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Properties

Label 30.2.493...181.1
Degree $30$
Signature $[2, 14]$
Discriminant $4.939\times 10^{51}$
Root discriminant $52.86$
Ramified primes $3, 7, 47$
Class number not computed
Class group not computed
Galois group $D_{30}$ (as 30T14)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 9*x^29 + 51*x^28 - 225*x^27 + 938*x^26 - 3065*x^25 + 9027*x^24 - 23897*x^23 + 55855*x^22 - 110855*x^21 + 206236*x^20 - 276419*x^19 + 363187*x^18 - 228654*x^17 - 26451*x^16 + 1021780*x^15 - 676090*x^14 + 4072978*x^13 + 297137*x^12 + 7819653*x^11 - 19045735*x^10 + 54647605*x^9 - 99924364*x^8 + 139085676*x^7 - 209111579*x^6 + 87062190*x^5 - 13968112*x^4 - 6858710*x^3 - 581573*x^2 - 428582*x - 60395)
 
gp: K = bnfinit(x^30 - 9*x^29 + 51*x^28 - 225*x^27 + 938*x^26 - 3065*x^25 + 9027*x^24 - 23897*x^23 + 55855*x^22 - 110855*x^21 + 206236*x^20 - 276419*x^19 + 363187*x^18 - 228654*x^17 - 26451*x^16 + 1021780*x^15 - 676090*x^14 + 4072978*x^13 + 297137*x^12 + 7819653*x^11 - 19045735*x^10 + 54647605*x^9 - 99924364*x^8 + 139085676*x^7 - 209111579*x^6 + 87062190*x^5 - 13968112*x^4 - 6858710*x^3 - 581573*x^2 - 428582*x - 60395, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-60395, -428582, -581573, -6858710, -13968112, 87062190, -209111579, 139085676, -99924364, 54647605, -19045735, 7819653, 297137, 4072978, -676090, 1021780, -26451, -228654, 363187, -276419, 206236, -110855, 55855, -23897, 9027, -3065, 938, -225, 51, -9, 1]);
 

\( x^{30} - 9 x^{29} + 51 x^{28} - 225 x^{27} + 938 x^{26} - 3065 x^{25} + 9027 x^{24} - 23897 x^{23} + 55855 x^{22} - 110855 x^{21} + 206236 x^{20} - 276419 x^{19} + 363187 x^{18} - 228654 x^{17} - 26451 x^{16} + 1021780 x^{15} - 676090 x^{14} + 4072978 x^{13} + 297137 x^{12} + 7819653 x^{11} - 19045735 x^{10} + 54647605 x^{9} - 99924364 x^{8} + 139085676 x^{7} - 209111579 x^{6} + 87062190 x^{5} - 13968112 x^{4} - 6858710 x^{3} - 581573 x^{2} - 428582 x - 60395 \)

