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Properties

Label 21.21.3261504404...9329.2
Degree $21$
Signature $[21, 0]$
Discriminant $3^{28}\cdot 7^{14}\cdot 71^{18}$
Root discriminant $611.44$
Ramified primes $3, 7, 71$
Class number $21$ (GRH)
Class group $[21]$ (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![-20341937701, 58460165631, 80675421420, -175186581586, -28704806013, 113556752970, -481181304, -33101161890, 1384500363, 5345836461, -210631296, -518275467, 11716858, 30750441, -107259, -1094568, -11901, 22227, 373, -234, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 3*x^20 - 234*x^19 + 373*x^18 + 22227*x^17 - 11901*x^16 - 1094568*x^15 - 107259*x^14 + 30750441*x^13 + 11716858*x^12 - 518275467*x^11 - 210631296*x^10 + 5345836461*x^9 + 1384500363*x^8 - 33101161890*x^7 - 481181304*x^6 + 113556752970*x^5 - 28704806013*x^4 - 175186581586*x^3 + 80675421420*x^2 + 58460165631*x - 20341937701)
 
gp: K = bnfinit(x^21 - 3*x^20 - 234*x^19 + 373*x^18 + 22227*x^17 - 11901*x^16 - 1094568*x^15 - 107259*x^14 + 30750441*x^13 + 11716858*x^12 - 518275467*x^11 - 210631296*x^10 + 5345836461*x^9 + 1384500363*x^8 - 33101161890*x^7 - 481181304*x^6 + 113556752970*x^5 - 28704806013*x^4 - 175186581586*x^3 + 80675421420*x^2 + 58460165631*x - 20341937701, 1)
 

Normalized defining polynomial

\( x^{21} - 3 x^{20} - 234 x^{19} + 373 x^{18} + 22227 x^{17} - 11901 x^{16} - 1094568 x^{15} - 107259 x^{14} + 30750441 x^{13} + 11716858 x^{12} - 518275467 x^{11} - 210631296 x^{10} + 5345836461 x^{9} + 1384500363 x^{8} - 33101161890 x^{7} - 481181304 x^{6} + 113556752970 x^{5} - 28704806013 x^{4} - 175186581586 x^{3} + 80675421420 x^{2} + 58460165631 x - 20341937701 \)

