Properties

Label 30.2.866...533.1
Degree $30$
Signature $[2, 14]$
Discriminant $8.662\times 10^{52}$
Root discriminant $58.15$
Ramified primes $3, 1213$
Class number $7$ (GRH)
Class group $[7]$ (GRH)
Galois group $D_{30}$ (as 30T14)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 8*x^29 + 43*x^28 - 163*x^27 + 446*x^26 - 385*x^25 - 2817*x^24 + 21864*x^23 - 95826*x^22 + 323313*x^21 - 894891*x^20 + 2178948*x^19 - 4695543*x^18 + 9008829*x^17 - 15016794*x^16 + 20604987*x^15 - 18291717*x^14 - 5218266*x^13 + 65252417*x^12 - 159581317*x^11 + 242005763*x^10 - 224361122*x^9 + 47652991*x^8 + 233622619*x^7 - 454603911*x^6 + 464765022*x^5 - 306287199*x^4 + 111399003*x^3 - 14270364*x^2 - 456489*x - 204363)
 
gp: K = bnfinit(x^30 - 8*x^29 + 43*x^28 - 163*x^27 + 446*x^26 - 385*x^25 - 2817*x^24 + 21864*x^23 - 95826*x^22 + 323313*x^21 - 894891*x^20 + 2178948*x^19 - 4695543*x^18 + 9008829*x^17 - 15016794*x^16 + 20604987*x^15 - 18291717*x^14 - 5218266*x^13 + 65252417*x^12 - 159581317*x^11 + 242005763*x^10 - 224361122*x^9 + 47652991*x^8 + 233622619*x^7 - 454603911*x^6 + 464765022*x^5 - 306287199*x^4 + 111399003*x^3 - 14270364*x^2 - 456489*x - 204363, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-204363, -456489, -14270364, 111399003, -306287199, 464765022, -454603911, 233622619, 47652991, -224361122, 242005763, -159581317, 65252417, -5218266, -18291717, 20604987, -15016794, 9008829, -4695543, 2178948, -894891, 323313, -95826, 21864, -2817, -385, 446, -163, 43, -8, 1]);
 

\(x^{30} - 8 x^{29} + 43 x^{28} - 163 x^{27} + 446 x^{26} - 385 x^{25} - 2817 x^{24} + 21864 x^{23} - 95826 x^{22} + 323313 x^{21} - 894891 x^{20} + 2178948 x^{19} - 4695543 x^{18} + 9008829 x^{17} - 15016794 x^{16} + 20604987 x^{15} - 18291717 x^{14} - 5218266 x^{13} + 65252417 x^{12} - 159581317 x^{11} + 242005763 x^{10} - 224361122 x^{9} + 47652991 x^{8} + 233622619 x^{7} - 454603911 x^{6} + 464765022 x^{5} - 306287199 x^{4} + 111399003 x^{3} - 14270364 x^{2} - 456489 x - 204363\)  Toggle raw display

