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Properties

Label 21.21.1838975925...5289.1
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![-1092874069, -399661124, 12032834277, 2645891174, -30297922744, 9213729365, 15440417206, -5827035514, -3526931287, 1426797465, 434797030, -184175217, -30630118, 13705276, 1228287, -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 + 1228287*x^14 + 13705276*x^13 - 30630118*x^12 - 184175217*x^11 + 434797030*x^10 + 1426797465*x^9 - 3526931287*x^8 - 5827035514*x^7 + 15440417206*x^6 + 9213729365*x^5 - 30297922744*x^4 + 2645891174*x^3 + 12032834277*x^2 - 399661124*x - 1092874069)
 
gp: K = bnfinit(x^21 - x^20 - 192*x^19 + 277*x^18 + 14834*x^17 - 26598*x^16 - 597785*x^15 + 1228287*x^14 + 13705276*x^13 - 30630118*x^12 - 184175217*x^11 + 434797030*x^10 + 1426797465*x^9 - 3526931287*x^8 - 5827035514*x^7 + 15440417206*x^6 + 9213729365*x^5 - 30297922744*x^4 + 2645891174*x^3 + 12032834277*x^2 - 399661124*x - 1092874069, 1)
 

Normalized defining polynomial

\( x^{21} - x^{20} - 192 x^{19} + 277 x^{18} + 14834 x^{17} - 26598 x^{16} - 597785 x^{15} + 1228287 x^{14} + 13705276 x^{13} - 30630118 x^{12} - 184175217 x^{11} + 434797030 x^{10} + 1426797465 x^{9} - 3526931287 x^{8} - 5827035514 x^{7} + 15440417206 x^{6} + 9213729365 x^{5} - 30297922744 x^{4} + 2645891174 x^{3} + 12032834277 x^{2} - 399661124 x - 1092874069 \)

