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Properties

Label 21.1.13061378461...4416.1
Degree $21$
Signature $[1, 10]$
Discriminant $2^{14}\cdot 19^{10}\cdot 113^{6}\cdot 8443^{6}\cdot 16073^{6}$
Root discriminant $5245.77$
Ramified primes $2, 19, 113, 8443, 16073$
Class number Not computed
Class group Not computed
Galois group 21T47

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1184058275409918144914281, -912333939839097504347263, 283467196064582172225401, -33863314750260563653093, -2263522908572035410110, 1071605630606569600064, -81390216175476678839, -9084224812998996279, 2520008448643564047, -274492120954513017, 20213240958670760, -1247533795041060, 71679958343819, -3705859606469, 155406985707, -4635953607, 80623634, -543456, 7807, -621, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 3*x^20 - 621*x^19 + 7807*x^18 - 543456*x^17 + 80623634*x^16 - 4635953607*x^15 + 155406985707*x^14 - 3705859606469*x^13 + 71679958343819*x^12 - 1247533795041060*x^11 + 20213240958670760*x^10 - 274492120954513017*x^9 + 2520008448643564047*x^8 - 9084224812998996279*x^7 - 81390216175476678839*x^6 + 1071605630606569600064*x^5 - 2263522908572035410110*x^4 - 33863314750260563653093*x^3 + 283467196064582172225401*x^2 - 912333939839097504347263*x + 1184058275409918144914281)
 
gp: K = bnfinit(x^21 - 3*x^20 - 621*x^19 + 7807*x^18 - 543456*x^17 + 80623634*x^16 - 4635953607*x^15 + 155406985707*x^14 - 3705859606469*x^13 + 71679958343819*x^12 - 1247533795041060*x^11 + 20213240958670760*x^10 - 274492120954513017*x^9 + 2520008448643564047*x^8 - 9084224812998996279*x^7 - 81390216175476678839*x^6 + 1071605630606569600064*x^5 - 2263522908572035410110*x^4 - 33863314750260563653093*x^3 + 283467196064582172225401*x^2 - 912333939839097504347263*x + 1184058275409918144914281, 1)
 

