Properties

Label 30.0.218...571.1
Degree $30$
Signature $[0, 15]$
Discriminant $-2.187\times 10^{41}$
Root discriminant $23.88$
Ramified primes $3, 7, 11$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{10}\times S_3$ (as 30T12)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - x^29 - 7*x^28 + 10*x^27 + 7*x^26 - 19*x^25 + 56*x^24 - 38*x^23 - 238*x^22 + 196*x^21 + 1084*x^20 + 255*x^19 - 3219*x^18 - 4472*x^17 + 2252*x^16 + 12483*x^15 + 13641*x^14 + 1966*x^13 - 11281*x^12 - 13340*x^11 - 5077*x^10 + 3378*x^9 + 6156*x^8 + 4833*x^7 + 3411*x^6 + 3672*x^5 + 4671*x^4 + 4536*x^3 + 2997*x^2 + 1215*x + 243)
 
gp: K = bnfinit(x^30 - x^29 - 7*x^28 + 10*x^27 + 7*x^26 - 19*x^25 + 56*x^24 - 38*x^23 - 238*x^22 + 196*x^21 + 1084*x^20 + 255*x^19 - 3219*x^18 - 4472*x^17 + 2252*x^16 + 12483*x^15 + 13641*x^14 + 1966*x^13 - 11281*x^12 - 13340*x^11 - 5077*x^10 + 3378*x^9 + 6156*x^8 + 4833*x^7 + 3411*x^6 + 3672*x^5 + 4671*x^4 + 4536*x^3 + 2997*x^2 + 1215*x + 243, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![243, 1215, 2997, 4536, 4671, 3672, 3411, 4833, 6156, 3378, -5077, -13340, -11281, 1966, 13641, 12483, 2252, -4472, -3219, 255, 1084, 196, -238, -38, 56, -19, 7, 10, -7, -1, 1]);
 

\( x^{30} - x^{29} - 7 x^{28} + 10 x^{27} + 7 x^{26} - 19 x^{25} + 56 x^{24} - 38 x^{23} - 238 x^{22} + 196 x^{21} + 1084 x^{20} + 255 x^{19} - 3219 x^{18} - 4472 x^{17} + 2252 x^{16} + 12483 x^{15} + 13641 x^{14} + 1966 x^{13} - 11281 x^{12} - 13340 x^{11} - 5077 x^{10} + 3378 x^{9} + 6156 x^{8} + 4833 x^{7} + 3411 x^{6} + 3672 x^{5} + 4671 x^{4} + 4536 x^{3} + 2997 x^{2} + 1215 x + 243 \)

