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Properties

Label 21.21.2018285563...0529.1
Degree $21$
Signature $[21, 0]$
Discriminant $19^{14}\cdot 43^{18}$
Root discriminant $178.90$
Ramified primes $19, 43$
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![12691, 122402, 73346, -1493691, -2093106, 4936501, 7111669, -7230861, -8919502, 5496701, 4808330, -2281327, -1226011, 503792, 153364, -56698, -9934, 3272, 320, -92, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 4*x^20 - 92*x^19 + 320*x^18 + 3272*x^17 - 9934*x^16 - 56698*x^15 + 153364*x^14 + 503792*x^13 - 1226011*x^12 - 2281327*x^11 + 4808330*x^10 + 5496701*x^9 - 8919502*x^8 - 7230861*x^7 + 7111669*x^6 + 4936501*x^5 - 2093106*x^4 - 1493691*x^3 + 73346*x^2 + 122402*x + 12691)
 
gp: K = bnfinit(x^21 - 4*x^20 - 92*x^19 + 320*x^18 + 3272*x^17 - 9934*x^16 - 56698*x^15 + 153364*x^14 + 503792*x^13 - 1226011*x^12 - 2281327*x^11 + 4808330*x^10 + 5496701*x^9 - 8919502*x^8 - 7230861*x^7 + 7111669*x^6 + 4936501*x^5 - 2093106*x^4 - 1493691*x^3 + 73346*x^2 + 122402*x + 12691, 1)
 

Normalized defining polynomial

\( x^{21} - 4 x^{20} - 92 x^{19} + 320 x^{18} + 3272 x^{17} - 9934 x^{16} - 56698 x^{15} + 153364 x^{14} + 503792 x^{13} - 1226011 x^{12} - 2281327 x^{11} + 4808330 x^{10} + 5496701 x^{9} - 8919502 x^{8} - 7230861 x^{7} + 7111669 x^{6} + 4936501 x^{5} - 2093106 x^{4} - 1493691 x^{3} + 73346 x^{2} + 122402 x + 12691 \)

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:  \(201828556349525896653055169768510327187913320529=19^{14}\cdot 43^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $178.90$
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:  $19, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(817=19\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{817}(704,·)$, $\chi_{817}(1,·)$, $\chi_{817}(514,·)$, $\chi_{817}(391,·)$, $\chi_{817}(520,·)$, $\chi_{817}(11,·)$, $\chi_{817}(140,·)$, $\chi_{817}(64,·)$, $\chi_{817}(216,·)$, $\chi_{817}(723,·)$, $\chi_{817}(790,·)$, $\chi_{817}(87,·)$, $\chi_{817}(600,·)$, $\chi_{817}(729,·)$, $\chi_{817}(666,·)$, $\chi_{817}(742,·)$, $\chi_{817}(102,·)$, $\chi_{817}(809,·)$, $\chi_{817}(752,·)$, $\chi_{817}(305,·)$, $\chi_{817}(121,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{3}$, $\frac{1}{7} a^{10} - \frac{1}{7} a^{4}$, $\frac{1}{7} a^{11} - \frac{1}{7} a^{5}$, $\frac{1}{7} a^{12} - \frac{1}{7} a^{6}$, $\frac{1}{7} a^{13} - \frac{1}{7} a$, $\frac{1}{49} a^{14} - \frac{2}{49} a^{8} + \frac{1}{49} a^{2}$, $\frac{1}{49} a^{15} - \frac{2}{49} a^{9} + \frac{1}{49} a^{3}$, $\frac{1}{49} a^{16} - \frac{2}{49} a^{10} + \frac{1}{49} a^{4}$, $\frac{1}{49} a^{17} - \frac{2}{49} a^{11} + \frac{1}{49} a^{5}$, $\frac{1}{49} a^{18} - \frac{2}{49} a^{12} + \frac{1}{49} a^{6}$, $\frac{1}{12691} a^{19} + \frac{27}{12691} a^{18} + \frac{3}{1813} a^{17} + \frac{71}{12691} a^{16} + \frac{3}{343} a^{15} + \frac{17}{1813} a^{14} - \frac{247}{12691} a^{13} + \frac{758}{12691} a^{12} + \frac{123}{1813} a^{11} + \frac{397}{12691} a^{10} - \frac{390}{12691} a^{9} - \frac{17}{1813} a^{8} - \frac{587}{12691} a^{7} + \frac{1077}{12691} a^{6} + \frac{18}{259} a^{5} - \frac{2036}{12691} a^{4} + \frac{426}{12691} a^{3} - \frac{6}{259} a^{2} - \frac{22}{259} a$, $\frac{1}{98195636016228310457646476864726073927630859} a^{20} + \frac{565432352110734388328705713147333489774}{98195636016228310457646476864726073927630859} a^{19} + \frac{412844832577986171967747826768937829953042}{98195636016228310457646476864726073927630859} a^{18} - \frac{798068004775347064862711543207986798047312}{98195636016228310457646476864726073927630859} a^{17} + \frac{687432011924703653373411001413767488896161}{98195636016228310457646476864726073927630859} a^{16} + \frac{964508758216345043195579470514832257720008}{98195636016228310457646476864726073927630859} a^{15} - \frac{895373220744696120053229096990389829843630}{98195636016228310457646476864726073927630859} a^{14} + \frac{5899745557969787235577867815259877923832546}{98195636016228310457646476864726073927630859} a^{13} - \frac{3697856318073619115379144536707301965018888}{98195636016228310457646476864726073927630859} a^{12} + \frac{3632711591108466855934307428789184450625390}{98195636016228310457646476864726073927630859} a^{11} - \frac{4419762333497889772213438357316100707913117}{98195636016228310457646476864726073927630859} a^{10} - \frac{5250739248185295603127450733753338707604057}{98195636016228310457646476864726073927630859} a^{9} + \frac{5083861958068680349786862567820193870288918}{98195636016228310457646476864726073927630859} a^{8} + \frac{1043943245658743409086005316439259497670707}{98195636016228310457646476864726073927630859} a^{7} - \frac{2573164863401372638691240529469676669901688}{98195636016228310457646476864726073927630859} a^{6} - \frac{10844632770750183322647404080063769306464415}{98195636016228310457646476864726073927630859} a^{5} - \frac{4049430074245513044886389471486519930167114}{98195636016228310457646476864726073927630859} a^{4} + \frac{17782376386464454384172832009806378029099629}{98195636016228310457646476864726073927630859} a^{3} + \frac{4942536303305987033631026313247235655866067}{14027948002318330065378068123532296275375837} a^{2} + \frac{751696755948520850992327707118749305202548}{2003992571759761437911152589076042325053691} a - \frac{1636930663114712762905543924525559386736}{7737423056987495899270859417282016699049}$

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:  \( 1907344761740974800 \) (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

3.3.361.1, 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 $21$ $21$ $21$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{21}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{3}$ $21$ $21$ R $21$ $21$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{21}$ $21$ R $21$ $21$ $21$

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
19Data not computed
43Data not computed