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:  \(4939009167239391016458724484733673047883402323239181\)\(\medspace = 3^{15}\cdot 7^{25}\cdot 47^{14}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $52.86$
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:  $3, 7, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{5} a^{17} - \frac{2}{5} a^{16} - \frac{2}{5} a^{14} - \frac{1}{5} a^{13} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{5} a^{18} + \frac{1}{5} a^{16} - \frac{2}{5} a^{15} - \frac{2}{5} a^{13} + \frac{1}{5} a^{6} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5} a$, $\frac{1}{5} a^{19} + \frac{1}{5} a^{13} + \frac{1}{5} a^{7} + \frac{1}{5} a$, $\frac{1}{5} a^{20} + \frac{1}{5} a^{14} + \frac{1}{5} a^{8} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{21} + \frac{1}{5} a^{15} + \frac{1}{5} a^{9} + \frac{1}{5} a^{3}$, $\frac{1}{5} a^{22} + \frac{1}{5} a^{16} + \frac{1}{5} a^{10} + \frac{1}{5} a^{4}$, $\frac{1}{25} a^{23} - \frac{2}{25} a^{22} - \frac{1}{25} a^{21} + \frac{1}{25} a^{19} - \frac{2}{25} a^{18} + \frac{3}{25} a^{16} - \frac{7}{25} a^{15} + \frac{7}{25} a^{14} + \frac{1}{25} a^{13} - \frac{1}{5} a^{12} + \frac{6}{25} a^{11} + \frac{8}{25} a^{10} - \frac{1}{25} a^{9} + \frac{2}{5} a^{8} + \frac{11}{25} a^{7} + \frac{3}{25} a^{6} - \frac{1}{5} a^{5} + \frac{8}{25} a^{4} + \frac{8}{25} a^{3} + \frac{12}{25} a^{2} + \frac{1}{25} a + \frac{2}{5}$, $\frac{1}{275} a^{24} - \frac{2}{275} a^{23} + \frac{24}{275} a^{22} - \frac{3}{55} a^{21} - \frac{14}{275} a^{20} - \frac{27}{275} a^{19} - \frac{1}{55} a^{18} + \frac{13}{275} a^{17} + \frac{93}{275} a^{16} + \frac{2}{275} a^{15} - \frac{9}{275} a^{14} + \frac{19}{55} a^{13} - \frac{19}{275} a^{12} + \frac{3}{25} a^{11} - \frac{26}{275} a^{10} - \frac{21}{55} a^{9} - \frac{4}{275} a^{8} - \frac{97}{275} a^{7} - \frac{2}{55} a^{6} - \frac{7}{275} a^{5} - \frac{17}{275} a^{4} + \frac{7}{275} a^{3} + \frac{91}{275} a^{2} + \frac{2}{5} a + \frac{1}{11}$, $\frac{1}{1375} a^{25} - \frac{2}{1375} a^{24} - \frac{4}{275} a^{23} + \frac{73}{1375} a^{22} + \frac{17}{275} a^{21} - \frac{82}{1375} a^{20} - \frac{104}{1375} a^{19} - \frac{64}{1375} a^{18} - \frac{127}{1375} a^{17} - \frac{26}{275} a^{16} + \frac{684}{1375} a^{15} - \frac{378}{1375} a^{14} - \frac{393}{1375} a^{13} + \frac{48}{125} a^{12} + \frac{52}{275} a^{11} + \frac{643}{1375} a^{10} - \frac{91}{275} a^{9} + \frac{233}{1375} a^{8} + \frac{1}{1375} a^{7} + \frac{521}{1375} a^{6} + \frac{258}{1375} a^{5} - \frac{124}{275} a^{4} - \frac{151}{1375} a^{3} - \frac{28}{125} a^{2} - \frac{74}{1375} a - \frac{8}{25}$, $\frac{1}{64625} a^{26} + \frac{2}{64625} a^{25} + \frac{82}{64625} a^{24} - \frac{502}{64625} a^{23} + \frac{4942}{64625} a^{22} + \frac{1633}{64625} a^{21} - \frac{1147}{64625} a^{20} - \frac{39}{2585} a^{19} - \frac{1758}{64625} a^{18} + \frac{22}{5875} a^{17} + \frac{227}{1375} a^{16} - \frac{21622}{64625} a^{15} + \frac{4646}{12925} a^{14} - \frac{23319}{64625} a^{13} + \frac{15407}{64625} a^{12} - \frac{1542}{5875} a^{11} - \frac{22193}{64625} a^{10} - \frac{32112}{64625} a^{9} - \frac{24807}{64625} a^{8} + \frac{2316}{12925} a^{7} + \frac{19667}{64625} a^{6} - \frac{25658}{64625} a^{5} + \frac{30149}{64625} a^{4} + \frac{19658}{64625} a^{3} - \frac{20171}{64625} a^{2} - \frac{8986}{64625} a - \frac{6}{25}$, $\frac{1}{323125} a^{27} - \frac{2}{323125} a^{26} - \frac{114}{323125} a^{25} + \frac{486}{323125} a^{24} - \frac{302}{64625} a^{23} - \frac{27394}{323125} a^{22} + \frac{24281}{323125} a^{21} + \frac{18794}{323125} a^{20} + \frac{11824}{323125} a^{19} + \frac{22361}{323125} a^{18} + \frac{19947}{323125} a^{17} + \frac{145792}{323125} a^{16} - \frac{125564}{323125} a^{15} - \frac{25203}{64625} a^{14} + \frac{80577}{323125} a^{13} + \frac{49861}{323125} a^{12} + \frac{19001}{64625} a^{11} - \frac{145534}{323125} a^{10} - \frac{41354}{323125} a^{9} - \frac{156481}{323125} a^{8} - \frac{141286}{323125} a^{7} - \frac{126369}{323125} a^{6} - \frac{90328}{323125} a^{5} - \frac{147703}{323125} a^{4} + \frac{24974}{64625} a^{3} + \frac{26437}{323125} a^{2} + \frac{140331}{323125} a + \frac{1}{1375}$, $\frac{1}{337665625} a^{28} - \frac{7}{7184375} a^{27} - \frac{87}{13506625} a^{26} + \frac{311}{1615625} a^{25} - \frac{179317}{337665625} a^{24} - \frac{4170244}{337665625} a^{23} + \frac{24621634}{337665625} a^{22} + \frac{14394187}{337665625} a^{21} - \frac{11856414}{337665625} a^{20} + \frac{183987}{17771875} a^{19} - \frac{4205123}{67533125} a^{18} - \frac{20350887}{337665625} a^{17} - \frac{115068533}{337665625} a^{16} - \frac{99223147}{337665625} a^{15} - \frac{101873963}{337665625} a^{14} - \frac{127490553}{337665625} a^{13} + \frac{84443773}{337665625} a^{12} + \frac{2557944}{17771875} a^{11} - \frac{150985146}{337665625} a^{10} + \frac{96429632}{337665625} a^{9} - \frac{108770349}{337665625} a^{8} + \frac{57625118}{337665625} a^{7} - \frac{7324756}{67533125} a^{6} - \frac{87245932}{337665625} a^{5} - \frac{122636784}{337665625} a^{4} - \frac{136466713}{337665625} a^{3} - \frac{89749373}{337665625} a^{2} + \frac{126060553}{337665625} a - \frac{463897}{1436875}$, $\frac{1}{386085290684648376204535419121173766378164174839797739560869427778954821746470380025387730194248909493234375} a^{29} + \frac{185617199514742470535389696442428497680903932441470533898643883846376119389310393830581381795704278}{386085290684648376204535419121173766378164174839797739560869427778954821746470380025387730194248909493234375} a^{28} - \frac{55879951630136560230737832605510880573215037212826709432055359757755149961781086292330676724329802333}{386085290684648376204535419121173766378164174839797739560869427778954821746470380025387730194248909493234375} a^{27} + \frac{2736043764718944110585976663447089510380456755018849553771076356699089829989664503966204601738967778784}{386085290684648376204535419121173766378164174839797739560869427778954821746470380025387730194248909493234375} a^{26} + \frac{51544685991679733079258197210717082331276463565593571120853644784753903610443337158419543953359580658021}{386085290684648376204535419121173766378164174839797739560869427778954821746470380025387730194248909493234375} a^{25} + \frac{664570138244472611213386065147362176217946567393446172808982118114463728646019742301254355446267092897657}{386085290684648376204535419121173766378164174839797739560869427778954821746470380025387730194248909493234375} a^{24} - \frac{7312902900375723214601844554763452945887882125559980768674705645138628556901609790414145158083261618950749}{386085290684648376204535419121173766378164174839797739560869427778954821746470380025387730194248909493234375} a^{23} - \frac{5764176010496969507393096614791025316485277162467466888451049914821178966250959651627432472782102398996201}{77217058136929675240907083824234753275632834967959547912173885555790964349294076005077546038849781898646875} a^{22} + \frac{991622828289122549631284635775563251722735661665362199290067068885481057473773329454879249482018821328658}{77217058136929675240907083824234753275632834967959547912173885555790964349294076005077546038849781898646875} a^{21} + \frac{4687150393225020460709350956904903871007477664906239275320581571755482992435565698680924202605799099476172}{77217058136929675240907083824234753275632834967959547912173885555790964349294076005077546038849781898646875} a^{20} - \frac{4125517239174872457960972970902172281796196634988717906728815272166765311070958636159334770225607483483714}{386085290684648376204535419121173766378164174839797739560869427778954821746470380025387730194248909493234375} a^{19} - \frac{38252004468896622390679899477318210040068676194112400018157189970033665766228984593119718448232326463726872}{386085290684648376204535419121173766378164174839797739560869427778954821746470380025387730194248909493234375} a^{18} - \frac{18827019025044358269088506774672305493871731716427562826270836125199138408693001573305858814549479892302502}{386085290684648376204535419121173766378164174839797739560869427778