141656232203186824076469340224503837618002850932109355070778094976380485}{1556442115134851948898199241611841913085058448358302573221612453547341056746011} a - \frac{146326702310874850060167959389751987186995212331592001399291161807501008558}{1352252054852173717548392043103251010499616375637100411139541662508550005861}$

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

Class group and class number

$C_{3128667}$, which has order $3128667$ (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:  \( 1038656.82438 \) (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

\(\Q(\sqrt{-31}) \), \(\Q(\zeta_{23})^+\)

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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ $22$ $22$ $22$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ R $22$ R $22$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{22}$ $22$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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

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
23Data not computed
31Data not computed