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:  \(32615044040601465591789077347406642010407599407890505169329=3^{28}\cdot 7^{14}\cdot 71^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $611.44$
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:  $3, 7, 71$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4473=3^{2}\cdot 7\cdot 71\)
Dirichlet character group:    $\lbrace$$\chi_{4473}(3712,·)$, $\chi_{4473}(1,·)$, $\chi_{4473}(3019,·)$, $\chi_{4473}(2956,·)$, $\chi_{4473}(529,·)$, $\chi_{4473}(2515,·)$, $\chi_{4473}(214,·)$, $\chi_{4473}(2647,·)$, $\chi_{4473}(1954,·)$, $\chi_{4473}(1891,·)$, $\chi_{4473}(1381,·)$, $\chi_{4473}(1255,·)$, $\chi_{4473}(1066,·)$, $\chi_{4473}(2860,·)$, $\chi_{4473}(403,·)$, $\chi_{4473}(2167,·)$, $\chi_{4473}(2104,·)$, $\chi_{4473}(316,·)$, $\chi_{4473}(1450,·)$, $\chi_{4473}(190,·)$, $\chi_{4473}(1663,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{10} - \frac{2}{5} a^{9} - \frac{2}{5} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{11} - \frac{2}{5} a^{10} - \frac{2}{5} a^{9} + \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a$, $\frac{1}{10} a^{14} - \frac{1}{10} a^{13} - \frac{1}{10} a^{12} - \frac{3}{10} a^{11} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{1}{10} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{3}{10} a^{4} + \frac{3}{10} a^{3} + \frac{1}{5} a^{2} + \frac{3}{10} a + \frac{1}{10}$, $\frac{1}{10} a^{15} - \frac{3}{10} a^{11} - \frac{1}{2} a^{8} + \frac{3}{10} a^{7} - \frac{3}{10} a^{5} - \frac{1}{10} a^{3} - \frac{1}{2} a^{2} - \frac{1}{5} a - \frac{1}{10}$, $\frac{1}{10} a^{16} - \frac{1}{10} a^{12} + \frac{1}{5} a^{10} + \frac{1}{10} a^{9} - \frac{1}{10} a^{8} + \frac{1}{5} a^{7} + \frac{1}{10} a^{6} - \frac{1}{2} a^{4} - \frac{3}{10} a^{3} - \frac{1}{5} a^{2} - \frac{1}{10} a + \frac{2}{5}$, $\frac{1}{10} a^{17} - \frac{1}{10} a^{13} + \frac{1}{5} a^{11} + \frac{1}{10} a^{10} - \frac{1}{10} a^{9} + \frac{1}{5} a^{8} + \frac{1}{10} a^{7} - \frac{1}{2} a^{5} - \frac{3}{10} a^{4} - \frac{1}{5} a^{3} - \frac{1}{10} a^{2} + \frac{2}{5} a$, $\frac{1}{850} a^{18} - \frac{31}{850} a^{17} - \frac{3}{850} a^{16} + \frac{1}{425} a^{15} + \frac{9}{850} a^{14} - \frac{81}{850} a^{13} + \frac{53}{850} a^{12} - \frac{29}{850} a^{11} + \frac{152}{425} a^{10} - \frac{41}{425} a^{9} - \frac{178}{425} a^{8} - \frac{197}{850} a^{7} - \frac{151}{425} a^{6} + \frac{7}{17} a^{5} + \frac{129}{425} a^{4} - \frac{167}{425} a^{3} + \frac{211}{850} a^{2} + \frac{371}{850} a - \frac{59}{425}$, $\frac{1}{850} a^{19} - \frac{29}{850} a^{17} - \frac{3}{425} a^{16} - \frac{7}{425} a^{15} + \frac{14}{425} a^{14} + \frac{7}{850} a^{13} - \frac{1}{850} a^{12} + \frac{1}{5} a^{11} + \frac{77}{850} a^{10} - \frac{174}{425} a^{9} + \frac{157}{850} a^{8} - \frac{7}{50} a^{7} - \frac{257}{850} a^{6} - \frac{141}{425} a^{5} + \frac{177}{425} a^{4} + \frac{57}{850} a^{3} - \frac{29}{425} a^{2} - \frac{46}{425} a - \frac{3}{850}$, $\frac{1}{925908012517598721336859988099319656431809683395680295135224198629725032498005970006950} a^{20} - \frac{295255596069147167759542003535460455698704377877171400749338401809725728152329892657}{925908012517598721336859988099319656431809683395680295135224198629725032498005970006950} a^{19} - \frac{56155550052349167554095470566204163975847902395326944636927012829232280940674374311}{185181602503519744267371997619863931286361936679136059027044839725945006499601194001390} a^{18} + \frac{5822963755438445473492390078549742284578626098432759302622123481946300757315025084503}{925908012517598721336859988099319656431809683395680295135224198629725032498005970006950} a^{17} - \frac{5502873209896205764747973305641004896188944306701125184067369331045238360409219626447}{462954006258799360668429994049659828215904841697840147567612099314862516249002985003475} a^{16} - \frac{17482313754898710383085535940247015259052200869262007045166093299117003958215416420128}{462954006258799360668429994049659828215904841697840147567612099314862516249002985003475} a^{15} + \frac{11742052465155190432495819368539230009457059754299197983321294565306419421222985097097}{925908012517598721336859988099319656431809683395680295135224198629725032498005970006950} a^{14} + \frac{17186894557639544682442017296276512282146626681481856285711544306601720057444076537508}{462954006258799360668429994049659828215904841697840147567612099314862516249002985003475} a^{13} - \frac{92286140229367444072451085071760115549734240422315628939450684452479107847828664014151}{925908012517598721336859988099319656431809683395680295135224198629725032498005970006950} a^{12} - \frac{165173227807010306307921790156205895712220870354071898680898557220047262277230160866899}{925908012517598721336859988099319656431809683395680295135224198629725032498005970006950} a^{11} + \frac{285494406837917159832521578695229311482252379417846378056058623722705927864634384097029}{925908012517598721336859988099319656431809683395680295135224198629725032498005970006950} a^{10} - \frac{70732010387687973141682971726441293628185123828620203094876804996227596218138517525477}{185181602503519744267371997619863931286361936679136059027044839725945006499601194001390} a^{9} - \frac{187414933283759679214018089360163950905844198102178278500861297242562292613536825286486}{462954006258799360668429994049659828215904841697840147567612099314862516249002985003475} a^{8} - \frac{5954970975915480534850462612568109255513344310636762990000247871672324956269905970146}{462954006258799360668429994049659828215904841697840147567612099314862516249002985003475} a^{7} - \frac{139776231085309094394928681368440597860205367172594562494303544788407832677787619515061}{925908012517598721336859988099319656431809683395680295135224198629725032498005970006950} a^{6} - \frac{3279106112073235841037807612687512831101803474691397993552983298685686364409295411166}{462954006258799360668429994049659828215904841697840147567612099314862516249002985003475} a^{5} + \frac{13031264425652226654151356841003886652851522437597889439824565674159527394240725008221}{925908012517598721336859988099319656431809683395680295135224198629725032498005970006950} a^{4} + \frac{354466615905570919896505381996369882811255948957255764864865989866810997029953874878087}{925908012517598721336859988099319656431809683395680295135224198629725032498005970006950} a^{3} + \frac{57122820752997909205139860400007596768207919771417200099248441194194623814138160417234}{462954006258799360668429994049659828215904841697840147567612099314862516249002985003475} a^{2} + \frac{1635806968435336379819578229529271521515812258761890199269117752771026547133186612763}{37036320500703948853474399523972786257272387335827211805408967945189001299920238800278} a + \frac{125760908851373906248541327827401311101000740270050202966902336620651273292783828184129}{925908012517598721336859988099319656431809683395680295135224198629725032498005970006950}$

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

Class group and class number

$C_{21}$, which has order $21$ (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:  \( 1039550977839431700000 \) (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

3.3.3969.1, 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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{3}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{7}$ R $21$ $21$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{7}$ $21$ $21$ $21$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{3}$ $21$ $21$ $21$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{3}$ $21$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{3}$

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
7Data 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}$