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Properties

Label 21.9.83450796621...3152.1
Degree $21$
Signature $[9, 6]$
Discriminant $2^{14}\cdot 3^{21}\cdot 37^{12}\cdot 59^{3}\cdot 109^{12}\cdot 10859^{3}$
Root discriminant $3695.62$
Ramified primes $2, 3, 37, 59, 109, 10859$
Class number Not computed
Class group Not computed
Galois group 21T138

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3724414958441731059478927509146752, -315068960394756661920290849696640, -44488899123058840085730493001568, 2440298711745375561122234921808, 266740346394447654507510010344, -4726551871709798292359565120, -792530253662409376623342416, -5646302360560715152043103, 945745018043095393911180, 16275661558142313035035, -582361703119054694040, -14213279062376569485, 203858863109865284, 6572542654341105, -41002370088498, -1795665131924, 4409577342, 291973068, -195536, -26208, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 26208*x^19 - 195536*x^18 + 291973068*x^17 + 4409577342*x^16 - 1795665131924*x^15 - 41002370088498*x^14 + 6572542654341105*x^13 + 203858863109865284*x^12 - 14213279062376569485*x^11 - 582361703119054694040*x^10 + 16275661558142313035035*x^9 + 945745018043095393911180*x^8 - 5646302360560715152043103*x^7 - 792530253662409376623342416*x^6 - 4726551871709798292359565120*x^5 + 266740346394447654507510010344*x^4 + 2440298711745375561122234921808*x^3 - 44488899123058840085730493001568*x^2 - 315068960394756661920290849696640*x + 3724414958441731059478927509146752)
 
gp: K = bnfinit(x^21 - 26208*x^19 - 195536*x^18 + 291973068*x^17 + 4409577342*x^16 - 1795665131924*x^15 - 41002370088498*x^14 + 6572542654341105*x^13 + 203858863109865284*x^12 - 14213279062376569485*x^11 - 582361703119054694040*x^10 + 16275661558142313035035*x^9 + 945745018043095393911180*x^8 - 5646302360560715152043103*x^7 - 792530253662409376623342416*x^6 - 4726551871709798292359565120*x^5 + 266740346394447654507510010344*x^4 + 2440298711745375561122234921808*x^3 - 44488899123058840085730493001568*x^2 - 315068960394756661920290849696640*x + 3724414958441731059478927509146752, 1)
 

