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Properties

Label 22.0.22262342333...3267.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,67^{21}$
Root discriminant $55.34$
Ramified prime $67$
Class number $67$ (GRH)
Class group $[67]$ (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![1073, 186, 5006, -5700, 355, 3508, -9804, 2341, 3984, -3927, 6412, 2088, -1439, 2902, 574, -361, 535, 59, -33, 40, 2, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + 2*x^20 + 40*x^19 - 33*x^18 + 59*x^17 + 535*x^16 - 361*x^15 + 574*x^14 + 2902*x^13 - 1439*x^12 + 2088*x^11 + 6412*x^10 - 3927*x^9 + 3984*x^8 + 2341*x^7 - 9804*x^6 + 3508*x^5 + 355*x^4 - 5700*x^3 + 5006*x^2 + 186*x + 1073)
 
gp: K = bnfinit(x^22 - x^21 + 2*x^20 + 40*x^19 - 33*x^18 + 59*x^17 + 535*x^16 - 361*x^15 + 574*x^14 + 2902*x^13 - 1439*x^12 + 2088*x^11 + 6412*x^10 - 3927*x^9 + 3984*x^8 + 2341*x^7 - 9804*x^6 + 3508*x^5 + 355*x^4 - 5700*x^3 + 5006*x^2 + 186*x + 1073, 1)
 

Normalized defining polynomial

\( x^{22} - x^{21} + 2 x^{20} + 40 x^{19} - 33 x^{18} + 59 x^{17} + 535 x^{16} - 361 x^{15} + 574 x^{14} + 2902 x^{13} - 1439 x^{12} + 2088 x^{11} + 6412 x^{10} - 3927 x^{9} + 3984 x^{8} + 2341 x^{7} - 9804 x^{6} + 3508 x^{5} + 355 x^{4} - 5700 x^{3} + 5006 x^{2} + 186 x + 1073 \)

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:  \(-222623423334106740048017938136372833267=-\,67^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.34$
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:  $67$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(67\)
Dirichlet character group:    $\lbrace$$\chi_{67}(64,·)$, $\chi_{67}(1,·)$, $\chi_{67}(66,·)$, $\chi_{67}(3,·)$, $\chi_{67}(5,·)$, $\chi_{67}(8,·)$, $\chi_{67}(9,·)$, $\chi_{67}(14,·)$, $\chi_{67}(15,·)$, $\chi_{67}(22,·)$, $\chi_{67}(24,·)$, $\chi_{67}(25,·)$, $\chi_{67}(27,·)$, $\chi_{67}(40,·)$, $\chi_{67}(42,·)$, $\chi_{67}(43,·)$, $\chi_{67}(45,·)$, $\chi_{67}(52,·)$, $\chi_{67}(53,·)$, $\chi_{67}(58,·)$, $\chi_{67}(59,·)$, $\chi_{67}(62,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{1073} a^{18} - \frac{14}{37} a^{17} - \frac{465}{1073} a^{16} + \frac{477}{1073} a^{15} + \frac{196}{1073} a^{14} - \frac{280}{1073} a^{13} - \frac{188}{1073} a^{12} - \frac{528}{1073} a^{11} + \frac{425}{1073} a^{10} + \frac{474}{1073} a^{9} + \frac{502}{1073} a^{8} + \frac{353}{1073} a^{7} - \frac{350}{1073} a^{6} - \frac{395}{1073} a^{5} + \frac{249}{1073} a^{4} + \frac{368}{1073} a^{3} + \frac{38}{1073} a^{2} + \frac{486}{1073} a$, $\frac{1}{1073} a^{19} - \frac{59}{1073} a^{17} + \frac{535}{1073} a^{16} - \frac{355}{1073} a^{15} - \frac{106}{1073} a^{14} - \frac{130}{1073} a^{13} + \frac{400}{1073} a^{12} - \frac{416}{1073} a^{11} + \frac{271}{1073} a^{10} - \frac{194}{1073} a^{9} + \frac{295}{1073} a^{8} + \frac{7}{29} a^{7} + \frac{214}{1073} a^{6} - \frac{244}{1073} a^{5} - \frac{473}{1073} a^{4} + \frac{299}{1073} a^{3} - \frac{181}{1073} a^{2} - \frac{4}{37} a$, $\frac{1}{1073} a^{20} + \frac{187}{1073} a^{17} + \frac{108}{1073} a^{16} + \frac{139}{1073} a^{15} - \frac{369}{1073} a^{14} - \frac{25}{1073} a^{13} + \frac{295}{1073} a^{12} + \frac{236}{1073} a^{11} + \frac{202}{1073} a^{10} + \frac{363}{1073} a^{9} - \frac{167}{1073} a^{8} - \frac{419}{1073} a^{7} - \frac{507}{1073} a^{6} - \frac{172}{1073} a^{5} - \frac{32}{1073} a^{4} + \frac{71}{1073} a^{3} - \frac{20}{1073} a^{2} - \frac{297}{1073} a$, $\frac{1}{282771446873143714790792225903047859} a^{21} - \frac{80533925868161970347019868387923}{282771446873143714790792225903047859} a^{20} - \frac{108023571629777619740718754586496}{282771446873143714790792225903047859} a^{19} + \frac{9024487331613800224898374625641}{282771446873143714790792225903047859} a^{18} + \frac{96052929383576115643125796395817107}{282771446873143714790792225903047859} a^{17} - \frac{70610044419220153651486542330462973}{282771446873143714790792225903047859} a^{16} - \frac{3273993997551090006709119055215263}{9750739547349783268648007789760271} a^{15} - \frac{98140927367370056505612999054810984}{282771446873143714790792225903047859} a^{14} - \frac{77762174972640680855014117005796415}{282771446873143714790792225903047859} a^{13} + \frac{133374860605491324843912348058095340}{282771446873143714790792225903047859} a^{12} + \frac{110722828079559091821076483743084594}{282771446873143714790792225903047859} a^{11} + \frac{101962984887965974444557322237424846}{282771446873143714790792225903047859} a^{10} - \frac{114145440394414143728449542589412032}{282771446873143714790792225903047859} a^{9} + \frac{17821229717532495917128033947726748}{282771446873143714790792225903047859} a^{8} - \frac{104803394079789110578831221204396886}{282771446873143714790792225903047859} a^{7} + \frac{2468144755135984650025708124980396}{7642471537111992291643033132514807} a^{6} - \frac{58870122401254469811799967581410399}{282771446873143714790792225903047859} a^{5} - \frac{21717101864859015627335270523510246}{282771446873143714790792225903047859} a^{4} + \frac{122040143707678877883237000268351754}{282771446873143714790792225903047859} a^{3} + \frac{120779925026331798613304262170420057}{282771446873143714790792225903047859} a^{2} + \frac{13375085356343952548539477525668814}{282771446873143714790792225903047859} a - \frac{36498034963818680776094827587832}{263533501279723872125621832155683}$

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

Class group and class number

$C_{67}$, which has order $67$ (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:  \( 338444542.043 \) (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

\(\Q(\sqrt{-67}) \), 11.11.1822837804551761449.1

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 $22$ $22$ $22$ $22$ $22$ $22$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/23.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{22}$ $22$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{22}$ $22$ $22$ ${\href{/LocalNumberField/47.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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
67Data not computed