954821746470380025387730194248909493234375} a^{17} + \frac{51642158268713573037023677400938401060016559138627906816180845982454688168205441513921704792799004971050612}{386085290684648376204535419121173766378164174839797739560869427778954821746470380025387730194248909493234375} a^{16} + \frac{86633256889694493385260551000716178051242065442673403083326801587381237022482192012191371995702992959827603}{386085290684648376204535419121173766378164174839797739560869427778954821746470380025387730194248909493234375} a^{15} - \frac{1477073505746753892270422842777016858634767369229359223800029140323588518583718282718188043836598543478626}{20320278457086756642343969427430198230429693412620933661098390935734464302445809475020406852328889973328125} a^{14} + \frac{8275155594951583799929674991105183049959990343861263082027892289909260975503978467377478508569621745591322}{35098662789513488745866856283743069670742197712708885414624493434450438340588216365944339108568082681203125} a^{13} + \frac{156730460540699190497468039404496865803234865185076003685633885321405723764961118569570854710462924591270092}{386085290684648376204535419121173766378164174839797739560869427778954821746470380025387730194248909493234375} a^{12} - \frac{171027134115545794732239738152057759602959233470872314325593120542831884787004173389437286608915199981706594}{386085290684648376204535419121173766378164174839797739560869427778954821746470380025387730194248909493234375} a^{11} - \frac{277469818949288588584953721834686226554432670509993972357041788146967195832496731927191096137878317415174}{7019732557902697749173371256748613934148439542541777082924898686890087668117643273188867821713616536240625} a^{10} + \frac{22518702443903495586228017348471375723287268667882365217154216301055814306723323286154876542089787541326154}{77217058136929675240907083824234753275632834967959547912173885555790964349294076005077546038849781898646875} a^{9} - \frac{21216637080438370134432365133475115522619509220551071583207026156268790024472881114372113220793528144275339}{77217058136929675240907083824234753275632834967959547912173885555790964349294076005077546038849781898646875} a^{8} + \frac{17236981683254036473221219246726958101524184886754227373402152021195509945844658867330904696095462203236941}{35098662789513488745866856283743069670742197712708885414624493434450438340588216365944339108568082681203125} a^{7} - \frac{784905910647477228558048750637318322253761976766014572103537942892598624824326862571780286455077684954566}{5762467025144005614993065957032444274300957833429817008371185489238131667857766866050563137227595664078125} a^{6} - \frac{2456126412348183392358429187172116594743311816347561459497410276051610278866822166575008926449640199479818}{386085290684648376204535419121173766378164174839797739560869427778954821746470380025387730194248909493234375} a^{5} + \frac{99906383902528560445060297426892770764773811500844843503634854290885328278764695420612383270953924408184}{397616159304478245318780040289571335095946627023478619527156980204896829810989062847979124813850576203125} a^{4} - \frac{8149574712065945671006655496998904381528647609105326179381772132908392239067443659526033408406223875962978}{29698868514203721246502724547782597413704936526138287658528417521458063211266952309645210014942223807171875} a^{3} + \frac{2952059334831942661429148492134294636195209154330126148517765955254543931492040135642696596434499352316884}{16786316986289059834979800831355381146876703253904249546124757729519774858542190435886423051923865630140625} a^{2} + \frac{70156580359670021592873781578439882585280853462953367213600257174564956899537317088560092064872992637017546}{386085290684648376204535419121173766378164174839797739560869427778954821746470380025387730194248909493234375} a - \frac{234445300840284729060559627011260449258417356247694968250174046307221419718456166167705007438016302611284}{1642916130572971813636320932430526665438996488679990381110082671399807752112639915001649915720208125503125}$