Normalized defining polynomial

\( x^{21} - 3 x^{20} - 621 x^{19} + 7807 x^{18} - 543456 x^{17} + 80623634 x^{16} - 4635953607 x^{15} + 155406985707 x^{14} - 3705859606469 x^{13} + 71679958343819 x^{12} - 1247533795041060 x^{11} + 20213240958670760 x^{10} - 274492120954513017 x^{9} + 2520008448643564047 x^{8} - 9084224812998996279 x^{7} - 81390216175476678839 x^{6} + 1071605630606569600064 x^{5} - 2263522908572035410110 x^{4} - 33863314750260563653093 x^{3} + 283467196064582172225401 x^{2} - 912333939839097504347263 x + 1184058275409918144914281 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1306137846196771424321284975759299211270747610069719686230406537457249361084416=2^{14}\cdot 19^{10}\cdot 113^{6}\cdot 8443^{6}\cdot 16073^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $5245.77$
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:  $2, 19, 113, 8443, 16073$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{12} - \frac{1}{2}$, $\frac{1}{38} a^{13} - \frac{3}{19} a^{12} + \frac{7}{38} a^{11} - \frac{5}{38} a^{10} + \frac{3}{38} a^{9} - \frac{15}{38} a^{8} + \frac{13}{38} a^{7} + \frac{1}{38} a^{6} - \frac{13}{38} a^{5} - \frac{13}{38} a^{4} + \frac{15}{38} a^{3} - \frac{5}{38} a^{2} + \frac{3}{38}$, $\frac{1}{38} a^{14} + \frac{9}{38} a^{12} - \frac{1}{38} a^{11} - \frac{4}{19} a^{10} + \frac{3}{38} a^{9} - \frac{1}{38} a^{8} - \frac{8}{19} a^{7} - \frac{7}{38} a^{6} - \frac{15}{38} a^{5} - \frac{3}{19} a^{4} + \frac{9}{38} a^{3} + \frac{4}{19} a^{2} - \frac{8}{19} a + \frac{9}{19}$, $\frac{1}{76} a^{15} + \frac{15}{76} a^{12} - \frac{7}{38} a^{11} + \frac{5}{38} a^{10} + \frac{5}{38} a^{9} - \frac{7}{38} a^{8} - \frac{5}{38} a^{7} - \frac{6}{19} a^{6} - \frac{11}{38} a^{5} - \frac{13}{38} a^{4} + \frac{25}{76} a^{3} + \frac{5}{38} a^{2} + \frac{9}{38} a - \frac{27}{76}$, $\frac{1}{76} a^{16} - \frac{1}{76} a^{13} + \frac{3}{38} a^{12} + \frac{3}{19} a^{11} + \frac{7}{38} a^{10} + \frac{7}{38} a^{9} - \frac{9}{19} a^{8} - \frac{1}{19} a^{7} - \frac{1}{2} a^{6} - \frac{2}{19} a^{5} + \frac{5}{76} a^{4} - \frac{1}{38} a^{3} - \frac{4}{19} a^{2} - \frac{27}{76} a + \frac{7}{19}$, $\frac{1}{1444} a^{17} + \frac{5}{1444} a^{16} + \frac{3}{1444} a^{15} - \frac{9}{1444} a^{14} + \frac{1}{1444} a^{13} - \frac{289}{1444} a^{12} + \frac{10}{361} a^{11} + \frac{89}{722} a^{10} + \frac{29}{361} a^{9} - \frac{299}{722} a^{8} + \frac{48}{361} a^{7} - \frac{335}{722} a^{6} - \frac{27}{76} a^{5} - \frac{311}{1444} a^{4} - \frac{479}{1444} a^{3} + \frac{543}{1444} a^{2} - \frac{153}{1444} a + \frac{675}{1444}$, $\frac{1}{1444} a^{18} - \frac{3}{1444} a^{16} - \frac{5}{1444} a^{15} + \frac{2}{361} a^{14} - \frac{9}{1444} a^{13} - \frac{141}{722} a^{12} - \frac{15}{361} a^{11} - \frac{45}{722} a^{10} + \frac{1}{19} a^{9} - \frac{31}{361} a^{8} - \frac{56}{361} a^{7} - \frac{659}{1444} a^{6} + \frac{117}{361} a^{5} - \frac{159}{1444} a^{4} + \frac{373}{1444} a^{3} + \frac{62}{361} a^{2} + \frac{433}{1444} a - \frac{60}{361}$, $\frac{1}{2888} a^{19} - \frac{1}{2888} a^{18} - \frac{3}{1444} a^{16} - \frac{2}{361} a^{15} + \frac{4}{361} a^{14} - \frac{23}{2888} a^{13} + \frac{153}{2888} a^{12} - \frac{63}{722} a^{11} + \frac{59}{361} a^{10} - \frac{77}{722} a^{9} - \frac{141}{722} a^{8} + \frac{179}{2888} a^{7} - \frac{275}{2888} a^{6} - \frac{3}{38} a^{5} + \frac{341}{1444} a^{4} - \frac{67}{361} a^{3} + \frac{121}{722} a^{2} + \frac{1091}{2888} a - \frac{965}{2888}$, $\frac{1}{64522509990976758857694962469857111905753902627677195181180666817432154615608602773285773825767107326984695918352111844363769041656402275894391768125609643940248818613776042409671853981150454970107127259218118499272} a^{20} + \frac{3432333850365319943256930641899929076600962249130202275107114983796338979575282741001325182447875435391111859431035490850134856979836922521090470865410466937860468869265816847605764107866235570781970985082090195}{32261254995488379428847481234928555952876951313838597590590333408716077307804301386642886912883553663492347959176055922181884520828201137947195884062804821970124409306888021204835926990575227485053563629609059249636} a^{19} - \frac{519168611254982139949410829696344062012137240442882044855639245238087370260876339737648241592537743254650259111695247816752376498028447380940110623891245019127388667194887319142358084975664107983935940740055805}{64522509990976758857694962469857111905753902627677195181180666817432154615608602773285773825767107326984695918352111844363769041656402275894391768125609643940248818613776042409671853981150454970107127259218118499272} a^{18} - \frac{6885257031124779040853415015582119656928349018450827111561247271817092440857350042563592318975337036376562177814505802465869284108419649300280597021588528296797381332965749715099220469229740731546096020252