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$$\chi_{559}(1,·)$, $\chi_{559}(391,·)$, $\chi_{559}(9,·)$, $\chi_{559}(183,·)$, $\chi_{559}(81,·)$, $\chi_{559}(274,·)$, $\chi_{559}(341,·)$, $\chi_{559}(412,·)$, $\chi_{559}(289,·)$, $\chi_{559}(354,·)$, $\chi_{559}(100,·)$, $\chi_{559}(165,·)$, $\chi_{559}(230,·)$, $\chi_{559}(529,·)$, $\chi_{559}(170,·)$, $\chi_{559}(365,·)$, $\chi_{559}(367,·)$, $\chi_{559}(497,·)$, $\chi_{559}(393,·)$, $\chi_{559}(508,·)$, $\chi_{559}(490,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{3}$, $\frac{1}{7} a^{10} - \frac{1}{7} a^{4}$, $\frac{1}{7} a^{11} - \frac{1}{7} a^{5}$, $\frac{1}{7} a^{12} - \frac{1}{7} a^{6}$, $\frac{1}{49} a^{13} - \frac{2}{49} a^{7} + \frac{2}{7} a^{6} + \frac{1}{49} a - \frac{2}{7}$, $\frac{1}{49} a^{14} - \frac{2}{49} a^{8} + \frac{1}{49} a^{2}$, $\frac{1}{49} a^{15} - \frac{2}{49} a^{9} + \frac{1}{49} a^{3}$, $\frac{1}{343} a^{16} + \frac{2}{343} a^{15} - \frac{1}{343} a^{14} - \frac{2}{343} a^{13} + \frac{3}{49} a^{12} + \frac{1}{49} a^{11} + \frac{12}{343} a^{10} + \frac{17}{343} a^{9} - \frac{5}{343} a^{8} - \frac{24}{343} a^{7} + \frac{3}{7} a^{6} - \frac{22}{49} a^{5} - \frac{62}{343} a^{4} + \frac{30}{343} a^{3} - \frac{92}{343} a^{2} + \frac{124}{343} a - \frac{3}{49}$, $\frac{1}{343} a^{17} + \frac{2}{343} a^{15} - \frac{3}{343} a^{13} + \frac{2}{49} a^{12} - \frac{2}{343} a^{11} - \frac{1}{49} a^{10} - \frac{4}{343} a^{9} - \frac{2}{49} a^{8} + \frac{6}{343} a^{7} + \frac{20}{49} a^{6} - \frac{97}{343} a^{5} + \frac{22}{49} a^{4} + \frac{149}{343} a^{3} - \frac{5}{49} a^{2} - \frac{52}{343} a + \frac{13}{49}$, $\frac{1}{12691} a^{18} - \frac{2}{1813} a^{17} - \frac{15}{12691} a^{16} - \frac{13}{12691} a^{15} + \frac{8}{1813} a^{14} - \frac{29}{12691} a^{13} - \frac{212}{12691} a^{12} - \frac{2}{259} a^{11} + \frac{331}{12691} a^{10} + \frac{145}{12691} a^{9} + \frac{78}{1813} a^{8} + \frac{653}{12691} a^{7} - \frac{391}{12691} a^{6} + \frac{100}{1813} a^{5} + \frac{2722}{12691} a^{4} - \frac{4101}{12691} a^{3} - \frac{2}{1813} a^{2} + \frac{2414}{12691} a - \frac{13}{49}$, $\frac{1}{88837} a^{19} + \frac{3}{88837} a^{18} + \frac{6}{88837} a^{17} - \frac{9}{88837} a^{16} - \frac{165}{88837} a^{15} - \frac{372}{88837} a^{14} - \frac{446}{88837} a^{13} + \frac{5363}{88837} a^{12} - \frac{3666}{88837} a^{11} - \frac{5}{2401} a^{10} - \frac{4241}{88837} a^{9} - \frac{1979}{88837} a^{8} - \frac{125}{1813} a^{7} - \frac{11386}{88837} a^{6} - \frac{34070}{88837} a^{5} + \frac{22489}{88837} a^{4} - \frac{24406}{88837} a^{3} - \frac{22688}{88837} a^{2} + \frac{3483}{88837} a + \frac{108}{343}$, $\frac{1}{3497583302298430942417292450860678584222586365831221245435606176314467} a^{20} + \frac{6431960815037217754393207212338266233326419133311782330228569041}{3497583302298430942417292450860678584222586365831221245435606176314467} a^{19} + \frac{59354874940178783245671628059669211814452411215931012234219779523}{3497583302298430942417292450860678584222586365831221245435606176314467} a^{18} + \frac{4225200127810174736197122730607868995702842035749804229665057718628}{3497583302298430942417292450860678584222586365831221245435606176314467} a^{17} - \frac{4706724953568178164736226949160984850789350665719363923854192769947}{3497583302298430942417292450860678584222586365831221245435606176314467} a^{16} + \frac{5896143898350309960609799509209194534805767880277882422893922450036}{3497583302298430942417292450860678584222586365831221245435606176314467} a^{15} + \frac{7675172645817416524669767316846232036293473123113013968040629425136}{3497583302298430942417292450860678584222586365831221245435606176314467} a^{14} - \frac{15476204600474842612356746596838978789063534769080651512242533592556}{3497583302298430942417292450860678584222586365831221245435606176314467} a^{13} + \frac{226297525067597544025156925379284709916679939935345804849979055702580}{3497583302298430942417292450860678584222586365831221245435606176314467} a^{12} + \frac{206848017814102729958637203553910682538824436673489704465611807386260}{3497583302298430942417292450860678584222586365831221245435606176314467} a^{11} - \frac{214111093762651368993798715010928173465285419582765596466420887622696}{3497583302298430942417292450860678584222586365831221245435606176314467} a^{10} + \frac{163028833080145408309159583567534864033714459974387325225375479159266}{3497583302298430942417292450860678584222586365831221245435606176314467} a^{9} + \frac{26347641425081385176653608033469435597223399244826757770003485586642}{3497583302298430942417292450860678584222586365831221245435606176314467} a^{8} - \frac{165215831849300586743396158032538023296119532816078543816421085690579}{3497583302298430942417292450860678584222586365831221245435606176314467} a^{7} + \frac{278875507446774216493396401989775096941723261835518305736135384367627}{3497583302298430942417292450860678584222586365831221245435606176314467} a^{6} - \frac{154194842552550352806226268469082891068349705221813545654176004010580}{3497583302298430942417292450860678584222586365831221245435606176314467} a^{5} + \frac{1218039595452688325129122977399125975292684784289225946987652653979237}{3497583302298430942417292450860678584222586365831221245435606176314467} a^{4} + \frac{1189532085781434279034005236129639614290226961261313126275822635134375}{3497583302298430942417292450860678584222586365831221245435606176314467} a^{3} + \frac{1235348843558959138707559171113224090805450490864871440498255697934152}{3497583302298430942417292450860678584222586365831221245435606176314467} a^{2} + \frac{205851150460886648678178390214171695515726505948653023449886739370157}{3497583302298430942417292450860678584222586365831221245435606176314467} a + \frac{22366261766680967375033581832850870824753588262883936453823634303}{13504182634356876225549391702164782178465584424058769287396162842913}$

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:  \( 1940642799270161000 \) (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

3.3.312481.2, 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 $21$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{3}$ $21$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{21}$ $21$ R ${\href{/LocalNumberField/17.7.0.1}{7} }^{3}$ $21$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{3}$ $21$ $21$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{21}$ $21$ 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