Normalized defining polynomial

\( x^{21} - 26208 x^{19} - 195536 x^{18} + 291973068 x^{17} + 4409577342 x^{16} - 1795665131924 x^{15} - 41002370088498 x^{14} + 6572542654341105 x^{13} + 203858863109865284 x^{12} - 14213279062376569485 x^{11} - 582361703119054694040 x^{10} + 16275661558142313035035 x^{9} + 945745018043095393911180 x^{8} - 5646302360560715152043103 x^{7} - 792530253662409376623342416 x^{6} - 4726551871709798292359565120 x^{5} + 266740346394447654507510010344 x^{4} + 2440298711745375561122234921808 x^{3} - 44488899123058840085730493001568 x^{2} - 315068960394756661920290849696640 x + 3724414958441731059478927509146752 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[9, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(834507966212271820403374346664412679514757637054312831716289049278398513152=2^{14}\cdot 3^{21}\cdot 37^{12}\cdot 59^{3}\cdot 109^{12}\cdot 10859^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $3695.62$
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:  $2, 3, 37, 59, 109, 10859$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{4033} a^{6} - \frac{2010}{4033} a^{4} - \frac{1952}{4033} a^{3}$, $\frac{1}{4033} a^{7} - \frac{2010}{4033} a^{5} - \frac{1952}{4033} a^{4}$, $\frac{1}{4033} a^{8} - \frac{1952}{4033} a^{5} + \frac{966}{4033} a^{4} + \frac{589}{4033} a^{3}$, $\frac{1}{16265089} a^{9} - \frac{2010}{16265089} a^{7} - \frac{1952}{16265089} a^{6} - \frac{159}{4033} a^{5} + \frac{1130}{4033} a^{4} - \frac{765}{4033} a^{3}$, $\frac{1}{16265089} a^{10} - \frac{2010}{16265089} a^{8} - \frac{1952}{16265089} a^{7} + \frac{1130}{4033} a^{5} - \frac{1748}{4033} a^{4} + \frac{173}{4033} a^{3}$, $\frac{1}{16265089} a^{11} - \frac{1952}{16265089} a^{8} + \frac{966}{16265089} a^{7} + \frac{589}{16265089} a^{6} - \frac{251}{4033} a^{5} + \frac{1991}{4033} a^{4} - \frac{1121}{4033} a^{3}$, $\frac{1}{65597103937} a^{12} - \frac{2010}{65597103937} a^{10} - \frac{1952}{65597103937} a^{9} - \frac{159}{16265089} a^{8} + \frac{1130}{16265089} a^{7} - \frac{765}{16265089} a^{6} - \frac{1646}{4033} a^{5} + \frac{1794}{4033} a^{4} + \frac{1138}{4033} a^{3}$, $\frac{1}{65597103937} a^{13} - \frac{2010}{65597103937} a^{11} - \frac{1952}{65597103937} a^{10} + \frac{1130}{16265089} a^{8} - \frac{1748}{16265089} a^{7} + \frac{173}{16265089} a^{6} - \frac{792}{4033} a^{5} - \frac{514}{4033} a^{4} - \frac{419}{4033} a^{3}$, $\frac{1}{65597103937} a^{14} - \frac{1952}{65597103937} a^{11} + \frac{966}{65597103937} a^{10} + \frac{589}{65597103937} a^{9} - \frac{251}{16265089} a^{8} + \frac{1991}{16265089} a^{7} - \frac{1121}{16265089} a^{6} - \frac{3}{109} a^{5} + \frac{732}{4033} a^{4} - \frac{1884}{4033} a^{3}$, $\frac{1}{529106240355842} a^{15} - \frac{1005}{264553120177921} a^{13} - \frac{976}{264553120177921} a^{12} + \frac{1937}{65597103937} a^{11} + \frac{565}{65597103937} a^{10} + \frac{1634}{65597103937} a^{9} - \frac{1799}{16265089} a^{8} - \frac{3283}{32530178} a^{7} + \frac{1904}{16265089} a^{6} - \frac{1511}{8066} a^{5} - \frac{1222}{4033} a^{4} + \frac{1087}{8066} a^{3} - \frac{1}{2} a$, $\frac{1}{1058212480711684} a^{16} + \frac{1514}{264553120177921} a^{14} - \frac{488}{264553120177921} a^{13} - \frac{1387}{131194207874} a^{11} + \frac{46}{65597103937} a^{10} - \frac{1341}{131194207874} a^{9} - \frac{5327}{65060356} a^{8} - \frac{647}{16265089} a^{7} - \frac{709}{65060356} a^{6} - \frac{2011}{4033} a^{5} + \frac{2403}{16132} a^{4} + \frac{182}{4033} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{2116424961423368} a^{17} - \frac{244}{264553120177921} a^{14} - \frac{1775}{529106240355842} a^{13} + \frac{2311}{1058212480711684} a^{12} + \frac{3967}{131194207874} a^{11} + \frac{107}{7091578804} a^{10} + \frac{4993}{524776831496} a^{9} - \frac{2599}{32530178} a^{8} - \frac{1749}{130120712} a^{7} + \frac{1375}{16265089} a^{6} + \frac{9051}{32264} a^{5} + \frac{3691}{8066} a^{4} + \frac{11569}{32264} a^{3}$, $\frac{1}{17071083738840886288} a^{18} + \frac{757}{2133885467355110786} a^{16} - \frac{122}{1066942733677555393} a^{15} - \frac{2053}{1058212480711684} a^{14} + \frac{6679}{2116424961423368} a^{13} - \frac{7249}{1058212480711684} a^{12} - \frac{2737}{524776831496} a^{11} + \frac{31825}{1049553662992} a^{10} - \frac{7451}{262388415748} a^{9} - \frac{29013}{260241424} a^{8} - \frac{2881}{32530178} a^{7} + \frac{16595}{260241424} a^{6} + \frac{11}{16132} a^{5} + \frac{19177}{64528} a^{4} + \frac{706}{4033} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{34142167477681772576} a^{19} + \frac{757}{4267770934710221572} a^{17} - \frac{61}{1066942733677555393} a^{16} - \frac{1}{2116424961423368} a^{15} - \frac{25585}{4232849922846736} a^{14} + \frac{14155}{2116424961423368} a^{13} - \frac{13569}{4232849922846736} a^{12} - \frac{21519}{2099107325984} a^{11} + \frac{9621}{524776831496} a^{10} + \frac{43163}{2099107325984} a^{9} - \frac{5029}{65060356} a^{8} + \frac{35331}{520482848} a^{7} + \frac{435}{3516776} a^{6} - \frac{14583}{129056} a^{5} + \frac{585}{4033} a^{4} - \frac{5869}{16132} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{9868164911693192118981216464827011585528600170357518855986935612111181055896165055197635139314479766603925018865651717246289642452565815378215459136} a^{20} - \frac{22262227037796663113103141432916331253057811409440535506893849768512375131559589606473066335056786764409329758344499482935982025}{2467041227923298029745304116206752896382150042589379713996733903027795263974041263799408784828619941650981254716412929311572410613141453844