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

Invariants

Degree:  $30$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[2, 14]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(86618284683850782149418819696007456920853807515055533\)\(\medspace = 3^{14}\cdot 1213^{15}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $58.15$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{9} - \frac{1}{3} a^{5} - \frac{4}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{10} - \frac{1}{9} a^{4}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{5}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{6}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{7}$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{8}$, $\frac{1}{27} a^{15} + \frac{1}{27} a^{13} + \frac{1}{27} a^{11} - \frac{1}{27} a^{9} - \frac{1}{27} a^{7} + \frac{8}{27} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{81} a^{16} - \frac{1}{81} a^{15} + \frac{1}{81} a^{14} + \frac{2}{81} a^{13} + \frac{1}{81} a^{12} + \frac{2}{81} a^{11} - \frac{1}{81} a^{10} + \frac{4}{81} a^{9} + \frac{8}{81} a^{8} - \frac{2}{81} a^{7} - \frac{10}{81} a^{6} - \frac{20}{81} a^{5} - \frac{1}{9} a^{4} - \frac{7}{27} a^{3} - \frac{2}{9} a^{2} - \frac{2}{9} a$, $\frac{1}{81} a^{17} + \frac{1}{27} a^{14} + \frac{1}{27} a^{13} + \frac{1}{27} a^{12} + \frac{1}{81} a^{11} + \frac{1}{27} a^{10} + \frac{1}{27} a^{9} + \frac{2}{27} a^{8} - \frac{4}{27} a^{7} - \frac{1}{27} a^{6} - \frac{2}{81} a^{5} - \frac{1}{27} a^{4} - \frac{1}{27} a^{3} - \frac{1}{9} a^{2} + \frac{1}{9} a$, $\frac{1}{81} a^{18} + \frac{1}{27} a^{14} + \frac{1}{81} a^{12} + \frac{1}{27} a^{10} - \frac{4}{27} a^{8} - \frac{2}{81} a^{6} - \frac{1}{27} a^{4} + \frac{1}{9} a^{2}$, $\frac{1}{81} a^{19} - \frac{2}{81} a^{13} + \frac{1}{81} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{81} a^{20} - \frac{2}{81} a^{14} + \frac{1}{81} a^{8}$, $\frac{1}{243} a^{21} - \frac{1}{243} a^{19} - \frac{2}{243} a^{15} + \frac{1}{27} a^{14} + \frac{2}{243} a^{13} - \frac{1}{27} a^{12} + \frac{1}{243} a^{9} - \frac{1}{27} a^{8} + \frac{26}{243} a^{7} + \frac{1}{27} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{4}{9} a$, $\frac{1}{243} a^{22} - \frac{1}{243} a^{20} + \frac{1}{243} a^{16} - \frac{1}{81} a^{15} + \frac{5}{243} a^{14} - \frac{4}{81} a^{13} + \frac{1}{81} a^{12} - \frac{1}{81} a^{11} - \frac{2}{243} a^{10} + \frac{4}{81} a^{9} - \frac{31}{243} a^{8} + \frac{4}{81} a^{7} - \frac{10}{81} a^{6} + \frac{37}{81} a^{5} - \frac{1}{9} a^{4} + \frac{11}{27} a^{3} + \frac{4}{9} a$, $\frac{1}{729} a^{23} + \frac{1}{729} a^{22} - \frac{1}{729} a^{20} - \frac{4}{729} a^{19} + \frac{4}{729} a^{17} + \frac{1}{729} a^{16} - \frac{1}{243} a^{15} - \frac{13}{729} a^{14} - \frac{40}{729} a^{13} + \frac{1}{243} a^{12} - \frac{23}{729} a^{11} + \frac{16}{729} a^{10} + \frac{10}{243} a^{9} + \frac{41}{729} a^{8} - \frac{64}{729} a^{7} + \frac{17}{243} a^{6} + \frac{23}{81} a^{5} + \frac{4}{81} a^{4} - \frac{4}{9} a^{3} - \frac{8}{27} a^{2} + \frac{2}{27} a + \frac{1}{3}$, $\frac{1}{2187} a^{24} - \frac{1}{2187} a^{22} - \frac{1}{2187} a^{21} - \frac{1}{729} a^{20} + \frac{4}{2187} a^{19} + \frac{13}{2187} a^{18} + \frac{2}{729} a^{17} - \frac{13}{2187} a^{16} + \frac{26}{2187} a^{15} + \frac{11}{243} a^{14} - \frac{83}{2187} a^{13} + \frac{1}{2187} a^{12} - \frac{8}{729} a^{11} - \frac{85}{2187} a^{10} + \frac{56}{2187} a^{9} - \frac{23}{729} a^{8} + \frac{160}{2187} a^{7} - \frac{14}{729} a^{6} + \frac{86}{243} a^{5} + \frac{35}{243} a^{4} - \frac{20}{81} a^{3} - \frac{29}{81} a^{2} - \frac{11}{81} a + \frac{2}{9}$, $\frac{1}{2187} a^{25} - \frac{1}{2187} a^{23} - \frac{1}{2187} a^{22} - \frac{1}{729} a^{21} + \frac{4}{2187} a^{20} + \frac{13}{2187} a^{19} + \frac{2}{729} a^{18} - \frac{13}{2187} a^{17} - \frac{1}{2187} a^{16} - \frac{4}{243} a^{15} - \frac{110}{2187} a^{14} + \frac{28}{2187} a^{13} - \frac{17}{729} a^{12} - \frac{58}{2187} a^{11} + \frac{83}{2187} a^{10} - \frac{5}{729} a^{9} - \frac{56}{2187} a^{8} - \frac{23}{729} a^{7} + \frac{35}{243} a^{6} - \frac{76}{243} a^{5} - \frac{38}{81} a^{4} + \frac{19}{81} a^{3} - \frac{20}{81} a^{2} - \frac{2}{9} a$, $\frac{1}{2187} a^{26} - \frac{1}{2187} a^{23} - \frac{4}{2187} a^{22} + \frac{1}{729} a^{21} + \frac{10}{2187} a^{20} + \frac{10}{2187} a^{19} + \frac{5}{2187} a^{17} + \frac{5}{2187} a^{16} + \frac{8}{729} a^{15} - \frac{62}{2187} a^{14} - \frac{107}{2187} a^{13} - \frac{1}{729} a^{12} + \frac{86}{2187} a^{11} + \frac{89}{2187} a^{10} + \frac{2}{81} a^{9} - \frac{64}{729} a^{8} - \frac{281}{2187} a^{7} + \frac{64}{729} a^{6} - \frac{58}{243} a^{5} - \frac{70}{243} a^{4} + \frac{26}{81} a^{3} - \frac{2}{81} a^{2} + \frac{7}{81} a + \frac{2}{9}$, $\frac{1}{72171} a^{27} - \frac{5}{72171} a^{26} - \frac{1}{6561} a^{25} + \frac{1}{24057} a^{24} + \frac{13}{24057} a^{23} - \frac{4}{2187} a^{22} - \frac{1}{24057} a^{21} + \frac{2}{2187} a^{20} + \frac{85}{24057} a^{19} - \frac{25}{8019} a^{18} + \frac{58}{24057} a^{17} + \frac{103}{24057} a^{16} + \frac{8}{2187} a^{15} + \frac{95}{2673} a^{14} + \frac{26}{2673} a^{13} - \frac{5}{729} a^{12} - \frac{959}{24057} a^{11} + \frac{1279}{24057} a^{10} - \frac{1126}{72171} a^{9} - \frac{2086}{72171} a^{8} - \frac{6265}{72171} a^{7} - \frac{1327}{24057} a^{6} - \frac{3988}{8019} a^{5} - \frac{536}{8019} a^{4} - \frac{262}{2673} a^{3} - \frac{115}{243} a^{2} + \frac{956}{2673} a - \frac{29}{297}$, $\frac{1}{11475189} a^{28} + \frac{1}{3825063} a^{27} - \frac{578}{3825063} a^{26} + \frac{2522}{11475189} a^{25} - \frac{661}{3825063} a^{24} + \frac{1369}{3825063} a^{23} - \frac{841}{425007} a^{22} - \frac{637}{1275021} a^{21} - \frac{9265}{3825063} a^{20} + \frac{554}{115911} a^{19} + \frac{887}{425007} a^{18} - \frac{1358}{3825063} a^{17} - \frac{15346}{3825063} a^{16} + \frac{667}{425007} a^{15} - \frac{55648}{3825063} a^{14} + \frac{188278}{3825063} a^{13} + \frac{44432}{1275021} a^{12} - \frac{157643}{3825063} a^{11} + \frac{454907}{11475189} a^{10} - \frac{73636}{3825063} a^{9} + \frac{585799}{3825063} a^{8} + \frac{497758}{11475189} a^{7} + \frac{367711}{3825063} a^{6} + \frac{608321}{1275021} a^{5} - \frac{17185}{1275021} a^{4} - \frac{209908}{425007} a^{3} - \frac{874}{8019} a^{2} - \frac{145634}{425007} a - \frac{17986}{47223}$, $\frac{1}{158260775658308551236052670238805547952710951055911894018436637792202778504827903985171} a^{29} - \frac{5150149656476718383227641894069859194527495080557920589078018633193491117666502}{158260775658308551236052670238805547952710951055911894018436637792202778504827903985171} a^{28} - \frac{409957973079390277017290456387199541867824015270510604582698342235484366860225}{331783596767942455421494067586594440152433859655999777816429010046546705460855144623} a^{27} - \frac{2158646174080002575292572401436485946113702019446933199046457491564763785811395028}{158260775658308551236052670238805547952710951055911894018436637792202778504827903985171} a^{26} + \frac{15194264901140247947351944243341298691781006724323741267855531601605658630360956253}{158260775658308551236052670238805547952710951055911894018436637792202778504827903985171} a^{25} - \frac{8288442743050031998996811158352476994224908590188241396787671852186583094911698590}{52753591886102850412017556746268515984236983685303964672812212597400926168275967995057} a^{24} + \frac{24699995941612377619212996202835679598989334204863088738335112514650920193649394554}{52753591886102850412017556746268515984236983685303964672812212597400926168275967995057} a^{23} - \frac{85062548054929545792984091717465684530648003272075240369818160566020956284029565}{110594532255980818473831355862198146717477953218666592605476336682182235153618381541} a^{22} + \frac{4964589985581868180504564359759722106697726607783397953493886397302187212478915963}{52753591886102850412017556746268515984236983685303964672812212597400926168275967995057} a^{21} + \frac{534645659878959702