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Properties

Label 21.21.3059490478...3201.1
Degree $21$
Signature $[21, 0]$
Discriminant $211^{20}$
Root discriminant $163.53$
Ramified prime $211$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{21}$ (as 21T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1759, -6036, 378406, -1452380, -2662713, 11921203, -1508898, -18320520, 4651760, 12688535, -2182156, -4477546, 286211, 794170, -6133, -75232, -1091, 3859, 67, -100, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 - 100*x^19 + 67*x^18 + 3859*x^17 - 1091*x^16 - 75232*x^15 - 6133*x^14 + 794170*x^13 + 286211*x^12 - 4477546*x^11 - 2182156*x^10 + 12688535*x^9 + 4651760*x^8 - 18320520*x^7 - 1508898*x^6 + 11921203*x^5 - 2662713*x^4 - 1452380*x^3 + 378406*x^2 - 6036*x - 1759)
 
gp: K = bnfinit(x^21 - x^20 - 100*x^19 + 67*x^18 + 3859*x^17 - 1091*x^16 - 75232*x^15 - 6133*x^14 + 794170*x^13 + 286211*x^12 - 4477546*x^11 - 2182156*x^10 + 12688535*x^9 + 4651760*x^8 - 18320520*x^7 - 1508898*x^6 + 11921203*x^5 - 2662713*x^4 - 1452380*x^3 + 378406*x^2 - 6036*x - 1759, 1)
 

