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Properties

Label 21.5.80227383150...8881.1
Degree $21$
Signature $[5, 8]$
Discriminant $3^{28}\cdot 5717^{6}\cdot 372877^{14}$
Root discriminant $265{,}469.95$
Ramified primes $3, 5717, 372877$
Class number Not computed
Class group Not computed
Galois group 21T125

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2088718429416644256490983451035101319180494475114679662929419, -48009417721498935817205943488991772635258940070864253182793, -466634026077513762177084801819135983678010730432212315894, -2370907189749956210416412828825202913422782720289202211, -6185059649097648674878532059998125319351942271945047, -4697851621310243859913919973268158827694595110354, 16125401227872560540508055585819213690726439925, 41954377021743556239213520604452789963666647, 12478872486479001442437106502409828400056, -73835406083373320874465772825189421539, -83049356543524546524930516403494435, 43024340680467379995387995868789, 118414763283624334546271313025, 19132280510434775639075655, -82398195321133725334215, -42614569075187792049, 29463941307234906, 26278041597381, -4389880921, -7830417, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7830417*x^19 - 4389880921*x^18 + 26278041597381*x^17 + 29463941307234906*x^16 - 42614569075187792049*x^15 - 82398195321133725334215*x^14 + 19132280510434775639075655*x^13 + 118414763283624334546271313025*x^12 + 43024340680467379995387995868789*x^11 - 83049356543524546524930516403494435*x^10 - 73835406083373320874465772825189421539*x^9 + 12478872486479001442437106502409828400056*x^8 + 41954377021743556239213520604452789963666647*x^7 + 16125401227872560540508055585819213690726439925*x^6 - 4697851621310243859913919973268158827694595110354*x^5 - 6185059649097648674878532059998125319351942271945047*x^4 - 2370907189749956210416412828825202913422782720289202211*x^3 - 466634026077513762177084801819135983678010730432212315894*x^2 - 48009417721498935817205943488991772635258940070864253182793*x - 2088718429416644256490983451035101319180494475114679662929419)
 
gp: K = bnfinit(x^21 - 7830417*x^19 - 4389880921*x^18 + 26278041597381*x^17 + 29463941307234906*x^16 - 42614569075187792049*x^15 - 82398195321133725334215*x^14 + 19132280510434775639075655*x^13 + 118414763283624334546271313025*x^12 + 43024340680467379995387995868789*x^11 - 83049356543524546524930516403494435*x^10 - 73835406083373320874465772825189421539*x^9 + 12478872486479001442437106502409828400056*x^8 + 41954377021743556239213520604452789963666647*x^7 + 16125401227872560540508055585819213690726439925*x^6 - 4697851621310243859913919973268158827694595110354*x^5 - 6185059649097648674878532059998125319351942271945047*x^4 - 2370907189749956210416412828825202913422782720289202211*x^3 - 466634026077513762177084801819135983678010730432212315894*x^2 - 48009417721498935817205943488991772635258940070864253182793*x - 2088718429416644256490983451035101319180494475114679662929419, 1)
 

