Properties

Label 30.0.214...923.1
Degree $30$
Signature $[0, 15]$
Discriminant $-2.142\times 10^{50}$
Root discriminant $47.61$
Ramified primes $3, 1213$
Class number $7$ (GRH)
Class group $[7]$ (GRH)
Galois group $D_{30}$ (as 30T14)

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Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 6*x^29 + 7*x^28 + 64*x^27 - 269*x^26 + 169*x^25 + 1697*x^24 - 5426*x^23 + 2993*x^22 + 23602*x^21 - 68143*x^20 + 32587*x^19 + 255113*x^18 - 722972*x^17 + 638195*x^16 + 912175*x^15 - 3578530*x^14 + 6806701*x^13 - 13186196*x^12 + 20272003*x^11 - 4641245*x^10 - 65229024*x^9 + 170166391*x^8 - 258098865*x^7 + 369027093*x^6 - 495862722*x^5 + 471214449*x^4 - 297134325*x^3 + 130881204*x^2 - 35973639*x + 5861241)
 
gp: K = bnfinit(x^30 - 6*x^29 + 7*x^28 + 64*x^27 - 269*x^26 + 169*x^25 + 1697*x^24 - 5426*x^23 + 2993*x^22 + 23602*x^21 - 68143*x^20 + 32587*x^19 + 255113*x^18 - 722972*x^17 + 638195*x^16 + 912175*x^15 - 3578530*x^14 + 6806701*x^13 - 13186196*x^12 + 20272003*x^11 - 4641245*x^10 - 65229024*x^9 + 170166391*x^8 - 258098865*x^7 + 369027093*x^6 - 495862722*x^5 + 471214449*x^4 - 297134325*x^3 + 130881204*x^2 - 35973639*x + 5861241, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5861241, -35973639, 130881204, -297134325, 471214449, -495862722, 369027093, -258098865, 170166391, -65229024, -4641245, 20272003, -13186196, 6806701, -3578530, 912175, 638195, -722972, 255113, 32587, -68143, 23602, 2993, -5426, 1697, 169, -269, 64, 7, -6, 1]);
 

\( x^{30} - 6 x^{29} + 7 x^{28} + 64 x^{27} - 269 x^{26} + 169 x^{25} + 1697 x^{24} - 5426 x^{23} + 2993 x^{22} + 23602 x^{21} - 68143 x^{20} + 32587 x^{19} + 255113 x^{18} - 722972 x^{17} + 638195 x^{16} + 912175 x^{15} - 3578530 x^{14} + 6806701 x^{13} - 13186196 x^{12} + 20272003 x^{11} - 4641245 x^{10} - 65229024 x^{9} + 170166391 x^{8} - 258098865 x^{7} + 369027093 x^{6} - 495862722 x^{5} + 471214449 x^{4} - 297134325 x^{3} + 130881204 x^{2} - 35973639 x + 5861241 \)

