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Properties

Label 21.21.4808896268...5521.1
Degree $21$
Signature $[21, 0]$
Discriminant $3^{28}\cdot 71^{18}$
Root discriminant $167.09$
Ramified primes $3, 71$
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![-5904559, -34962030, -40148160, 109853705, 266883405, 66234471, -231646188, -139946343, 72399831, 64179378, -9596163, -13945275, 326899, 1686642, 48774, -119292, -5637, 4893, 220, -108, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 3*x^20 - 108*x^19 + 220*x^18 + 4893*x^17 - 5637*x^16 - 119292*x^15 + 48774*x^14 + 1686642*x^13 + 326899*x^12 - 13945275*x^11 - 9596163*x^10 + 64179378*x^9 + 72399831*x^8 - 139946343*x^7 - 231646188*x^6 + 66234471*x^5 + 266883405*x^4 + 109853705*x^3 - 40148160*x^2 - 34962030*x - 5904559)
 
gp: K = bnfinit(x^21 - 3*x^20 - 108*x^19 + 220*x^18 + 4893*x^17 - 5637*x^16 - 119292*x^15 + 48774*x^14 + 1686642*x^13 + 326899*x^12 - 13945275*x^11 - 9596163*x^10 + 64179378*x^9 + 72399831*x^8 - 139946343*x^7 - 231646188*x^6 + 66234471*x^5 + 266883405*x^4 + 109853705*x^3 - 40148160*x^2 - 34962030*x - 5904559, 1)
 

Normalized defining polynomial

\( x^{21} - 3 x^{20} - 108 x^{19} + 220 x^{18} + 4893 x^{17} - 5637 x^{16} - 119292 x^{15} + 48774 x^{14} + 1686642 x^{13} + 326899 x^{12} - 13945275 x^{11} - 9596163 x^{10} + 64179378 x^{9} + 72399831 x^{8} - 139946343 x^{7} - 231646188 x^{6} + 66234471 x^{5} + 266883405 x^{4} + 109853705 x^{3} - 40148160 x^{2} - 34962030 x - 5904559 \)