553864784} a^{19} - \frac{13204267686207680733888918430743352997447948550918976245125718434527637261786250879981722006123836501628241491642223336609911109}{2467041227923298029745304116206752896382150042589379713996733903027795263974041263799408784828619941650981254716412929311572410613141453844553864784} a^{18} - \frac{130959147503494971238971778369841582358587814354823710272262165790865794222682294211488183589527228228284676853122131753135127857789}{616760306980824507436326029051688224095537510647344928499183475756948815993510315949852196207154985412745313679103232327893102653285363461138466196} a^{17} + \frac{326821363009457626179334878379715293160432369860642702626194563518656350922968831667933418622537347388272809210852815972039089444455}{2467041227923298029745304116206752896382150042589379713996733903027795263974041263799408784828619941650981254716412929311572410613141453844553864784} a^{16} + \frac{1554885012109695950594666580852834879628552523568684650229666319593565170547266878040320045183951725983488076862957497828657528749255}{4934082455846596059490608232413505792764300085178759427993467806055590527948082527598817569657239883301962509432825858623144821226282907689107729568} a^{15} - \frac{3462734693904784733543286974533644002035616796493292186617480014075108757033816284141558047837680289052939184122398128536300067927359}{611713669209843300209596854997955094565373181896697176790660526414033043385579286833476019049992546900813601467000478381247808235343777298426448} a^{14} - \frac{1956065512640039370781985998004562743990125592401137125506475772084283488226296535661044364730011393219041397626547542137899614718237}{1223427338419686600419193709995910189130746363793394353581321052828066086771158573666952038099985093801627202934000956762495616470687554596852896} a^{13} + \frac{2384026820903649627348121550749873124471954566534663454452079135362831387467706119055970301555405262045744314465953326942623070064265}{2446854676839373200838387419991820378261492727586788707162642105656132173542317147333904076199970187603254405868001913524991232941375109193705792} a^{12} + \frac{4404665436410219797468631241863528548813381162154998817385346789985792216656869185922450521990111835674845876958638084969760613889}{2049690288933338136755539954155095779298400298539405233816488719463188972683400080529804850021084655982782589136249182022797757136541697544} a^{11} - \frac{16797166826629218529249720449495277152804563024184555206082241991083425432247559432619702696101460075025806994804112357409655020989393}{606708325524268088479639826429908350672326488367663949209680660961103935914286423836822235606241058170903646384329757878748136112416342473024} a^{10} - \frac{322347113017990488237534703307659870010914684986959589472961970027172358464446356405754695011103668398114814600437542216600660932109}{151677081381067022119909956607477087668081622091915987302420165240275983978571605959205558901560264542725911596082439469687034028104085618256} a^{9} + \frac{123234791141551858822495752986244375104354759831326687741574696943732741105720781292407646460410412113165400642314380683890536619827}{1009637479925294322642960386259513960617400553433608883106453405314051584352392133750288701445026614608845558345544822128094455827371072} a^{8} - \frac{112095104970645851908918805332016977444061026375430300852782405844681589809151234199394225780812062060399281177309969298066592719427}{4701124515902151689806284298520861879124771326925241361964423668493552689640825872774781766103405174272437131046443078033939809946196554} a^{7} + \frac{8823233050058398280675757584015278235383810058379435829231787314124290089310470947177415057579003167066827123477199731590045912223973}{150435984508868854073801097552667580131992682461607723582861557391793686068506427928793016515308965576717988193486178497086073918278289728} a^{6} - \frac{2332366228148839498112025566798039779831883867618131220776974219993132595295593517091827030671896339313617636725457277539610769693401}{9325315181556462563463990673981377394742913616514240241932900904524775977467544503396542060210077211549590143409755671775729848641104} a^{5} + \frac{3418589229472252006441972488021553301886271984592729402517063848306587292128020722780970053147543382626453863518745713045763293910687}{9325315181556462563463990673981377394742913616514240241932900904524775977467544503396542060210077211549590143409755671775729848641104} a^{4} - \frac{1728764334702139427380538515679701335386626529410290357143595601572164714812428051764007486023042328298776670636373370283970782728219}{4662657590778231281731995336990688697371456808257120120966450452262387988733772251698271030105038605774795071704877835887864924320552} a^{3} + \frac{41581851909899793556379027241441725408874844039927489849038104774045068015044343750198986829216154028872853074837296251626666663}{144515794407954106178155074912927370982254426241542279970445402066153855341364128802946659747862590062447156945973153852215005093} a^{2} - \frac{38358230901343189838515535963175198888973399423107638067478858230153403649094031918114583893865818918084656130509683384203061634}{144515794407954106178155074912927370982254426241542279970445402066153855341364128802946659747862590062447156945973153852215005093} a + \frac{44786316634607122633741044769657500258415609841066317059703831966444784266077088748533771859711563448267533885254189951633114508}{144515794407954106178155074912927370982254426241542279970445402066153855341364128802946659747862590062447156945973153852215005093}$