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:  \(-218673142739125286650364496602923076372571\)\(\medspace = -\,3^{10}\cdot 7^{10}\cdot 11^{27}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $23.88$
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:  $3, 7, 11$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $10$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{3} a^{22} - \frac{1}{3} a^{21} - \frac{1}{3} a^{20} + \frac{1}{3} a^{19} + \frac{1}{3} a^{18} - \frac{1}{3} a^{17} - \frac{1}{3} a^{16} + \frac{1}{3} a^{15} - \frac{1}{3} a^{14} + \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{23} + \frac{1}{3} a^{21} - \frac{1}{3} a^{19} + \frac{1}{3} a^{17} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{24} - \frac{1}{9} a^{23} - \frac{1}{9} a^{22} + \frac{4}{9} a^{21} + \frac{1}{9} a^{20} - \frac{4}{9} a^{19} - \frac{1}{9} a^{18} + \frac{1}{9} a^{17} - \frac{1}{9} a^{16} + \frac{4}{9} a^{15} - \frac{2}{9} a^{14} + \frac{1}{9} a^{11} + \frac{2}{9} a^{10} - \frac{1}{3} a^{9} + \frac{4}{9} a^{7} - \frac{4}{9} a^{6} + \frac{4}{9} a^{5} + \frac{2}{9} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{25} + \frac{1}{9} a^{23} + \frac{2}{9} a^{21} - \frac{2}{9} a^{19} - \frac{1}{3} a^{18} - \frac{1}{3} a^{17} - \frac{1}{3} a^{16} - \frac{1}{9} a^{15} + \frac{1}{9} a^{14} + \frac{1}{3} a^{13} + \frac{1}{9} a^{12} + \frac{1}{3} a^{11} + \frac{2}{9} a^{10} + \frac{1}{3} a^{9} + \frac{4}{9} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{4}{9} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{27} a^{26} - \frac{1}{27} a^{25} - \frac{1}{27} a^{24} + \frac{4}{27} a^{23} + \frac{1}{27} a^{22} + \frac{5}{27} a^{21} + \frac{8}{27} a^{20} - \frac{8}{27} a^{19} - \frac{1}{27} a^{18} + \frac{13}{27} a^{17} - \frac{2}{27} a^{16} + \frac{1}{3} a^{15} + \frac{1}{3} a^{14} + \frac{10}{27} a^{13} + \frac{11}{27} a^{12} - \frac{4}{9} a^{11} - \frac{1}{3} a^{10} + \frac{4}{27} a^{9} + \frac{5}{27} a^{8} - \frac{5}{27} a^{7} + \frac{2}{27} a^{6} + \frac{4}{9} a^{4}$, $\frac{1}{16118811} a^{27} - \frac{175403}{16118811} a^{26} - \frac{218239}{5372937} a^{25} - \frac{519343}{16118811} a^{24} + \frac{550706}{5372937} a^{23} - \frac{2171594}{16118811} a^{22} - \frac{2681012}{5372937} a^{21} - \frac{1741096}{16118811} a^{20} - \frac{1864496}{16118811} a^{19} + \frac{6992216}{16118811} a^{18} + \frac{250573}{1790979} a^{17} - \frac{4271440}{16118811} a^{16} + \frac{675887}{1790979} a^{15} - \frac{6204290}{16118811} a^{14} + \frac{6597991}{16118811} a^{13} + \frac{3996766}{16118811} a^{12} + \frac{440647}{1790979} a^{11} - \frac{2066660}{16118811} a^{10} - \frac{5198174}{16118811} a^{9} + \frac{1452506}{16118811} a^{8} + \frac{6909142}{16118811} a^{7} + \frac{1708003}{16118811} a^{6} - \frac{535498}{1790979} a^{5} - \frac{1955864}{5372937} a^{4} - \frac{203687}{1790979} a^{3} - \frac{589217}{1790979} a^{2} - \frac{5286}{596993} a + \frac{11975}{596993}$, $\frac{1}{1333518366271850477416683} a^{28} + \frac{10977046988423792}{1333518366271850477416683} a^{27} + \frac{16309318526368448113565}{1333518366271850477416683} a^{26} - \frac{19908387478860157598174}{1333518366271850477416683} a^{25} - \frac{72357581942627208779798}{1333518366271850477416683} a^{24} - \frac{84853911085445036528983}{1333518366271850477416683} a^{23} + \frac{172667324619154738850228}{1333518366271850477416683} a^{22} + \frac{386013291403372209402091}{1333518366271850477416683} a^{21} - \frac{632894806546798574807569}{1333518366271850477416683} a^{20} + \frac{176942586394782718352518}{1333518366271850477416683} a^{19} - \frac{642524618350511710500287}{1333518366271850477416683} a^{18} - \frac{185621089848261291058085}{444506122090616825805561} a^{17} + \frac{136919766368507892523766}{444506122090616825805561} a^{16} + \frac{357239689204906294399612}{1333518366271850477416683} a^{15} - \frac{432222795977897830555441}{1333518366271850477416683} a^{14} + \frac{113454723600494365222133}{444506122090616825805561} a^{13} + \frac{70969151963881104568135}{444506122090616825805561} a^{12} - \frac{52020236019806855530295}{1333518366271850477416683} a^{11} - \frac{5451411102505778553064}{1333518366271850477416683} a^{10} - \frac{465000351090244064007116}{1333518366271850477416683} a^{9} - \frac{624499941357410486426332}{1333518366271850477416683} a^{8} - \frac{18518884131376722321074}{148168707363538941935187} a^{7} - \frac{73119468936340559216104}{444506122090616825805561} a^{6} - \frac{4456741546280316292514}{16463189707059882437243} a^{5} + \frac{22813190751193174067096}{148168707363538941935187} a^{4} + \frac{2640504131038370496125}{49389569121179647311729} a^{3} - \frac{5088190024893386561783}{49389569121179647311729} a^{2} - \frac{4309558803614081968073}{16463189707059882437243} a + \frac{458138978976263008360}{16463189707059882437243}$, $\frac{1}{1333518366271850477416683} a^{29} - \frac{12444758502189401}{1333518366271850477416683} a^{27} - \frac{2218761977979068313146}{148168707363538941935187} a^{26} + \frac{60775728820146114223049}{1333518366271850477416683} a^{25} + \frac{987285122501941335676}{49389569121179647311729} a^{24} - \frac{170672407578492158952593}{1333518366271850477416683} a^{23} + \frac{73919302866596826534125}{444506122090616825805561} a^{22} + \frac{34935396063055151624452}{444506122090616825805561} a^{21} + \frac{18660079588158131316023}{444506122090616825805561} a^{20} - \frac{86228114004983172078415}{1333518366271850477416683} a^{19} + \frac{590341409444573689033891}{1333518366271850477416683} a^{18} - \frac{185985905414224771137349}{444506122090616825805561} a^{17} - \frac{511833211750613682425729}{1333518366271850477416683} a^{16} + \frac{189154150978511306033815}{444506122090616825805561} a^{15} - \frac{205874286892989675314977}{1333518366271850477416683} a^{14} + \frac{155516539903186604475244}{444506122090616825805561} a^{13} - \frac{89842465140936175565948}{1333518366271850477416683} a^{12} + \frac{46386620640676287921239}{148168707363538941935187} a^{11} - \frac{44957085751331710666931}{148168707363538941935187} a^{10} + \frac{75211017231453293935798}{444506122090616825805561} a^{9} - \frac{264546298717554619631326}{1333518366271850477416683} a^{8} + \frac{37719157853539427239756}{148168707363538941935187} a^{7} - \frac{104251017533585648580313}{444506122090616825805561} a^{6} + \frac{1813881169196508772761}{16463189707059882437243} a^{5} + \frac{22454549488422209773237}{148168707363538941935187} a^{4} + \frac{6085639160847094091213}{49389569121179647311729} a^{3} + \frac{7073550056467715781296}{16463189707059882437243} a^{2} - \frac{383593736138215441566}{16463189707059882437243} a - \frac{4235109983577811421314}{16463189707059882437243}$