702695}{32261254995488379428847481234928555952876951313838597590590333408716077307804301386642886912883553663492347959176055922181884520828201137947195884062804821970124409306888021204835926990575227485053563629609059249636} a^{17} + \frac{154280990843872115186362432311136416770893109859040016896594544127674437069048467119658986090810656356185833073050005047130466478609104476618629694440897049581585745037724393301514425016609758511789683963597826375}{32261254995488379428847481234928555952876951313838597590590333408716077307804301386642886912883553663492347959176055922181884520828201137947195884062804821970124409306888021204835926990575227485053563629609059249636} a^{16} + \frac{1008556257535180804808541750218201787012554272891056954164657141236971675587092604469743511853532902608466267428999411853476154014111217801014950006528530856015419741152220716574116967228711462217364227134620603}{393429938969370480839603429694250682352157942851690214519394309862391186680540260812718133083945776384053023892390925880266884400343916316429218098326888072806395235449853917132145451104575944939677605239134868898} a^{15} + \frac{757382564944260609034147944838614853792351661952646199072985513628897711539450525208198437120337413136665274138698046980945245355063473556366794516264291352672724246876591174960278923626510553890804715159255186257}{64522509990976758857694962469857111905753902627677195181180666817432154615608602773285773825767107326984695918352111844363769041656402275894391768125609643940248818613776042409671853981150454970107127259218118499272} a^{14} - \frac{197198629607330943629722166125376517593374131300074651617223150256001643055745977230555529002544229111805670275792664770581397557029890095447576389856128875459458937156622571953343295384494170197291203102761602535}{16130627497744189714423740617464277976438475656919298795295166704358038653902150693321443456441776831746173979588027961090942260414100568973597942031402410985062204653444010602417963495287613742526781814804529624818} a^{13} + \frac{10738246026187917028846437206427776429603608291347039474477064044336196047861581515474111509054546799351710333996727839521449968058093253255886247213926446436895194158360761400635962265288031973943867399981268386193}{64522509990976758857694962469857111905753902627677195181180666817432154615608602773285773825767107326984695918352111844363769041656402275894391768125609643940248818613776042409671853981150454970107127259218118499272} a^{12} - \frac{290674206026177002831425529150835377560894387047091910953368693430256715605068732743831208440365463290176286335106653522970093988279675928266374310439619547340688687997885899976820092325674911038589295000617894358}{8065313748872094857211870308732138988219237828459649397647583352179019326951075346660721728220888415873086989794013980545471130207050284486798971015701205492531102326722005301208981747643806871263390907402264812409} a^{11} - \frac{829786191762595583867810739045792694400951516656695667206099682727479518942438416882838850274889739515328804781739964842695647802749365005834578685289435742017274926409780728208598130640676505942984735153372302058}{8065313748872094857211870308732138988219237828459649397647583352179019326951075346660721728220888415873086989794013980545471130207050284486798971015701205492531102326722005301208981747643806871263390907402264812409} a^{10} - \frac{1547260517129463469453085800750881964843267583595982397808283351132166041366580257497208966608061259037326998055919860689608595050425759063721330604692159641256665693199694636192400108834241780545692922375660370772}{8065313748872094857211870308732138988219237828459649397647583352179019326951075346660721728220888415873086989794013980545471130207050284486798971015701205492531102326722005301208981747643806871263390907402264812409} a^{9} - \frac{15022521567273321266715275934235671794921094070429254207713876941792917327289373469738577879728014131693156551582689005862521714012781810393589217728582356178981562087941701169170865968695502796836003708816131815605}{64522509990976758857694962469857111905753902627677195181180666817432154615608602773285773825767107326984695918352111844363769041656402275894391768125609643940248818613776042409671853981150454970107127259218118499272} a^{8} - \frac{1233178153958055815764428073803632861298803458435193820402053268560182482734437905521354287993535392459465472586322209868089109491849903819145591879619817223601153931909161679361907447301980762135572443470840558825}{32261254995488379428847481234928555952876951313838597590590333408716077307804301386642886912883553663492347959176055922181884520828201137947195884062804821970124409306888021204835926990575227485053563629609059249636} a^{7} + \frac{11988320075481639643704132150300226033528241738494049809