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:  $14$
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

21T138:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7348320
The 118 conjugacy class representatives for t21n138 are not computed
Character table for t21n138 is not computed

Intermediate fields

7.3.640681.1

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

Sibling fields

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 R ${\href{/LocalNumberField/5.14.0.1}{14} }{,}\,{\href{/LocalNumberField/5.7.0.1}{7} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.9.0.1}{9} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.14.0.1}{14} }{,}\,{\href{/LocalNumberField/19.7.0.1}{7} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ R $18{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.14.0.1}{14} }{,}\,{\href{/LocalNumberField/53.7.0.1}{7} }$ R

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$2.7.0.1$x^{7} - x + 1$$1$$7$$0$$C_7$$[\ ]^{7}$
2.14.14.15$x^{14} + 2 x^{13} + x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{9} + 4 x^{8} - 2 x^{6} + 4 x^{5} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 1$$2$$7$$14$$C_{14}$$[2]^{7}$
3Data not computed
$37$$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
37.3.2.3$x^{3} - 148$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.3$x^{3} - 148$$3$$1$$2$$C_3$$[\ ]_{3}$
37.6.4.2$x^{6} - 37 x^{3} + 6845$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
37.6.4.1$x^{6} + 518 x^{3} + 171125$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
59Data not computed
$109$109.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
109.9.6.1$x^{9} + 3270 x^{6} + 3552419 x^{3} + 1295029000$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
109.9.6.3$x^{9} - 218 x^{6} + 11881 x^{3} - 129502900$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$
10859Data not computed