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

Invariants

Degree:  $30$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 15]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-214224941509935982232692876412219596671526317019923\)\(\medspace = -\,3^{15}\cdot 1213^{14}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $47.61$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $3, 1213$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{9} - \frac{1}{3} a^{5} - \frac{4}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{10} - \frac{1}{9} a^{4}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{5}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{6}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{7}$, $\frac{1}{27} a^{14} + \frac{1}{27} a^{12} + \frac{1}{27} a^{10} - \frac{4}{27} a^{8} - \frac{1}{27} a^{6} - \frac{10}{27} a^{4} - \frac{2}{9} a^{2} - \frac{1}{3}$, $\frac{1}{27} a^{15} + \frac{1}{27} a^{13} + \frac{1}{27} a^{11} - \frac{1}{27} a^{9} - \frac{1}{27} a^{7} + \frac{8}{27} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{27} a^{16} + \frac{1}{27} a^{10} + \frac{1}{9} a^{8} + \frac{7}{27} a^{4} + \frac{2}{9} a^{2} + \frac{1}{3}$, $\frac{1}{27} a^{17} + \frac{1}{27} a^{11} - \frac{11}{27} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{81} a^{18} - \frac{2}{81} a^{12} + \frac{1}{81} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{81} a^{19} - \frac{2}{81} a^{13} + \frac{1}{81} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{243} a^{20} - \frac{1}{243} a^{18} - \frac{2}{243} a^{14} - \frac{7}{243} a^{12} + \frac{1}{27} a^{10} + \frac{28}{243} a^{8} + \frac{35}{243} a^{6} - \frac{10}{27} a^{4} - \frac{4}{9} a^{2} - \frac{4}{9}$, $\frac{1}{243} a^{21} - \frac{1}{243} a^{19} - \frac{2}{243} a^{15} - \frac{7}{243} a^{13} + \frac{1}{27} a^{11} + \frac{1}{243} a^{9} + \frac{35}{243} a^{7} - \frac{1}{27} a^{5} - \frac{1}{9} a$, $\frac{1}{243} a^{22} - \frac{1}{243} a^{18} - \frac{2}{243} a^{16} + \frac{11}{243} a^{12} - \frac{8}{243} a^{10} + \frac{1}{9} a^{8} + \frac{17}{243} a^{6} + \frac{10}{27} a^{4} + \frac{2}{9} a^{2} + \frac{2}{9}$, $\frac{1}{243} a^{23} - \frac{1}{243} a^{19} - \frac{2}{243} a^{17} + \frac{11}{243} a^{13} - \frac{8}{243} a^{11} + \frac{17}{243} a^{7} - \frac{8}{27} a^{5} - \frac{1}{3} a^{3} - \frac{4}{9} a$, $\frac{1}{243} a^{24} - \frac{1}{81} a^{12} + \frac{29}{243} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{2}{9}$, $\frac{1}{729} a^{25} - \frac{1}{729} a^{24} + \frac{1}{729} a^{23} - \frac{1}{729} a^{22} + \frac{1}{729} a^{21} - \frac{1}{729} a^{20} + \frac{4}{729} a^{19} - \frac{4}{729} a^{18} - \frac{11}{729} a^{17} + \frac{11}{729} a^{16} - \frac{11}{729} a^{15} + \frac{11}{729} a^{14} - \frac{20}{729} a^{13} + \frac{20}{729} a^{12} - \frac{17}{729} a^{11} + \frac{17}{729} a^{10} - \frac{17}{729} a^{9} + \frac{17}{729} a^{8} + \frac{32}{243} a^{7} - \frac{32}{243} a^{6} + \frac{10}{27} a^{5} - \frac{10}{27} a^{4} + \frac{1}{27} a^{3} - \frac{1}{27} a^{2} - \frac{4}{9} a + \frac{4}{9}$, $\frac{1}{190269} a^{26} - \frac{5}{190269} a^{25} - \frac{52}{190269} a^{24} + \frac{148}{190269} a^{23} + \frac{197}{190269} a^{22} + \frac{265}{190269} a^{21} + \frac{353}{190269} a^{20} + \frac{187}{190269} a^{19} + \frac{251}{190269} a^{18} + \frac{1315}{190269} a^{17} - \frac{1006}{190269} a^{16} + \frac{3133}{190269} a^{15} - \frac{2212}{190269} a^{14} - \frac{1637}{190269} a^{13} - \frac{3658}{190269} a^{12} + \frac{7609}{190269} a^{11} + \frac{7802}{190269} a^{10} + \frac{2245}{190269} a^{9} - \frac{15017}{190269} a^{8} + \frac{4400}{63423} a^{7} - \frac{7622}{63423} a^{6} - \frac{91}{2349} a^{5} + \frac{398}{7047} a^{4} - \frac{1883}{7047} a^{3} - \frac{491}{7047} a^{2} - \frac{472}{2349} a - \frac{35}{2349}$, $\frac{1}{570807} a^{27} - \frac{77}{570807} a^{25} - \frac{895}{570807} a^{24} + \frac{154}{570807} a^{23} + \frac{467}{570807} a^{22} - \frac{671}{570807} a^{21} + \frac{386}{570807} a^{20} - \frac{2729}{570807} a^{19} + \frac{221}{570807} a^{18} - \frac{6959}{570807} a^{17} + \frac{6716}{570807} a^{16} - \frac{10037}{570807} a^{15} - \frac{9565}{570807} a^{14} - \frac{18107}{570807} a^{13} + \frac{24554}{570807} a^{12} - \frac{11312}{570807} a^{11} + \frac{19331}{570807} a^{10} + \frac{7349}{190269} a^{9} + \frac{21113}{570807} a^{8} - \frac{10417}{190269} a^{7} + \frac{26249}{190269} a^{6} - \frac{7753}{21141} a^{5} + \frac{10025}{21141} a^{4} + \frac{613}{7047} a^{3} + \frac{9440}{21141} a^{2} - \frac{2134}{7047} a + \frac{2435}{7047}$, $\frac{1}{73088383646521610433} a^{28} + \frac{40603480685353}{73088383646521610433} a^{27} + \frac{140510969455810}{73088383646521610433} a^{26} - \frac{3593470931818390}{8120931516280178937} a^{25} + \frac{43344732077861570}{24362794548840536811} a^{24} + \frac{8072113333467452}{8120931516280178937} a^{23} - \frac{18139774488748958}{24362794548840536811} a^{22} + \frac{33121564846427449}{24362794548840536811} a^{21} - \frac{33177932570745661}{24362794548840536811} a^{20} + \frac{96714802157807923}{24362794548840536811} a^{19} + \frac{143579207773478320}{24362794548840536811} a^{18} + \frac{11265702873657571}{738266501480016267} a^{17} - \frac{25649876119966727}{8120931516280178937} a^{16} + \frac{126369730807153094}{8120931516280178937} a^{15} + \frac{289095770692474402}{24362794548840536811} a^{14} + \frac{251280129581908843}{24362794548840536811} a^{13} - \frac{792005236562230319}{24362794548840536811} a^{12} - \frac{384216333092265340}{8120931516280178937} a^{11} + \frac{3940668448513343564}{73088383646521610433} a^{10} + \frac{3848785483282393073}{73088383646521610433} a^{9} - \frac{7967958507063032323}{73088383646521610433} a^{8} - \frac{3503938415497216892}{24362794548840536811} a^{7} - \frac{1116612095756974942}{24362794548840536811} a^{6} - \frac{518571677240487689}{2706977172093392979} a^{5} - \frac{332333995757749270}{2706977172093392979} a^{4} + \frac{679860529047295301}{2706977172093392979} a^{3} + \frac{810277279553627753}{2706977172093392979} a^{2} - \frac{289862539096528643}{902325724031130993} a - \frac{108917835362633200}{902325724031130993}$, $\frac{1}{7541979875366180765901279988583133951992155268388404417581132237407} a^{29} + \frac{18454677806117421280433171457294712510791353044}{7541979875366180765901279988583133951992155268388404417581132237407} a^{28} + \frac{522597302263249754407884660730736028112825790129440325783941}{685634534124198251445570908053012177453832297126218583416466567037} a^{27} - \frac{338246370468810153244446301188490020133280149564681729874626}{228544844708066083815190302684337392484610765708739527805488855679} a^{26} - \frac{7645290621715503800684508289036197719808028321642168038576355}{28037099908424463813759405162019085323390911778395555455691941403} a^{25} - \frac{14173491403977914444354803653444326624187952494062226437947727911}{7541979875366180765901279988583133951992155268388404417581132237407} a^{24} + \frac{13590995714670974602006317342019401036600092066712006350978394940}{7541979875366180765901279988583133951992155268388404417581132237407} a^{23} - \frac{389601215632647496118514686749245553379184809810960787471268751}{260068271564351060893147585813211515585936388565117393709694215083} a^{22} + \frac{14036485309484571690384256565155043557598956218005095031656743937}{7541979875366180765901279988583133951992155268388404417581132237407} a^{21} + \frac{2678696306874978213842808384601224703016755769222450787933472076}{7541979875366180765901279988583133951992155268388404417581132237407} a^{20} - \frac{4930812186963116135223240535179702092726270328650596339869374294}{7541979875366180765901279988583133951992155268388404417581132237407} a^{19} - \frac{43838458190475843813193210766305719246258737096084617589463540703}{7541979875366180765901279988583133951992155268388404417581132237407} a^{18} - \frac{44718147706286799822182926793688098017667349205216827958916165045}{7541979875366180765901279988583133951992155268388404417581132237407} a^{17} - \frac{28184446829172517072555521951596421348409005382710651531703088314}{7541979875366180765901279988583133951992155268388404417581132237407} a^{16} - \frac{30428392854872823967619157935849521032701077127701724212249192109}{7541979875366180765901279988583133951992155268388404417581132237407} a^{15} - \frac{137814415535272452668679680538045233352232215975916635446301086668}{7541979875366180765901279988583133951992155268388404417581132237407} a^{14} - \frac{278807665078577921713896479336420286218197330928775253226381512859}{7541979875366180765901279988583133951992155268388404417581132237407} a^{13} - \frac{158764286111329653428264762990745199451762063327828090604221059258}{7541979875366180765901279988583133951992155268388404417581132237407} a^{12} - \frac{887211243258779888047693859584015148833448124530776693833106721}{279332587976525213551899258836412368592302046977348311762264156941} a^{11} - \frac{79341383623419202524944317451973961844631689639643905752587799427}{7541979875366180765901279988583133951992155268388404417581132237407} a^{10} + \frac{355338196593255692558445849574128802456689875625966807940901250575}{7541979875366180765901279988583133951992155268388404417581132237407} a^{9} - \frac{382744145221012787104476580758155598819069249168174566398163505976}{7541979875366180765901279988583133951992155268388404417581132237407} a^{8} + \frac{179519472071972853489299081441034974966437986087847778224478604586}{2513993291788726921967093329527711317330718422796134805860377412469} a^{7} + \frac{400790965185815516032669315455774893521686709696509309748820888993}{2513993291788726921967093329527711317330718422796134805860377412469} a^{6} - \frac{136755088500445050607366643437333991011740491350933223439684152825}{279332587976525213551899258836412368592302046977348311762264156941} a^{5} - \frac{64294610651276383578727284345043287361740972872707491206986117001}{279332587976525213551899258836412368592302046977348311762264156941} a^{4} - \frac{64970478507813954934174596755587663913011151164268042469318432739}{279332587976525213551899258836412368592302046977348311762264156941} a^{3} + \frac{115936378727795976813159511356986993682291655444683232139842552178}{279332587976525213551899258836412368592302046977348311762264156941} a^{2} + \frac{9556362032931673154877618255132939796170111107201075297020505176}{93110862658841737850633086278804122864100682325782770587421385647} a - \frac{41990629343679401372618032448547533384721463412802625075287173}{346137035906474861898264261259494880535690268869080931551752363}$