329688063138880909557550223213717906227480679161624026654054964}{90486435482166124205862018432707574587027416269818121222662457285421828762051403079} a^{20} - \frac{81725258981222584842439388640598526578731985841098549149940119849661468425325425953}{17584530628700950137339185582089505328078994561767988224270737532466975389425322665019} a^{19} - \frac{3333520413648989067273560636182983346890704556571310505448628647684215216169565060}{1425772753678455416541015047196446377952350910413620666832762502632457464007458594461} a^{18} + \frac{155261309039196881531016223884142807065661277919120090776860433013432026965709704006}{52753591886102850412017556746268515984236983685303964672812212597400926168275967995057} a^{17} + \frac{2153143026620958367639431578660818193161242928306707808849496294245865794970867066}{995350790303827366264482202759783320457301578967999333449287030139640116382565433869} a^{16} + \frac{842717102797680326981020411316101449661501800873965390889322272967184021881185145502}{52753591886102850412017556746268515984236983685303964672812212597400926168275967995057} a^{15} - \frac{780678969814359440052025597784981869349377801161317265179124969282728154597619473190}{52753591886102850412017556746268515984236983685303964672812212597400926168275967995057} a^{14} + \frac{570869938488942730268838157564626746360744673631923239086555456256267138686698695114}{52753591886102850412017556746268515984236983685303964672812212597400926168275967995057} a^{13} - \frac{1499552484179422391995728918171176087329282854525051508318551116775163282998884027843}{52753591886102850412017556746268515984236983685303964672812212597400926168275967995057} a^{12} + \frac{7738264498288044955292083667360731878168838913977178798581240641170924009617209027872}{158260775658308551236052670238805547952710951055911894018436637792202778504827903985171} a^{11} + \frac{4671000685217555102450452573325787473699051704355116584857179282933662822123063093628}{158260775658308551236052670238805547952710951055911894018436637792202778504827903985171} a^{10} - \frac{1951209774314889650167666014977089026931535060055449905666211730905350658596580106739}{52753591886102850412017556746268515984236983685303964672812212597400926168275967995057} a^{9} + \frac{8402960704606455827459537994486146229007582606405952618710490968849685818151915905780}{158260775658308551236052670238805547952710951055911894018436637792202778504827903985171} a^{8} + \frac{16801742945224246569241890277971091732233824504795827488073826007839492167356397167269}{158260775658308551236052670238805547952710951055911894018436637792202778504827903985171} a^{7} - \frac{3059293665868966461141920804431640024202448832884584931582228487695800778722031601971}{52753591886102850412017556746268515984236983685303964672812212597400926168275967995057} a^{6} + \frac{2656521348149599642079314476242007401477296071476261506265895228006173462589842764145}{5861510209566983379113061860696501776026331520589329408090245844155658463141774221673} a^{5} + \frac{7467814579289960490652009895827140801769207797228409590080091105310093062880755947458}{17584530628700950137339185582089505328078994561767988224270737532466975389425322665019} a^{4} + \frac{2114857652960170053726178902026330935859876699343891984895874411040506357104134835556}{5861510209566983379113061860696501776026331520589329408090245844155658463141774221673} a^{3} - \frac{37198477453971199324389348497472465144777953294656037490332840910822611994041273918}{651278912174109264345895762299611308447370168954369934232249538239517607015752691297} a^{2} - \frac{2250363202225784213120108213989340612448808087737674281205303832699285730970801942032}{5861510209566983379113061860696501776026331520589329408090245844155658463141774221673} a + \frac{11200392239124752539998341490742681119828908935439835203115365731129496625450946253}{22457893523245147046410198699986596843012764446702411525249984077224745069508713493}$  Toggle raw display