Normalized defining polynomial

\( x^{21} - 7830417 x^{19} - 4389880921 x^{18} + 26278041597381 x^{17} + 29463941307234906 x^{16} - 42614569075187792049 x^{15} - 82398195321133725334215 x^{14} + 19132280510434775639075655 x^{13} + 118414763283624334546271313025 x^{12} + 43024340680467379995387995868789 x^{11} - 83049356543524546524930516403494435 x^{10} - 73835406083373320874465772825189421539 x^{9} + 12478872486479001442437106502409828400056 x^{8} + 41954377021743556239213520604452789963666647 x^{7} + 16125401227872560540508055585819213690726439925 x^{6} - 4697851621310243859913919973268158827694595110354 x^{5} - 6185059649097648674878532059998125319351942271945047 x^{4} - 2370907189749956210416412828825202913422782720289202211 x^{3} - 466634026077513762177084801819135983678010730432212315894 x^{2} - 48009417721498935817205943488991772635258940070864253182793 x - 2088718429416644256490983451035101319180494475114679662929419 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[5, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(802273831501677373604859665315711644435432601942694083317197283802609114639108233270158494363951147372096290648881=3^{28}\cdot 5717^{6}\cdot 372877^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $265{,}469.95$
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:  $3, 5717, 372877$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3355893} a^{3} - \frac{1}{3} a - \frac{4}{9}$, $\frac{1}{3355893} a^{4} - \frac{1}{3} a^{2} - \frac{4}{9} a$, $\frac{1}{3355893} a^{5} - \frac{4}{9} a^{2} - \frac{1}{3}$, $\frac{1}{11262017827449} a^{6} + \frac{1}{10067679} a^{4} + \frac{1}{30203037} a^{3} - \frac{2}{9} a^{2} - \frac{13}{27} a - \frac{20}{81}$, $\frac{1}{11262017827449} a^{7} + \frac{1}{10067679} a^{5} + \frac{1}{30203037} a^{4} - \frac{13}{27} a^{2} + \frac{7}{81} a - \frac{2}{9}$, $\frac{1}{11262017827449} a^{8} + \frac{1}{30203037} a^{5} + \frac{1}{10067679} a^{3} + \frac{34}{81} a^{2} - \frac{2}{9} a - \frac{13}{27}$, $\frac{1}{37794126793011306957} a^{9} - \frac{1}{33786053482347} a^{6} - \frac{1}{10067679} a^{5} + \frac{1}{10067679} a^{4} + \frac{10}{90609111} a^{3} - \frac{2}{27} a^{2} + \frac{5}{27} a - \frac{82}{729}$, $\frac{1}{37794126793011306957} a^{10} - \frac{1}{33786053482347} a^{7} + \frac{1}{10067679} a^{5} + \frac{10}{90609111} a^{4} - \frac{1}{10067679} a^{3} - \frac{4}{27} a^{2} + \frac{161}{729} a - \frac{2}{27}$, $\frac{1}{37794126793011306957} a^{11} - \frac{1}{33786053482347} a^{8} + \frac{10}{90609111} a^{5} - \frac{1}{10067679} a^{4} + \frac{1}{10067679} a^{3} - \frac{325}{729} a^{2} - \frac{2}{27} a - \frac{4}{27}$, $\frac{1}{126833045545779093937847601} a^{12} - \frac{1}{113382380379033920871} a^{10} + \frac{2}{340147141137101762613} a^{9} - \frac{1}{33786053482347} a^{8} + \frac{4}{101358160447041} a^{7} + \frac{5}{304074481341123} a^{6} - \frac{10}{90609111} a^{5} - \frac{7}{271827333} a^{4} - \frac{94}{2446445997} a^{3} - \frac{52}{243} a^{2} - \frac{98}{2187} a + \frac{724}{6561}$, $\frac{1}{126833045545779093937847601} a^{13} - \frac{1}{113382380379033920871} a^{11} + \frac{2}{340147141137101762613} a^{10} + \frac{4}{101358160447041} a^{8} + \frac{5}{304074481341123} a^{7} - \frac{1}{33786053482347} a^{6} - \frac{7}{271827333} a^{5} - \frac{94}{2446445997} a^{4} - \frac{10}{90609111} a^{3} + \frac{631}{2187} a^{2} + \frac{2911}{6561} a + \frac{2}{243}$, $\frac{1}{126833045545779093937847601} a^{14} + \frac{2}{340147141137101762613} a^{11} - \frac{1}{113382380379033920871} a^{9} + \frac{5}{304074481341123} a^{8} - \frac{1}{33786053482347} a^{7} + \frac{4}{101358160447041} a^{6} - \frac{94}{2446445997} a^{5} - \frac{10}{90609111} a^{4} + \frac{20}{271827333} a^{3} - \frac{1463}{6561} a^{2} + \frac{29}{243} a + \frac{307}{2187}$, $\frac{1}{425638129715761240892365199262693} a^{15} + \frac{1}{380499136637337281813542803} a^{13} - \frac{2}{1141497409912011845440628409} a^{12} + \frac{4}{340147141137101762613} a^{11} - \frac{7}{1020441423411305287839} a^{10} + \frac{7}{3061324270233915863517} a^{9} - \frac{7}{304074481341123} a^{8} + \frac{14}{912223444023369} a^{7} - \frac{316}{8210010996210321} a^{6} + \frac{64}{815481999} a^{5} - \frac{1012}{7339337991} a^{4} + \frac{1667}{22018013973} a^{3} + \frac{419}{6561} a^{2} + \frac{6241}{19683} a + \frac{14924}{59049}$, $\frac{1}{425638129715761240892365199262693} a^{16} + \frac{1}{380499136637337281813542803} a^{14} - \frac{2}{1141497409912011845440628409} a^{13} - \frac{7}{1020441423411305287839} a^{11} + \frac{34}{3061324270233915863517} a^{10} + \frac{4}{340147141137101762613} a^{9} - \frac{40}{912223444023369} a^{8} + \frac{89}{8210010996210321} a^{7} - \frac{7}{304074481341123} a^{6} + \frac{1013}{7339337991} a^{5} - \frac{1411}{22018013973} a^{4} + \frac{91}{815481999} a^{3} - \frac{8015}{19683} a^{2} - \frac{5758}{59049} a - \frac{2821}{6561}$, $\frac{1}{425638129715761240892365199262693} a^{17} - \frac{2}{1141497409912011845440628409} a^{14} + \frac{1}{380499136637337281813542803} a^{12} - \frac{20}{3061324270233915863517} a^{11} + \frac{4}{340147141137101762613} a^{10} - \frac{7}{1020441423411305287839} a^{9} + \frac{8}{8210010996210321} a^{8} - \frac{7}{304074481341123} a^{7} + \frac{14}{912223444023369} a^{6} - \frac{1087}{22018013973} a^{5} - \frac{98}{815481999} a^{4} + \frac{446}{7339337991} a^{3} - \frac{3031}{59049} a^{2} + \frac{176}{6561} a + \frac{409}{19683}$, $\frac{1}{19259982777874993135189828650887322566859241933976057148449555253249353102619117} a^{18} - \frac{2}{5739152820985351182290325898616947133552601925620410766508215623456812569} a^{16} - \frac{845991038654820688999571940924985902941}{17217458462956053546870977695850841400657805776861232299524646870370437707} a^{15} + \frac{5}{5130514728257442518838049275066529654151013091556027650322774555199} a^{14} + \frac{4229955193274103444997859704624929514705}{15391544184772327556514147825199588962453039274668082950968323665597} a^{13} + \frac{88589273025450121702249560538267024444922}{46174632554316982669542443475598766887359117824004248852904970996791} a^{12} - \frac{8459910386548206889995719409249859029410}{13759268413598700158062978609746725204694880863008519298113787} a^{11} - \frac{354357092101800486808998242153068315912733}{41277805240796100474188935829240175614084642589025557894341361} a^{10} + \frac{2520421154672868260570319407758109997396115}{371500247167164904267700422463161580526761783301230021049072249} a^{9} + \frac{177178546050900243404499121076534176413778}{12300095754184087655413606997970488217021413551929563277} a^{8} - \frac{1643535120423827377130322291622093198278574}{110700861787656788898722462981734393953192721967366069493} a^{7} + \frac{10064047466461157307051945500778783438364340}{332102585362970366696167388945203181859578165902098208479} a^{6} + \frac{12119175592157665007131689147943250100527137}{98961017339638172819028314950805398700011640985603} a^{5} + \frac{6785541445716419169224187907631493482453561}{296883052018914518457084944852416196100034922956809} a^{4} - \frac{77430947567654634628357476588607092557110658}{890649156056743555371254834557248588300104768870427} a^{3} - \frac{51654603721868327825036095758408597287488774}{265398555930342104283794159872572989752684239} a^{2} - \frac{258926579190925399649028756973348046899124728}{796195667791026312851382479617718969258052717} a - \frac{1972389402169298744848209123345561681537271793}{7165761010119236815662442316559470723322474453}$, $\frac{1}{19259982777874993135189828650887322566859241933976057148449555253249353102619117} a^{19} - \frac{2}{5739152820985351182290325898616947133552601925620410766508215623456812569} a^{17} - \frac{845991038654820688999571940924985902941}{17217458462956053546870977695850841400657805776861232299524646870370437707} a^{16} + \frac{5}{5130514728257442518838049275066529654151013091556027650322774555199} a^{15} + \frac{4229955193274103444997859704624929514705}{15391544184772327556514147825199588962453039274668082950968323665597} a^{14} + \frac{88589273025450121702249560538267024444922}{46174632554316982669542443475598766887359117824004248852904970996791} a^{13} + \frac{19369840301061867025860022589489035044646}{5130514728257442518838049275066529654151013091556027650322774555199} a^{12} - \frac{354357092101800486808998242153068315912733}{41277805240796100474188935829240175614084642589025557894341361} a^{11} - \frac{4032629609039282462486326515021469996497323}{371500247167164904267700422463161580526761783301230021049072249} a^{10} - \frac{17053748816874034595807483089513828161727}{4586422804532900052687659536582241734898293621002839766037929} a^{9} + \frac{1632990261432247984398000669767696798668145}{110700861787656788898722462981734393953192721967366069493} a^{8} + \frac{6787522084605081945523622539388993441417621}{332102585362970366696167388945203181859578165902098208479} a^{7} - \frac{732640339379563462119986042977045418112176}{36900287262552262966240820993911464651064240655788689831} a^{6} + \frac{10062066827572494530752510869021283479400280}{296883052018914518457084944852416196100034922956809} a^{5} - \frac{64324846040230333182244184743047932569323782}{890649156056743555371254834557248588300104768870427} a^{4} - \frac{14621934219674756091945024886665000117294184}{98961017339638172819028314950805398700011640985603} a^{3} + \frac{117873839722523266926728383586477802749747957}{796195667791026312851382479617718969258052717} a^{2} + \frac{2870315112213980639490652213588547933949978889}{7165761010119236815662442316559470723322474453} a + \frac{2174}{6561}$, $\frac{1}{510986992803046799893929968505764201150465504042950355307466894665645381954268381072358206369371783} a^{20} + \frac{10201987633562210746}{510986992803046799893929968505764201150465504042950355307466894665645381954268381072358206369371783} a^{19} - \frac{20}{1370390216621156037765616995700362857324172593222296776973283132683553509479716853204563988579} a^{18} + \frac{5182252087869773470906522392700263441781042286071341632773}{456796738873718679255205665233454285774724197740765592324427710894517836493238951068187996193} a^{17} + \frac{358426403648751415863090146413992672319464591538436936897758}{456796738873718679255205665233454285774724197740765592324427710894517836493238951068187996193} a^{16} - \frac{2948764841479272468435063623887003053329426603183429428766}{12345857807397802142032585546850115831749302641642313306065613807959941526844295974815891789} a^{15} - \frac{494063515680008997818550415944164356597349504714521791210136}{1225060110636265254373977652774116627667365371800260118817807778153433535705444291463909} a^{14} - \frac{4565054103686687582095711932644495662064892338075087798480524}{1225060110636265254373977652774116627667365371800260118817807778153433535705444291463909} a^{13} - \frac{302994663244355082371948528741245821782158529548208453388571}{175008587233752179196282521824873803952480767400037159831115396879061933672206327351987} a^{12} + \frac{15941661096008926128888214540756471423804216050423175993379935}{9856280574851213035724737536298430536080520159196679753520392339726774800044875051} a^{11} + \frac{2243746499093120254695839796717049724278881417343661393222998}{266385961482465217181749663143200825299473517816126479824875468641264183784996623} a^{10} - \frac{120278752012213729504232608475914226959033898556968284203586430}{9856280574851213035724737536298430536080520159196679753520392339726774800044875051} a^{9} + \frac{287132713816711605527563629635131027011649649197752840773065700}{8811020412317567665945908468742981855572141447176664828277056812949734809821} a^{8} + \frac{50880703615904251202157090414689523021863553260255012295590087}{1258717201759652523706558352677568836510305921025237832611008116135676401403} a^{7} - \frac{256720391962825065573250794103378696380920001828521082512373859}{8811020412317567665945908468742981855572141447176664828277056812949734809821} a^{6} + \frac{2066319510786549167843352796805080262016529331850702044831293912}{23629830781511242758190793394988110973785300372982685519023851867907473} a^{5} - \frac{2168178275304098939331601545347482366251304540852185126637611737}{23629830781511242758190793394988110973785300372982685519023851867907473} a^{4} + \frac{3231918842803554475729944009744295363857630190396189415602902199}{23629830781511242758190793394988110973785300372982685519023851867907473} a^{3} - \frac{37638460871045981852273326169596972041044848590129810702182120402}{190114950357715086408044422651341683507848167408952701714161923647} a^{2} - \frac{2907883522674256415244924665316410942080670228349639627695939474}{190114950357715086408044422651341683507848167408952701714161923647} a + \frac{54822019548671617475026676570194381045233318065847930128029933063}{190114950357715086408044422651341683507848167408952701714161923647}$

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:  $12$
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

21T125:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 2939328
The 120 conjugacy class representatives for t21n125 are not computed
Character table for t21n125 is not computed

Intermediate fields

7.3.32684089.2

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

Sibling fields

Degree 24 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 $21$ R ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.9.0.1}{9} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.9.0.1}{9} }{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{3}$ $21$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $21$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
5717Data not computed
372877Data not computed