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Properties

Label 21.21.1838975925...5289.2
Degree $21$
Signature $[21, 0]$
Discriminant $13^{14}\cdot 43^{20}$
Root discriminant $198.75$
Ramified primes $13, 43$
Class number $3$ (GRH)
Class group $[3]$ (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![-357231187, 732657661, 9354769810, -21439276738, -20453992557, 19001737751, 11518541404, -7272568707, -2924408106, 1521281881, 392209615, -186891957, -29260009, 13738816, 1210958, -597785, -26598, 14834, 277, -192, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 - 192*x^19 + 277*x^18 + 14834*x^17 - 26598*x^16 - 597785*x^15 + 1210958*x^14 + 13738816*x^13 - 29260009*x^12 - 186891957*x^11 + 392209615*x^10 + 1521281881*x^9 - 2924408106*x^8 - 7272568707*x^7 + 11518541404*x^6 + 19001737751*x^5 - 20453992557*x^4 - 21439276738*x^3 + 9354769810*x^2 + 732657661*x - 357231187)
 
gp: K = bnfinit(x^21 - x^20 - 192*x^19 + 277*x^18 + 14834*x^17 - 26598*x^16 - 597785*x^15 + 1210958*x^14 + 13738816*x^13 - 29260009*x^12 - 186891957*x^11 + 392209615*x^10 + 1521281881*x^9 - 2924408106*x^8 - 7272568707*x^7 + 11518541404*x^6 + 19001737751*x^5 - 20453992557*x^4 - 21439276738*x^3 + 9354769810*x^2 + 732657661*x - 357231187, 1)
 

Normalized defining polynomial

\( x^{21} - x^{20} - 192 x^{19} + 277 x^{18} + 14834 x^{17} - 26598 x^{16} - 597785 x^{15} + 1210958 x^{14} + 13738816 x^{13} - 29260009 x^{12} - 186891957 x^{11} + 392209615 x^{10} + 1521281881 x^{9} - 2924408106 x^{8} - 7272568707 x^{7} + 11518541404 x^{6} + 19001737751 x^{5} - 20453992557 x^{4} - 21439276738 x^{3} + 9354769810 x^{2} + 732657661 x - 357231187 \)

