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Properties

Label 22.0.74332715131...2288.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,2^{33}\cdot 89^{21}$
Root discriminant $205.27$
Ramified primes $2, 89$
Class number $25518776$ (GRH)
Class group $[25518776]$ (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![249530368, 0, 1447148544, 0, 3194544640, 0, 3411995136, 0, 1874827008, 0, 542401600, 0, 86100736, 0, 7832000, 0, 412248, 0, 12104, 0, 178, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 178*x^20 + 12104*x^18 + 412248*x^16 + 7832000*x^14 + 86100736*x^12 + 542401600*x^10 + 1874827008*x^8 + 3411995136*x^6 + 3194544640*x^4 + 1447148544*x^2 + 249530368)
 
gp: K = bnfinit(x^22 + 178*x^20 + 12104*x^18 + 412248*x^16 + 7832000*x^14 + 86100736*x^12 + 542401600*x^10 + 1874827008*x^8 + 3411995136*x^6 + 3194544640*x^4 + 1447148544*x^2 + 249530368, 1)
 

Normalized defining polynomial

\( x^{22} + 178 x^{20} + 12104 x^{18} + 412248 x^{16} + 7832000 x^{14} + 86100736 x^{12} + 542401600 x^{10} + 1874827008 x^{8} + 3411995136 x^{6} + 3194544640 x^{4} + 1447148544 x^{2} + 249530368 \)

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:  \(-743327151318017467503714837392191185817512378892288=-\,2^{33}\cdot 89^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $205.27$
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:  $2, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(712=2^{3}\cdot 89\)
Dirichlet character group:    $\lbrace$$\chi_{712}(1,·)$, $\chi_{712}(259,·)$, $\chi_{712}(601,·)$, $\chi_{712}(449,·)$, $\chi_{712}(139,·)$, $\chi_{712}(11,·)$, $\chi_{712}(401,·)$, $\chi_{712}(467,·)$, $\chi_{712}(203,·)$, $\chi_{712}(153,·)$, $\chi_{712}(217,·)$, $\chi_{712}(667,·)$, $\chi_{712}(97,·)$, $\chi_{712}(355,·)$, $\chi_{712}(105,·)$, $\chi_{712}(619,·)$, $\chi_{712}(625,·)$, $\chi_{712}(235,·)$, $\chi_{712}(251,·)$, $\chi_{712}(121,·)$, $\chi_{712}(443,·)$, $\chi_{712}(345,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{128} a^{14}$, $\frac{1}{128} a^{15}$, $\frac{1}{256} a^{16}$, $\frac{1}{256} a^{17}$, $\frac{1}{512} a^{18}$, $\frac{1}{18944} a^{19} - \frac{1}{2368} a^{17} - \frac{3}{2368} a^{15} - \frac{7}{2368} a^{13} + \frac{3}{296} a^{11} + \frac{1}{296} a^{9} - \frac{11}{296} a^{7} - \frac{3}{74} a^{5} + \frac{3}{37} a^{3} - \frac{5}{37} a$, $\frac{1}{578679728652378903373025806959616} a^{20} + \frac{30196258937315018612460416715}{144669932163094725843256451739904} a^{18} - \frac{7167019085376803714833586781}{9041870760193420365203528233744} a^{16} - \frac{212754313462033386859526894341}{72334966081547362921628225869952} a^{14} - \frac{299307107707611056087291351}{89523472873202181833698299344} a^{12} - \frac{226412214882176503970265119967}{18083741520386840730407056467488} a^{10} - \frac{48772164136066661728258885697}{2260467690048355091300882058436} a^{8} + \frac{105817446868772446636931793063}{4520935380096710182601764116872} a^{6} - \frac{46696014175742546069421026205}{565116922512088772825220514609} a^{4} - \frac{128846533052382569398229622575}{1130233845024177545650441029218} a^{2} - \frac{3033044518664110085629264282}{15273430338164561427708662557}$, $\frac{1}{578679728652378903373025806959616} a^{21} - \frac{350601739014104242956908399}{144669932163094725843256451739904} a^{19} + \frac{506650703299234857054692236}{565116922512088772825220514609} a^{17} + \frac{153808014653916087405481007027}{72334966081547362921628225869952} a^{15} - \frac{2558227173835207123847957117}{358093891492808727334793197376} a^{13} + \frac{170696973910102093150160106515}{18083741520386840730407056467488} a^{11} + \frac{247840823262505634490515671365}{9041870760193420365203528233744} a^{9} + \frac{106365729617962188315446215481}{2260467690048355091300882058436} a^{7} + \frac{44944567853244822496830949137}{565116922512088772825220514609} a^{5} + \frac{34854030671878364580991495333}{565116922512088772825220514609} a^{3} + \frac{193245959572719155385890472706}{565116922512088772825220514609} a$

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

Class group and class number

$C_{25518776}$, which has order $25518776$ (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:  \( 866679281.3791491 \) (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

\(\Q(\sqrt{-178}) \), 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 R $22$ $22$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/23.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{22}$ $22$ $22$ $22$ $22$ $22$

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
2Data not computed
89Data not computed