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Properties

Label 21.21.1593043650...6681.1
Degree $21$
Signature $[21, 0]$
Discriminant $29^{18}\cdot 31^{14}$
Root discriminant $176.90$
Ramified primes $29, 31$
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![144611, 3735328, 17526365, 6060357, -71499935, -29226462, 114123433, 12733514, -75328115, 4524633, 23337412, -3325067, -3821466, 712451, 346443, -73530, -17149, 3920, 425, -102, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 4*x^20 - 102*x^19 + 425*x^18 + 3920*x^17 - 17149*x^16 - 73530*x^15 + 346443*x^14 + 712451*x^13 - 3821466*x^12 - 3325067*x^11 + 23337412*x^10 + 4524633*x^9 - 75328115*x^8 + 12733514*x^7 + 114123433*x^6 - 29226462*x^5 - 71499935*x^4 + 6060357*x^3 + 17526365*x^2 + 3735328*x + 144611)
 
gp: K = bnfinit(x^21 - 4*x^20 - 102*x^19 + 425*x^18 + 3920*x^17 - 17149*x^16 - 73530*x^15 + 346443*x^14 + 712451*x^13 - 3821466*x^12 - 3325067*x^11 + 23337412*x^10 + 4524633*x^9 - 75328115*x^8 + 12733514*x^7 + 114123433*x^6 - 29226462*x^5 - 71499935*x^4 + 6060357*x^3 + 17526365*x^2 + 3735328*x + 144611, 1)
 

Normalized defining polynomial

\( x^{21} - 4 x^{20} - 102 x^{19} + 425 x^{18} + 3920 x^{17} - 17149 x^{16} - 73530 x^{15} + 346443 x^{14} + 712451 x^{13} - 3821466 x^{12} - 3325067 x^{11} + 23337412 x^{10} + 4524633 x^{9} - 75328115 x^{8} + 12733514 x^{7} + 114123433 x^{6} - 29226462 x^{5} - 71499935 x^{4} + 6060357 x^{3} + 17526365 x^{2} + 3735328 x + 144611 \)

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:  \(159304365023255555687695016934897349651835476681=29^{18}\cdot 31^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $176.90$
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:  $29, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(899=29\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{899}(1,·)$, $\chi_{899}(645,·)$, $\chi_{899}(842,·)$, $\chi_{899}(459,·)$, $\chi_{899}(397,·)$, $\chi_{899}(335,·)$, $\chi_{899}(342,·)$, $\chi_{899}(25,·)$, $\chi_{899}(284,·)$, $\chi_{899}(94,·)$, $\chi_{899}(36,·)$, $\chi_{899}(807,·)$, $\chi_{899}(552,·)$, $\chi_{899}(745,·)$, $\chi_{899}(683,·)$, $\chi_{899}(749,·)$, $\chi_{899}(687,·)$, $\chi_{899}(625,·)$, $\chi_{899}(373,·)$, $\chi_{899}(315,·)$, $\chi_{899}(893,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{34} a^{18} + \frac{5}{34} a^{17} - \frac{2}{17} a^{14} + \frac{1}{17} a^{13} - \frac{11}{34} a^{12} + \frac{8}{17} a^{11} + \frac{8}{17} a^{10} + \frac{7}{34} a^{9} - \frac{1}{34} a^{8} - \frac{3}{17} a^{7} - \frac{1}{2} a^{6} - \frac{8}{17} a^{5} + \frac{3}{17} a^{4} - \frac{3}{17} a^{3} + \frac{9}{34} a^{2} + \frac{9}{34} a - \frac{15}{34}$, $\frac{1}{34} a^{19} - \frac{4}{17} a^{17} - \frac{2}{17} a^{15} + \frac{5}{34} a^{14} + \frac{13}{34} a^{13} - \frac{7}{17} a^{12} - \frac{13}{34} a^{11} + \frac{6}{17} a^{10} + \frac{15}{34} a^{9} - \frac{1}{34} a^{8} + \frac{13}{34} a^{7} + \frac{1}{34} a^{6} + \frac{1}{34} a^{5} - \frac{1}{17} a^{4} - \frac{6}{17} a^{3} + \frac{15}{34} a^{2} - \frac{9}{34} a - \frac{5}{17}$, $\frac{1}{191039108129151735074665391998546933538290391932549743264067642} a^{20} - \frac{1173830730856315328300532164754219167798106351040679071476207}{95519554064575867537332695999273466769145195966274871632033821} a^{19} + \frac{258543551806188969416200171895551892309265085795572053319311}{191039108129151735074665391998546933538290391932549743264067642} a^{18} + \frac{1527334579006198388328846355813411623266593586500496649187135}{11237594595832455004392081882267466678722964231326455486121626} a^{17} - \frac{24503450266062725092312199182882086283614373260845781383193201}{191039108129151735074665391998546933538290391932549743264067642} a^{16} + \frac{31421203547177961117007495836233457865595765751189623009743}{547389994639403252362938085955721872602551266282377487862658} a^{15} + \frac{32942840224152517859228005576530364592686979290228491590801877}{191039108129151735074665391998546933538290391932549743264067642} a^{14} - \frac{2557617981324728158780123796544762523470465141667027590551847}{5618797297916227502196040941133733339361482115663227743060813} a^{13} - \frac{11588707443841951160092595547873110080882031460299307030837527}{191039108129151735074665391998546933538290391932549743264067642} a^{12} - \frac{37505395145352787340555681067007914483156876054688165980573123}{95519554064575867537332695999273466769145195966274871632033821} a^{11} + \frac{66364931898248500195274279955012375597456917664209378977944863}{191039108129151735074665391998546933538290391932549743264067642} a^{10} - \frac{51828895129050527086881452287578291667069301124502889876530585}{191039108129151735074665391998546933538290391932549743264067642} a^{9} - \frac{93282707110722061893857042421477857840608456298367017905935391}{191039108129151735074665391998546933538290391932549743264067642} a^{8} - \frac{80635037927027310355805415018504957000303352171426485064247243}{191039108129151735074665391998546933538290391932549743264067642} a^{7} - \frac{81786845182915608170378715807273932051841856241294088108803571}{191039108129151735074665391998546933538290391932549743264067642} a^{6} - \frac{33338477099023704666141703740588595659674647124377156404279006}{95519554064575867537332695999273466769145195966274871632033821} a^{5} + \frac{68215525606255009570457813244804037025291661390880091752298395}{191039108129151735074665391998546933538290391932549743264067642} a^{4} + \frac{43967335735511088309341000309060851970168231017768044719324197}{95519554064575867537332695999273466769145195966274871632033821} a^{3} + \frac{920722818332009031294584773858749593246983111310982616725746}{95519554064575867537332695999273466769145195966274871632033821} a^{2} + \frac{16677300382397707163359853982456544412939816546507186472420945}{95519554064575867537332695999273466769145195966274871632033821} a - \frac{2522259156157470745549317213051601775331580730450859657964811}{11237594595832455004392081882267466678722964231326455486121626}$

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:  \( 51659455204894376 \) (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

3.3.961.1, 7.7.594823321.1

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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{3}$ $21$ $21$ $21$ $21$ $21$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{7}$ $21$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{3}$ R R $21$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{7}$ $21$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{3}$ $21$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{7}$

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
$29$29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$
31Data not computed