Normalized defining polynomial

\( x^{21} - x^{20} - 100 x^{19} + 67 x^{18} + 3859 x^{17} - 1091 x^{16} - 75232 x^{15} - 6133 x^{14} + 794170 x^{13} + 286211 x^{12} - 4477546 x^{11} - 2182156 x^{10} + 12688535 x^{9} + 4651760 x^{8} - 18320520 x^{7} - 1508898 x^{6} + 11921203 x^{5} - 2662713 x^{4} - 1452380 x^{3} + 378406 x^{2} - 6036 x - 1759 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[21, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(30594904784676758060124060477990514093520773201=211^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $163.53$
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:  $211$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(211\)
Dirichlet character group:    $\lbrace$$\chi_{211}(1,·)$, $\chi_{211}(43,·)$, $\chi_{211}(196,·)$, $\chi_{211}(199,·)$, $\chi_{211}(73,·)$, $\chi_{211}(14,·)$, $\chi_{211}(144,·)$, $\chi_{211}(148,·)$, $\chi_{211}(161,·)$, $\chi_{211}(34,·)$, $\chi_{211}(101,·)$, $\chi_{211}(171,·)$, $\chi_{211}(173,·)$, $\chi_{211}(178,·)$, $\chi_{211}(179,·)$, $\chi_{211}(180,·)$, $\chi_{211}(117,·)$, $\chi_{211}(54,·)$, $\chi_{211}(185,·)$, $\chi_{211}(58,·)$, $\chi_{211}(123,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{23} a^{15} + \frac{7}{23} a^{14} - \frac{1}{23} a^{13} + \frac{4}{23} a^{12} - \frac{4}{23} a^{11} + \frac{4}{23} a^{10} + \frac{1}{23} a^{9} - \frac{4}{23} a^{8} - \frac{3}{23} a^{7} + \frac{7}{23} a^{6} - \frac{4}{23} a^{5} - \frac{1}{23} a^{4} - \frac{4}{23} a - \frac{3}{23}$, $\frac{1}{23} a^{16} - \frac{4}{23} a^{14} + \frac{11}{23} a^{13} - \frac{9}{23} a^{12} + \frac{9}{23} a^{11} - \frac{4}{23} a^{10} - \frac{11}{23} a^{9} + \frac{2}{23} a^{8} + \frac{5}{23} a^{7} - \frac{7}{23} a^{6} + \frac{4}{23} a^{5} + \frac{7}{23} a^{4} - \frac{4}{23} a^{2} + \frac{2}{23} a - \frac{2}{23}$, $\frac{1}{23} a^{17} - \frac{7}{23} a^{14} + \frac{10}{23} a^{13} + \frac{2}{23} a^{12} + \frac{3}{23} a^{11} + \frac{5}{23} a^{10} + \frac{6}{23} a^{9} - \frac{11}{23} a^{8} + \frac{4}{23} a^{7} + \frac{9}{23} a^{6} - \frac{9}{23} a^{5} - \frac{4}{23} a^{4} - \frac{4}{23} a^{3} + \frac{2}{23} a^{2} + \frac{5}{23} a + \frac{11}{23}$, $\frac{1}{174731} a^{18} + \frac{2471}{174731} a^{17} - \frac{1859}{174731} a^{16} + \frac{1057}{174731} a^{15} + \frac{21057}{174731} a^{14} - \frac{55520}{174731} a^{13} + \frac{7279}{174731} a^{12} - \frac{36477}{174731} a^{11} + \frac{40153}{174731} a^{10} + \frac{66895}{174731} a^{9} + \frac{40887}{174731} a^{8} + \frac{17554}{174731} a^{7} + \frac{36458}{174731} a^{6} + \frac{67955}{174731} a^{5} - \frac{32636}{174731} a^{4} - \frac{68877}{174731} a^{3} - \frac{39896}{174731} a^{2} + \frac{37006}{174731} a + \frac{61724}{174731}$, $\frac{1}{174731} a^{19} + \frac{288}{174731} a^{17} - \frac{1539}{174731} a^{16} - \frac{3}{2461} a^{15} + \frac{1550}{7597} a^{14} - \frac{72994}{174731} a^{13} - \frac{33190}{174731} a^{12} - \frac{16764}{174731} a^{11} - \frac{25512}{174731} a^{10} + \frac{23674}{174731} a^{9} + \frac{79056}{174731} a^{8} - \frac{36576}{174731} a^{7} + \frac{57866}{174731} a^{6} - \frac{55741}{174731} a^{5} + \frac{61673}{174731} a^{4} - \frac{2435}{174731} a^{3} - \frac{80202}{174731} a^{2} - \frac{18580}{174731} a - \frac{40617}{174731}$, $\frac{1}{76279165896974106214518174122144111593126557521} a^{20} + \frac{120639252783435482010763258431689309070973}{76279165896974106214518174122144111593126557521} a^{19} - \frac{13162162658013302610758620730851510134101}{76279165896974106214518174122144111593126557521} a^{18} - \frac{560340345359049602221542488518218237059681357}{76279165896974106214518174122144111593126557521} a^{17} - \frac{824303666062581331599897659010830304460069653}{76279165896974106214518174122144111593126557521} a^{16} - \frac{856771311039027405264335652551250767889400946}{76279165896974106214518174122144111593126557521} a^{15} - \frac{25220854628456945072304432385139811837775073228}{76279165896974106214518174122144111593126557521} a^{14} - \frac{19388058659488023455836172169895605552546303585}{76279165896974106214518174122144111593126557521} a^{13} + \frac{8711007399560667179758381928646387361257861088}{76279165896974106214518174122144111593126557521} a^{12} + \frac{2077750150825155234430514698500878420901681566}{76279165896974106214518174122144111593126557521} a^{11} + \frac{22422981721374619200945854918490640807759464687}{76279165896974106214518174122144111593126557521} a^{10} - \frac{889937688440745513284258676346990876527175032}{76279165896974106214518174122144111593126557521} a^{9} + \frac{13340601067712906893971534256379319162203478587}{76279165896974106214518174122144111593126557521} a^{8} + \frac{839777568542428201172473801672502768583946232}{3316485473781482878892094527049743982309850327} a^{7} + \frac{14244565740758928677578555581101394037316323621}{76279165896974106214518174122144111593126557521} a^{6} + \frac{6144969335749890489328683490970357844578827053}{76279165896974106214518174122144111593126557521} a^{5} - \frac{20089847717542981465373679660059290172095420456}{76279165896974106214518174122144111593126557521} a^{4} - \frac{16836969185469647782066742360009561833632917702}{76279165896974106214518174122144111593126557521} a^{3} - \frac{20052154970494914671393187856738661405202448385}{76279165896974106214518174122144111593126557521} a^{2} + \frac{22000617327927959924617881083842439901566056694}{76279165896974106214518174122144111593126557521} a + \frac{9896406234151521391612600833464484403057770129}{76279165896974106214518174122144111593126557521}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $20$
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:  \( 66095963290529660 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{21}$ (as 21T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 21
The 21 conjugacy class representatives for $C_{21}$
Character table for $C_{21}$ is not computed

Intermediate fields

3.3.44521.1, 7.7.88245939632761.1

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 $21$ $21$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{3}$ $21$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{3}$ $21$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{21}$ $21$ $21$ $21$ $21$ $21$ $21$ $21$ $21$

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
211Data not computed