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Properties

Label 22.0.16541106704...7696.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,2^{22}\cdot 3^{11}\cdot 67^{21}$
Root discriminant $191.72$
Ramified primes $2, 3, 67$
Class number $23097372$ (GRH)
Class group $[2, 11548686]$ (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![11868849, 0, 462885111, 0, 2090236185, 0, 3362840550, 0, 2478977622, 0, 883227969, 0, 149068836, 0, 12981384, 0, 613251, 0, 15678, 0, 201, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 201*x^20 + 15678*x^18 + 613251*x^16 + 12981384*x^14 + 149068836*x^12 + 883227969*x^10 + 2478977622*x^8 + 3362840550*x^6 + 2090236185*x^4 + 462885111*x^2 + 11868849)
 
gp: K = bnfinit(x^22 + 201*x^20 + 15678*x^18 + 613251*x^16 + 12981384*x^14 + 149068836*x^12 + 883227969*x^10 + 2478977622*x^8 + 3362840550*x^6 + 2090236185*x^4 + 462885111*x^2 + 11868849, 1)
 

Normalized defining polynomial

\( x^{22} + 201 x^{20} + 15678 x^{18} + 613251 x^{16} + 12981384 x^{14} + 149068836 x^{12} + 883227969 x^{10} + 2478977622 x^{8} + 3362840550 x^{6} + 2090236185 x^{4} + 462885111 x^{2} + 11868849 \)

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:  \(-165411067048459529582956967098503557995819954077696=-\,2^{22}\cdot 3^{11}\cdot 67^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $191.72$
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:  $2, 3, 67$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(804=2^{2}\cdot 3\cdot 67\)
Dirichlet character group:    $\lbrace$$\chi_{804}(1,·)$, $\chi_{804}(193,·)$, $\chi_{804}(493,·)$, $\chi_{804}(265,·)$, $\chi_{804}(779,·)$, $\chi_{804}(397,·)$, $\chi_{804}(527,·)$, $\chi_{804}(563,·)$, $\chi_{804}(611,·)$, $\chi_{804}(277,·)$, $\chi_{804}(407,·)$, $\chi_{804}(25,·)$, $\chi_{804}(539,·)$, $\chi_{804}(349,·)$, $\chi_{804}(803,·)$, $\chi_{804}(625,·)$, $\chi_{804}(455,·)$, $\chi_{804}(685,·)$, $\chi_{804}(241,·)$, $\chi_{804}(179,·)$, $\chi_{804}(311,·)$, $\chi_{804}(119,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{9} a^{4}$, $\frac{1}{9} a^{5}$, $\frac{1}{27} a^{6}$, $\frac{1}{27} a^{7}$, $\frac{1}{81} a^{8}$, $\frac{1}{81} a^{9}$, $\frac{1}{243} a^{10}$, $\frac{1}{243} a^{11}$, $\frac{1}{729} a^{12}$, $\frac{1}{729} a^{13}$, $\frac{1}{2187} a^{14}$, $\frac{1}{2187} a^{15}$, $\frac{1}{190269} a^{16} + \frac{1}{63423} a^{14} + \frac{13}{21141} a^{12} + \frac{4}{7047} a^{10} - \frac{11}{2349} a^{8} + \frac{11}{783} a^{6} + \frac{5}{261} a^{4} - \frac{7}{87} a^{2} - \frac{5}{29}$, $\frac{1}{190269} a^{17} + \frac{1}{63423} a^{15} + \frac{13}{21141} a^{13} + \frac{4}{7047} a^{11} - \frac{11}{2349} a^{9} + \frac{11}{783} a^{7} + \frac{5}{261} a^{5} - \frac{7}{87} a^{3} - \frac{5}{29} a$, $\frac{1}{2048626323} a^{18} - \frac{574}{227625147} a^{16} + \frac{13399}{227625147} a^{14} - \frac{14164}{25291683} a^{12} - \frac{18677}{25291683} a^{10} - \frac{4927}{936729} a^{8} - \frac{21790}{2810187} a^{6} + \frac{16753}{936729} a^{4} + \frac{16682}{104081} a^{2} + \frac{7629}{104081}$, $\frac{1}{2048626323} a^{19} - \frac{574}{227625147} a^{17} + \frac{13399}{227625147} a^{15} - \frac{14164}{25291683} a^{13} - \frac{18677}{25291683} a^{11} - \frac{4927}{936729} a^{9} - \frac{21790}{2810187} a^{7} + \frac{16753}{936729} a^{5} + \frac{16682}{104081} a^{3} + \frac{7629}{104081} a$, $\frac{1}{695002794010614820344717} a^{20} + \frac{4810300708832}{77222532667846091149413} a^{18} - \frac{178006461792639605}{77222532667846091149413} a^{16} - \frac{2346707019078143843}{25740844222615363716471} a^{14} - \frac{5754079320484931552}{8580281407538454572157} a^{12} + \frac{2913900756043386233}{2860093802512818190719} a^{10} + \frac{33945429182939610}{11769933343674148933} a^{8} + \frac{1230209212891240634}{105929400093067340397} a^{6} + \frac{3036278358663837739}{105929400093067340397} a^{4} - \frac{4557385933435253828}{35309800031022446799} a^{2} - \frac{1757122307640656399}{11769933343674148933}$, $\frac{1}{695002794010614820344717} a^{21} + \frac{4810300708832}{77222532667846091149413} a^{19} - \frac{178006461792639605}{77222532667846091149413} a^{17} - \frac{2346707019078143843}{25740844222615363716471} a^{15} - \frac{5754079320484931552}{8580281407538454572157} a^{13} + \frac{2913900756043386233}{2860093802512818190719} a^{11} + \frac{33945429182939610}{11769933343674148933} a^{9} + \frac{1230209212891240634}{105929400093067340397} a^{7} + \frac{3036278358663837739}{105929400093067340397} a^{5} - \frac{4557385933435253828}{35309800031022446799} a^{3} - \frac{1757122307640656399}{11769933343674148933} a$

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

Class group and class number

$C_{2}\times C_{11548686}$, which has order $23097372$ (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:  \( 338444542.042557 \) (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

\(\Q(\sqrt{-201}) \), 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 R R ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ $22$ $22$ $22$ $22$ ${\href{/LocalNumberField/23.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{22}$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/47.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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
3Data not computed
67Data not computed