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

Class group and class number

not computed

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 $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) $ not computed

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

\(\Q(\sqrt{21}) \), 3.1.2303.1, 5.1.2209.1, 6.2.1002419901.1, 10.2.19929110051781.1, 15.1.143108492101942920287.1

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

Sibling fields

Degree 30 sibling: 30.0.232133430860251377773560050782482633250519909192241507.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $30$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{15}$ $15^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{15}$ $15^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ R $30$ $15^{2}$

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$3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
7Data not computed
$47$$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.987.2t1.a.a$1$ $ 3 \cdot 7 \cdot 47 $ \(\Q(\sqrt{-987}) \) $C_2$ (as 2T1) $1$ $-1$
1.47.2t1.a.a$1$ $ 47 $ \(\Q(\sqrt{-47}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.21.2t1.a.a$1$ $ 3 \cdot 7 $ \(\Q(\sqrt{21}) \) $C_2$ (as 2T1) $1$ $1$
* 2.20727.6t3.a.a$2$ $ 3^{2} \cdot 7^{2} \cdot 47 $ 6.0.47113735347.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.2303.3t2.a.a$2$ $ 7^{2} \cdot 47 $ 3.1.2303.1 $S_3$ (as 3T2) $1$ $0$
* 2.47.5t2.a.b$2$ $ 47 $ 5.1.2209.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.47.5t2.a.a$2$ $ 47 $ 5.1.2209.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.20727.10t3.b.b$2$ $ 3^{2} \cdot 7^{2} \cdot 47 $ 10.0.936668172433707.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.20727.10t3.b.a$2$ $ 3^{2} \cdot 7^{2} \cdot 47 $ 10.0.936668172433707.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.20727.30t14.b.c$2$ $ 3^{2} \cdot 7^{2} \cdot 47 $ 30.2.4939009167239391016458724484733673047883402323239181.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.2303.15t2.a.a$2$ $ 7^{2} \cdot 47 $ 15.1.143108492101942920287.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.2303.15t2.a.c$2$ $ 7^{2} \cdot 47 $ 15.1.143108492101942920287.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.20727.30t14.b.a$2$ $ 3^{2} \cdot 7^{2} \cdot 47 $ 30.2.4939009167239391016458724484733673047883402323239181.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.2303.15t2.a.b$2$ $ 7^{2} \cdot 47 $ 15.1.143108492101942920287.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.20727.30t14.b.b$2$ $ 3^{2} \cdot 7^{2} \cdot 47 $ 30.2.4939009167239391016458724484733673047883402323239181.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.2303.15t2.a.d$2$ $ 7^{2} \cdot 47 $ 15.1.143108492101942920287.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.20727.30t14.b.d$2$ $ 3^{2} \cdot 7^{2} \cdot 47 $ 30.2.4939009167239391016458724484733673047883402323239181.1 $D_{30}$ (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 *.