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:  \(1838975925801931725577355822978754285318753465289=13^{14}\cdot 43^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $198.75$
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:  $13, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(559=13\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{559}(1,·)$, $\chi_{559}(68,·)$, $\chi_{559}(391,·)$, $\chi_{559}(74,·)$, $\chi_{559}(139,·)$, $\chi_{559}(146,·)$, $\chi_{559}(274,·)$, $\chi_{559}(152,·)$, $\chi_{559}(380,·)$, $\chi_{559}(282,·)$, $\chi_{559}(224,·)$, $\chi_{559}(425,·)$, $\chi_{559}(170,·)$, $\chi_{559}(365,·)$, $\chi_{559}(178,·)$, $\chi_{559}(183,·)$, $\chi_{559}(185,·)$, $\chi_{559}(315,·)$, $\chi_{559}(508,·)$, $\chi_{559}(445,·)$, $\chi_{559}(126,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{509} a^{19} - \frac{235}{509} a^{18} - \frac{137}{509} a^{17} + \frac{226}{509} a^{16} + \frac{146}{509} a^{15} + \frac{27}{509} a^{14} - \frac{117}{509} a^{13} - \frac{80}{509} a^{12} + \frac{27}{509} a^{11} + \frac{249}{509} a^{10} - \frac{141}{509} a^{9} + \frac{134}{509} a^{8} + \frac{62}{509} a^{7} + \frac{106}{509} a^{6} - \frac{235}{509} a^{5} + \frac{49}{509} a^{4} - \frac{131}{509} a^{3} - \frac{217}{509} a^{2} - \frac{143}{509} a - \frac{214}{509}$, $\frac{1}{55203909301187356764850778511665189264303648494138209029836811635708717648360173387982974041} a^{20} - \frac{41319123850416522117961906650696177734622374263223528707322873283710861352068566111118987}{55203909301187356764850778511665189264303648494138209029836811635708717648360173387982974041} a^{19} + \frac{16453675243112503524961231940020303489617872126263494549887633059682390614026529790086740498}{55203909301187356764850778511665189264303648494138209029836811635708717648360173387982974041} a^{18} - \frac{12885632374449033007033674179085681564024341954553820065349557774490352293062882553954704310}{55203909301187356764850778511665189264303648494138209029836811635708717648360173387982974041} a^{17} + \frac{26875475332317959571595801983201566552553430370247909565667147676261184711397620219765763403}{55203909301187356764850778511665189264303648494138209029836811635708717648360173387982974041} a^{16} - \frac{13943214686410813771776662919163263196929069667740687310192697273420913423625451002793423967}{55203909301187356764850778511665189264303648494138209029836811635708717648360173387982974041} a^{15} + \frac{7030422503795344505042445211697988637760630436773103108803592872891106801360358935587183224}{55203909301187356764850778511665189264303648494138209029836811635708717648360173387982974041} a^{14} - \frac{21391910807754410617072178541255362815317048973074614715561214836676931780344111602936647079}{55203909301187356764850778511665189264303648494138209029836811635708717648360173387982974041} a^{13} - \frac{19621562956139840215447045262729442429012266437839429665846336428863435428444042985038279046}{55203909301187356764850778511665189264303648494138209029836811635708717648360173387982974041} a^{12} + \frac{18576323629386168716993894670949919412178336167911638872940791146478093676084311263158398718}{55203909301187356764850778511665189264303648494138209029836811635708717648360173387982974041} a^{11} - \frac{1807308240409031898441347946852368244177126784352203240278101316134564491467409876320078238}{55203909301187356764850778511665189264303648494138209029836811635708717648360173387982974041} a^{10} - \frac{7456819284927031773733476657786935629821702699983882022704087549146829863670273347885981302}{55203909301187356764850778511665189264303648494138209029836811635708717648360173387982974041} a^{9} + \frac{1479556486109644023339542975318201378829998518698137814603772647086110137560846453122738101}{55203909301187356764850778511665189264303648494138209029836811635708717648360173387982974041} a^{8} + \frac{6888987742827220058540995683435670195128811951883551574870181028218371744907396147889292526}{55203909301187356764850778511665189264303648494138209029836811635708717648360173387982974041} a^{7} - \frac{890647279564285940697475558022090828615022241863605140465641989686896178019588460783203476}{55203909301187356764850778511665189264303648494138209029836811635708717648360173387982974041} a^{6} - \frac{20457877126565247128708077245903958933262927559137437845312273275624923158519010521625983393}{55203909301187356764850778511665189264303648494138209029836811635708717648360173387982974041} a^{5} + \frac{13396709998321580744602513912049088947702903085283470787443859761203670038367819221741435002}{55203909301187356764850778511665189264303648494138209029836811635708717648360173387982974041} a^{4} - \frac{11804739040938759501523706266616156381970418293672253513608493234939740413449863336806748362}{55203909301187356764850778511665189264303648494138209029836811635708717648360173387982974041} a^{3} - \frac{6204611457191275943687350046957923743660182310622793319454908003994738325489646932121442749}{55203909301187356764850778511665189264303648494138209029836811635708717648360173387982974041} a^{2} - \frac{10294554160945100286042589477211916217914583463028652593310273939934726090806841428812350593}{55203909301187356764850778511665189264303648494138209029836811635708717648360173387982974041} a - \frac{21761310254144959005922048585218205785553685882244229160102319734997127862717930979598515153}{55203909301187356764850778511665189264303648494138209029836811635708717648360173387982974041}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 47107157976503970 \) (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

3.3.312481.1, 7.7.6321363049.1

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 $21$ $21$ $21$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{7}$ $21$ R $21$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{3}$ $21$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ $21$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ $21$ R ${\href{/LocalNumberField/47.7.0.1}{7} }^{3}$ $21$ $21$

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
13Data not computed
43Data not computed