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:  \(48088962682434277721537576300296135799190535521=3^{28}\cdot 71^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $167.09$
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:  $3, 71$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(639=3^{2}\cdot 71\)
Dirichlet character group:    $\lbrace$$\chi_{639}(1,·)$, $\chi_{639}(517,·)$, $\chi_{639}(385,·)$, $\chi_{639}(463,·)$, $\chi_{639}(400,·)$, $\chi_{639}(529,·)$, $\chi_{639}(403,·)$, $\chi_{639}(214,·)$, $\chi_{639}(91,·)$, $\chi_{639}(598,·)$, $\chi_{639}(37,·)$, $\chi_{639}(613,·)$, $\chi_{639}(103,·)$, $\chi_{639}(616,·)$, $\chi_{639}(427,·)$, $\chi_{639}(172,·)$, $\chi_{639}(304,·)$, $\chi_{639}(250,·)$, $\chi_{639}(187,·)$, $\chi_{639}(316,·)$, $\chi_{639}(190,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{85} a^{12} + \frac{5}{17} a^{11} - \frac{42}{85} a^{10} + \frac{21}{85} a^{9} - \frac{39}{85} a^{8} + \frac{11}{85} a^{7} - \frac{11}{85} a^{6} + \frac{14}{85} a^{5} + \frac{21}{85} a^{4} - \frac{19}{85} a^{3} - \frac{32}{85} a^{2} - \frac{1}{85} a + \frac{2}{5}$, $\frac{1}{85} a^{13} + \frac{13}{85} a^{11} - \frac{2}{5} a^{10} + \frac{31}{85} a^{9} - \frac{2}{5} a^{8} - \frac{31}{85} a^{7} + \frac{2}{5} a^{6} + \frac{11}{85} a^{5} - \frac{2}{5} a^{4} + \frac{18}{85} a^{3} + \frac{2}{5} a^{2} - \frac{26}{85} a$, $\frac{1}{85} a^{14} - \frac{19}{85} a^{11} - \frac{18}{85} a^{10} + \frac{33}{85} a^{9} - \frac{2}{5} a^{8} - \frac{24}{85} a^{7} - \frac{16}{85} a^{6} + \frac{39}{85} a^{5} + \frac{26}{85} a^{3} - \frac{7}{17} a^{2} + \frac{13}{85} a - \frac{1}{5}$, $\frac{1}{85} a^{15} + \frac{32}{85} a^{11} + \frac{5}{17} a^{9} + \frac{23}{85} a^{7} + \frac{11}{85} a^{5} + \frac{29}{85} a^{3} - \frac{36}{85} a - \frac{2}{5}$, $\frac{1}{85} a^{16} - \frac{7}{17} a^{11} + \frac{9}{85} a^{10} + \frac{8}{85} a^{9} - \frac{4}{85} a^{8} - \frac{12}{85} a^{7} + \frac{23}{85} a^{6} - \frac{23}{85} a^{5} + \frac{37}{85} a^{4} + \frac{13}{85} a^{3} - \frac{32}{85} a^{2} - \frac{2}{85} a + \frac{1}{5}$, $\frac{1}{85} a^{17} + \frac{2}{5} a^{11} - \frac{1}{5} a^{10} - \frac{2}{5} a^{9} - \frac{1}{5} a^{8} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{18}{85} a$, $\frac{1}{425} a^{18} - \frac{1}{425} a^{17} - \frac{2}{425} a^{15} + \frac{1}{425} a^{14} + \frac{2}{425} a^{12} - \frac{84}{425} a^{11} - \frac{8}{25} a^{10} - \frac{77}{425} a^{9} + \frac{24}{425} a^{8} - \frac{133}{425} a^{7} - \frac{4}{425} a^{6} - \frac{42}{85} a^{5} + \frac{178}{425} a^{4} + \frac{151}{425} a^{3} + \frac{53}{425} a^{2} + \frac{2}{17} a - \frac{6}{25}$, $\frac{1}{7225} a^{19} + \frac{8}{7225} a^{18} + \frac{1}{7225} a^{17} + \frac{8}{7225} a^{16} + \frac{38}{7225} a^{15} - \frac{36}{7225} a^{14} + \frac{42}{7225} a^{13} - \frac{21}{7225} a^{12} - \frac{2167}{7225} a^{11} - \frac{1271}{7225} a^{10} - \frac{1404}{7225} a^{9} + \frac{3388}{7225} a^{8} + \frac{959}{7225} a^{7} - \frac{3361}{7225} a^{6} - \frac{1852}{7225} a^{5} - \frac{1012}{7225} a^{4} - \frac{463}{7225} a^{3} + \frac{1957}{7225} a^{2} - \frac{31}{425} a - \frac{7}{25}$, $\frac{1}{7931706429483875339033686820377564523812413601030675} a^{20} - \frac{77163016685290701589060691498149799260934475621}{1586341285896775067806737364075512904762482720206135} a^{19} + \frac{4792090653521813408458398718431049158003900437564}{7931706429483875339033686820377564523812413601030675} a^{18} + \frac{24463082673904275683347648971099405411761099646358}{7931706429483875339033686820377564523812413601030675} a^{17} - \frac{45433054894675152651417936562940553473610385979936}{7931706429483875339033686820377564523812413601030675} a^{16} - \frac{23024305013710511119095955978718562722092082682954}{7931706429483875339033686820377564523812413601030675} a^{15} + \frac{24770861944826584995980418657371628657162705771327}{7931706429483875339033686820377564523812413601030675} a^{14} + \frac{18945965978091290503962260493097070925637780863153}{7931706429483875339033686820377564523812413601030675} a^{13} + \frac{1795079981580830797410039804056808304099197479512}{1586341285896775067806737364075512904762482720206135} a^{12} + \frac{2271464088786390186935612752608874976034532669372072}{7931706429483875339033686820377564523812413601030675} a^{11} + \frac{647026569542212749370844183265419629995627809860022}{7931706429483875339033686820377564523812413601030675} a^{10} + \frac{2852503854116603101672821478688070180695526105153291}{7931706429483875339033686820377564523812413601030675} a^{9} - \frac{3541755517156870720276439997736357448428816455189092}{7931706429483875339033686820377564523812413601030675} a^{8} + \frac{2491121257758623642664274849308703565201267549980021}{7931706429483875339033686820377564523812413601030675} a^{7} + \frac{2975071323606132148385056123112230211977564198346583}{7931706429483875339033686820377564523812413601030675} a^{6} + \frac{3634148010731057021958308531844040227714200371732174}{7931706429483875339033686820377564523812413601030675} a^{5} - \frac{1049697022951635120466033226104115834093678414214466}{7931706429483875339033686820377564523812413601030675} a^{4} - \frac{3641323931960213019864647729131790994609115188645657}{7931706429483875339033686820377564523812413601030675} a^{3} - \frac{3307239419950111567211438849764709509750869760499872}{7931706429483875339033686820377564523812413601030675} a^{2} - \frac{132981508412189270730269197219641410802498324592816}{466570966440227961119628636492797913165436094178275} a - \frac{1278032462918442110025141584415079432831776275698}{5489070193414446601407395723444681331358071696215}$

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:  \( 82999008884616240 \) (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

\(\Q(\zeta_{9})^+\), 7.7.128100283921.1

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 $21$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{7}$ $21$ $21$ $21$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{21}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{3}$ $21$ $21$ $21$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{3}$ $21$ $21$ $21$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{3}$ $21$

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
3Data not computed
$71$71.7.6.1$x^{7} - 71$$7$$1$$6$$C_7$$[\ ]_{7}$
71.7.6.1$x^{7} - 71$$7$$1$$6$$C_7$$[\ ]_{7}$
71.7.6.1$x^{7} - 71$$7$$1$$6$$C_7$$[\ ]_{7}$