356021023480507513844828034658561207125532039741863082885652618696515074330128121167124051875609659352615206587494780743690235147452860132445799123605820672405}{64522509990976758857694962469857111905753902627677195181180666817432154615608602773285773825767107326984695918352111844363769041656402275894391768125609643940248818613776042409671853981150454970107127259218118499272} a^{6} + \frac{12652653459326727610580552465704731308778462492827146033684194044907572103868027939210996721420368158959749776785163912430798118675052096615928376000339194319484897247476880152735194810261600483382822230303153403701}{32261254995488379428847481234928555952876951313838597590590333408716077307804301386642886912883553663492347959176055922181884520828201137947195884062804821970124409306888021204835926990575227485053563629609059249636} a^{5} + \frac{5728236432023763447383050898926917660212605444562987152995825324614129246401729894173354118415074064641948363321342895418914960059057654480890696342927565311619488634963725689756643069619022524494255570504731627809}{32261254995488379428847481234928555952876951313838597590590333408716077307804301386642886912883553663492347959176055922181884520828201137947195884062804821970124409306888021204835926990575227485053563629609059249636} a^{4} + \frac{552371316321905071469768985867565727669546059489804376293667128695023398396894156376380004606567623953287835277230503272771905949794953364938652875897832029478864044286389908879890604807122654772914061865037420832}{8065313748872094857211870308732138988219237828459649397647583352179019326951075346660721728220888415873086989794013980545471130207050284486798971015701205492531102326722005301208981747643806871263390907402264812409} a^{3} + \frac{4335270678732316737920873755200234729377375585869134650071558090948719329871528892647812927896395844779022043719120480047180627187751731611187978858444933329658603317315519762265881788409074985965174993143469497115}{64522509990976758857694962469857111905753902627677195181180666817432154615608602773285773825767107326984695918352111844363769041656402275894391768125609643940248818613776042409671853981150454970107127259218118499272} a^{2} - \frac{3099110644079496565393095368901431501522980257357616612024230744845639965936266377438456953874111421971869638053426306903198059870651900875316963111203403637592292064288075783453666266802951166609432279072563404408}{8065313748872094857211870308732138988219237828459649397647583352179019326951075346660721728220888415873086989794013980545471130207050284486798971015701205492531102326722005301208981747643806871263390907402264812409} a - \frac{9295943106774925609879526761552357028483449603343752463419892867139663586076608379194254944894649593801519998394386489237038790341583905197089068640688084378658122882153557226873511773825393101421705762875899594417}{64522509990976758857694962469857111905753902627677195181180666817432154615608602773285773825767107326984695918352111844363769041656402275894391768125609643940248818613776042409671853981150454970107127259218118499272}$

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

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $10$
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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

21T47:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 8232
The 43 conjugacy class representatives for t21n47
Character table for t21n47 is not computed

Intermediate fields

3.1.76.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 28 sibling: data not computed
Degree 42 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.7.0.1}{7} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ $21$ $21$ $21$ ${\href{/LocalNumberField/13.14.0.1}{14} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ $21$ R ${\href{/LocalNumberField/23.7.0.1}{7} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/31.14.0.1}{14} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{7}$ $21$ $21$ ${\href{/LocalNumberField/53.14.0.1}{14} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{7}$

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
$2$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.3.1$x^{4} + 76$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
19.4.3.1$x^{4} + 76$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
19.4.3.1$x^{4} + 76$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$113$113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.4.0.1$x^{4} - x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
113.4.0.1$x^{4} - x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
113.4.0.1$x^{4} - x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
113.7.6.3$x^{7} - 1017$$7$$1$$6$$C_7$$[\ ]_{7}$
8443Data not computed
16073Data not computed