↑

Properties

Label 22.0.36295271060...6256.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,2^{22}\cdot 89^{21}$
Root discriminant $145.15$
Ramified primes $2, 89$
Class number $230484$ (GRH)
Class group $[230484]$ (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![121841, 0, 1413231, 0, 6239345, 0, 13328106, 0, 14647086, 0, 8475025, 0, 2690648, 0, 489500, 0, 51531, 0, 3026, 0, 89, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 89*x^20 + 3026*x^18 + 51531*x^16 + 489500*x^14 + 2690648*x^12 + 8475025*x^10 + 14647086*x^8 + 13328106*x^6 + 6239345*x^4 + 1413231*x^2 + 121841)
 
gp: K = bnfinit(x^22 + 89*x^20 + 3026*x^18 + 51531*x^16 + 489500*x^14 + 2690648*x^12 + 8475025*x^10 + 14647086*x^8 + 13328106*x^6 + 6239345*x^4 + 1413231*x^2 + 121841, 1)
 

Normalized defining polynomial

\( x^{22} + 89 x^{20} + 3026 x^{18} + 51531 x^{16} + 489500 x^{14} + 2690648 x^{12} + 8475025 x^{10} + 14647086 x^{8} + 13328106 x^{6} + 6239345 x^{4} + 1413231 x^{2} + 121841 \)

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:  \(-362952710604500716554548260445405852449957216256=-\,2^{22}\cdot 89^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $145.15$
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:  \(356=2^{2}\cdot 89\)
Dirichlet character group:    $\lbrace$$\chi_{356}(1,·)$, $\chi_{356}(259,·)$, $\chi_{356}(263,·)$, $\chi_{356}(11,·)$, $\chi_{356}(269,·)$, $\chi_{356}(139,·)$, $\chi_{356}(203,·)$, $\chi_{356}(153,·)$, $\chi_{356}(217,·)$, $\chi_{356}(93,·)$, $\chi_{356}(97,·)$, $\chi_{356}(355,·)$, $\chi_{356}(105,·)$, $\chi_{356}(235,·)$, $\chi_{356}(45,·)$, $\chi_{356}(111,·)$, $\chi_{356}(245,·)$, $\chi_{356}(311,·)$, $\chi_{356}(121,·)$, $\chi_{356}(87,·)$, $\chi_{356}(251,·)$, $\chi_{356}(345,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{37} a^{19} - \frac{4}{37} a^{17} - \frac{6}{37} a^{15} - \frac{7}{37} a^{13} + \frac{12}{37} a^{11} + \frac{2}{37} a^{9} - \frac{11}{37} a^{7} - \frac{6}{37} a^{5} + \frac{6}{37} a^{3} - \frac{5}{37} a$, $\frac{1}{565116922512088772825220514609} a^{20} + \frac{60392517874630037224920833430}{565116922512088772825220514609} a^{18} - \frac{114672305366028859437337388496}{565116922512088772825220514609} a^{16} - \frac{212754313462033386859526894341}{565116922512088772825220514609} a^{14} - \frac{1197228430830444224349165404}{5595217054575136364606143709} a^{12} - \frac{226412214882176503970265119967}{565116922512088772825220514609} a^{10} - \frac{195088656544266646913035542788}{565116922512088772825220514609} a^{8} + \frac{105817446868772446636931793063}{565116922512088772825220514609} a^{6} - \frac{186784056702970184277684104820}{565116922512088772825220514609} a^{4} - \frac{128846533052382569398229622575}{565116922512088772825220514609} a^{2} - \frac{3033044518664110085629264282}{15273430338164561427708662557}$, $\frac{1}{565116922512088772825220514609} a^{21} - \frac{701203478028208485913816798}{565116922512088772825220514609} a^{19} + \frac{129702580044604123406001212416}{565116922512088772825220514609} a^{17} + \frac{153808014653916087405481007027}{565116922512088772825220514609} a^{15} - \frac{2558227173835207123847957117}{5595217054575136364606143709} a^{13} + \frac{170696973910102093150160106515}{565116922512088772825220514609} a^{11} + \frac{247840823262505634490515671365}{565116922512088772825220514609} a^{9} + \frac{212731459235924376630892430962}{565116922512088772825220514609} a^{7} + \frac{179778271412979289987323796548}{565116922512088772825220514609} a^{5} + \frac{69708061343756729161982990666}{565116922512088772825220514609} a^{3} + \frac{193245959572719155385890472706}{565116922512088772825220514609} a$

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

Class group and class number

$C_{230484}$, which has order $230484$ (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

\(\Q(\sqrt{-89}) \), 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 ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/23.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{11}$ $22$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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
2Data not computed
89Data not computed