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:  $15$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
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:  \( 15542059406474578 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{14}\cdot 15542059406474578 \cdot 7}{2\sqrt{86618284683850782149418819696007456920853807515055533}}\approx 110.496867882122$ (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

\(\Q(\sqrt{1213}) \), 3.1.3639.1, 5.1.13242321.1, 6.2.16062935373.1, 10.2.212710546411520733.1, 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: 30.0.214224941509935982232692876412219596671526317019923.1

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 $30$ $15^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $15^{2}$ $30$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{6}$ $30$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ $15^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{5}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{10}$ $30$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{3}$

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$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.2$x^{2} + 3$$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$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.3.2t1.a.a$1$ $ 3 $ \(\Q(\sqrt{-3}) \) $C_2$ (as 2T1) $1$ $-1$
1.3639.2t1.a.a$1$ $ 3 \cdot 1213 $ \(\Q(\sqrt{-3639}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.1213.2t1.a.a$1$ $ 1213 $ \(\Q(\sqrt{1213}) \) $C_2$ (as 2T1) $1$ $1$
* 2.3639.6t3.b.a$2$ $ 3 \cdot 1213 $ 6.0.39726963.2 $D_{6}$ (as 6T3) $1$ $0$
* 2.3639.3t2.a.a$2$ $ 3 \cdot 1213 $ 3.1.3639.1 $S_3$ (as 3T2) $1$ $0$
* 2.3639.5t2.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) $1$ $0$
* 2.3639.10t3.a.b$2$ $ 3 \cdot 1213 $ 10.0.526077196401123.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.3639.10t3.a.a$2$ $ 3 \cdot 1213 $ 10.0.526077196401123.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.3639.15t2.a.d$2$ $ 3 \cdot 1213 $ 15.1.8450344007000266933623879.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.3639.30t14.a.c$2$ $ 3 \cdot 1213 $ 30.2.86618284683850782149418819696007456920853807515055533.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.3639.15t2.a.c$2$ $ 3 \cdot 1213 $ 15.1.8450344007000266933623879.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.3639.30t14.a.d$2$ $ 3 \cdot 1213 $ 30.2.86618284683850782149418819696007456920853807515055533.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.3639.15t2.a.b$2$ $ 3 \cdot 1213 $ 15.1.8450344007000266933623879.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.3639.30t14.a.b$2$ $ 3 \cdot 1213 $ 30.2.86618284683850782149418819696007456920853807515055533.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.3639.15t2.a.a$2$ $ 3 \cdot 1213 $ 15.1.8450344007000266933623879.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.3639.30t14.a.a$2$ $ 3 \cdot 1213 $ 30.2.86618284683850782149418819696007456920853807515055533.1 $D_{30}$ (as 30T14) $1$ $0$

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 *.