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

Class group and class number

$C_{7}$, which has order $7$ (assuming GRH)

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $14$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -\frac{57077206969750426954283184170484360609363132859}{58901492849240563040544616077258251589422606427776553} a^{29} + \frac{267831654181433608882105781655654457496861935046}{58901492849240563040544616077258251589422606427776553} a^{28} - \frac{57960733985396156971599527111747575024948332623}{58901492849240563040544616077258251589422606427776553} a^{27} - \frac{1231172811667900150920528024535990607212398233209}{19633830949746854346848205359086083863140868809258851} a^{26} + \frac{39163748016767733080357575044143537665512228415}{218964657432121052195333145268618035648411176311437} a^{25} + \frac{3584021617667990373249092006744098474236791524942}{58901492849240563040544616077258251589422606427776553} a^{24} - \frac{90907753884451422258832686931504123627355003175013}{58901492849240563040544616077258251589422606427776553} a^{23} + \frac{192032106913158289147884416046265716797478969418174}{58901492849240563040544616077258251589422606427776553} a^{22} + \frac{67456949598384154996825650832947287044876445073677}{58901492849240563040544616077258251589422606427776553} a^{21} - \frac{1237356090549657003766010063246751004264605934264673}{58901492849240563040544616077258251589422606427776553} a^{20} + \frac{2292318434178381577687079377315368886102655841398471}{58901492849240563040544616077258251589422606427776553} a^{19} + \frac{966277348727986817917568303348008391510243022589768}{58901492849240563040544616077258251589422606427776553} a^{18} - \frac{13049698953658612961804920416575542479762930084182629}{58901492849240563040544616077258251589422606427776553} a^{17} + \frac{24466096873150481085568281812614704857839628306589625}{58901492849240563040544616077258251589422606427776553} a^{16} - \frac{6211934229335449781302276679058871189187336596653928}{58901492849240563040544616077258251589422606427776553} a^{15} - \frac{57502307327824827659053973395786632676774096842321359}{58901492849240563040544616077258251589422606427776553} a^{14} + \frac{129427449021541203477629770006976985967738773274327844}{58901492849240563040544616077258251589422606427776553} a^{13} - \frac{227348901123478183714244662536771328395966684263314685}{58901492849240563040544616077258251589422606427776553} a^{12} + \frac{200179391759866945576641815423258098460285154324395}{25075135312575803763535383600365368918442999756397} a^{11} - \frac{569049970345156070664706103066666392211492313393425326}{58901492849240563040544616077258251589422606427776553} a^{10} - \frac{422062326950517017386118360568537687859504490422033385}{58901492849240563040544616077258251589422606427776553} a^{9} + \frac{3115091059989811849778862751614289806219708679672055839}{58901492849240563040544616077258251589422606427776553} a^{8} - \frac{1909080137742186431222959267782483370181921797973210905}{19633830949746854346848205359086083863140868809258851} a^{7} + \frac{2550805200369783669066272383228654377788539359153889678}{19633830949746854346848205359086083863140868809258851} a^{6} - \frac{433089733908956101054160631759843065576600983849275277}{2181536772194094927427578373231787095904540978806539} a^{5} + \frac{513193087899700829554646837618122774273206873420339359}{2181536772194094927427578373231787095904540978806539} a^{4} - \frac{375205819819123972844853114874296358423855288965197780}{2181536772194094927427578373231787095904540978806539} a^{3} + \frac{190128118449125091416651584659578759364656169957462187}{2181536772194094927427578373231787095904540978806539} a^{2} - \frac{20774004819729407609733228362595648467987367820441639}{727178924064698309142526124410595698634846992935513} a + \frac{16125261378914570686283125133470436597907762299627}{2703267375705198175251026484797753526523594769277} \) (order $6$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 828216417792195.4 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{15}\cdot 828216417792195.4 \cdot 7}{6\sqrt{214224941509935982232692876412219596671526317019923}}\approx 61.9945674018352$ (assuming GRH)

Galois group

$D_{30}$ (as 30T14):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 60
The 18 conjugacy class representatives for $D_{30}$
Character table for $D_{30}$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.3639.1, 5.1.13242321.1, 6.0.39726963.2, 10.0.526077196401123.1, 15.1.8450344007000266933623879.1

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

Sibling fields

Degree 30 sibling: 30.2.86618284683850782149418819696007456920853807515055533.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.10.0.1}{10} }^{3}$ R $30$ $15^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{15}$ $15^{2}$ $30$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{6}$ $30$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{15}$ $15^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{5}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{10}$ $30$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{3}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
1213Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.1213.2t1.a.a$1$ $ 1213 $ \(\Q(\sqrt{1213}) \) $C_2$ (as 2T1) $1$ $1$
1.3639.2t1.a.a$1$ $ 3 \cdot 1213 $ \(\Q(\sqrt{-3639}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.3.2t1.a.a$1$ $ 3 $ \(\Q(\sqrt{-3}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.3639.6t3.a.a$2$ $ 3 \cdot 1213 $ 6.2.16062935373.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.3639.3t2.a.a$2$ $ 3 \cdot 1213 $ 3.1.3639.1 $S_3$ (as 3T2) $1$ $0$
* 2.3639.10t3.b.a$2$ $ 3 \cdot 1213 $ 10.2.212710546411520733.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.3639.10t3.b.b$2$ $ 3 \cdot 1213 $ 10.2.212710546411520733.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.3639.5t2.a.b$2$ $ 3 \cdot 1213 $ 5.1.13242321.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.3639.5t2.a.a$2$ $ 3 \cdot 1213 $ 5.1.13242321.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.3639.30t14.b.b$2$ $ 3 \cdot 1213 $ 30.0.214224941509935982232692876412219596671526317019923.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.3639.30t14.b.d$2$ $ 3 \cdot 1213 $ 30.0.214224941509935982232692876412219596671526317019923.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.3639.30t14.b.c$2$ $ 3 \cdot 1213 $ 30.0.214224941509935982232692876412219596671526317019923.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.3639.30t14.b.a$2$ $ 3 \cdot 1213 $ 30.0.214224941509935982232692876412219596671526317019923.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.3639.15t2.a.d$2$ $ 3 \cdot 1213 $ 15.1.8450344007000266933623879.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.3639.15t2.a.b$2$ $ 3 \cdot 1213 $ 15.1.8450344007000266933623879.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.3639.15t2.a.c$2$ $ 3 \cdot 1213 $ 15.1.8450344007000266933623879.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.3639.15t2.a.a$2$ $ 3 \cdot 1213 $ 15.1.8450344007000266933623879.1 $D_{15}$ (as 15T2) $1$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.