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

Class group and class number

Trivial group, which has order $1$ (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{14985944126668949796770}{444506122090616825805561} a^{29} - \frac{57916689768438880663813}{1333518366271850477416683} a^{28} - \frac{310995119440734449425775}{1333518366271850477416683} a^{27} + \frac{570759361142080836846808}{1333518366271850477416683} a^{26} + \frac{205542305643285951787943}{1333518366271850477416683} a^{25} - \frac{1145929077705595095381430}{1333518366271850477416683} a^{24} + \frac{3048528716784902297297278}{1333518366271850477416683} a^{23} - \frac{2451991912176888140242832}{1333518366271850477416683} a^{22} - \frac{11063715601165292668878010}{1333518366271850477416683} a^{21} + \frac{13911367511948149567990186}{1333518366271850477416683} a^{20} + \frac{45894595032825835522510586}{1333518366271850477416683} a^{19} - \frac{7409899204825608944141599}{1333518366271850477416683} a^{18} - \frac{49827236241543549134194300}{444506122090616825805561} a^{17} - \frac{49021381004627799702597941}{444506122090616825805561} a^{16} + \frac{174808008743312504409384794}{1333518366271850477416683} a^{15} + \frac{515661522125556506978389783}{1333518366271850477416683} a^{14} + \frac{15018328043639418064459739}{49389569121179647311729} a^{13} - \frac{11975206705689880210213799}{148168707363538941935187} a^{12} - \frac{494137991370875464772033209}{1333518366271850477416683} a^{11} - \frac{400469526139081526844869225}{1333518366271850477416683} a^{10} - \frac{41400014075348365025775271}{1333518366271850477416683} a^{9} + \frac{186951589892709425110358449}{1333518366271850477416683} a^{8} + \frac{68122579637294031654713024}{444506122090616825805561} a^{7} + \frac{14557323152659319920512899}{148168707363538941935187} a^{6} + \frac{10475848704164554417644293}{148168707363538941935187} a^{5} + \frac{13955815672782903681447167}{148168707363538941935187} a^{4} + \frac{1959045578039528792102941}{16463189707059882437243} a^{3} + \frac{5088934067583265898729254}{49389569121179647311729} a^{2} + \frac{911351614229557878857748}{16463189707059882437243} a + \frac{251388423805714886048849}{16463189707059882437243} \) (order $22$)
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:  \( 2830009383.1065817 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{15}\cdot 2830009383.1065817 \cdot 1}{22\sqrt{218673142739125286650364496602923076372571}}\approx 0.258324440068225$ (assuming GRH)

Galois group

$C_{10}\times S_3$ (as 30T12):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 60
The 30 conjugacy class representatives for $C_{10}\times S_3$
Character table for $C_{10}\times S_3$ is not computed

Intermediate fields

\(\Q(\sqrt{-11}) \), 3.1.231.1, \(\Q(\zeta_{11})^+\), 6.0.586971.1, \(\Q(\zeta_{11})\), 15.5.140994243189740741031.1

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

Sibling fields

Degree 30 sibling: Deg 30

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 $15^{2}$ R R $30$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{3}$ $30$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{10}$ $30$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ $15^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{15}$ $15^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ $15^{2}$

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$3.5.0.1$x^{5} - x + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
3.5.0.1$x^{5} - x + 1$$1$$5$$0$$C_5$$[\ ]^{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
11Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.21.2t1.a.a$1$ $ 3 \cdot 7 $ \(\Q(\sqrt{21}) \) $C_2$ (as 2T1) $1$ $1$
* 1.11.2t1.a.a$1$ $ 11 $ \(\Q(\sqrt{-11}) \) $C_2$ (as 2T1) $1$ $-1$
1.231.2t1.a.a$1$ $ 3 \cdot 7 \cdot 11 $ \(\Q(\sqrt{-231}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.11.5t1.a.a$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
* 1.11.10t1.a.a$1$ $ 11 $ \(\Q(\zeta_{11})\) $C_{10}$ (as 10T1) $0$ $-1$
1.231.10t1.a.a$1$ $ 3 \cdot 7 \cdot 11 $ 10.10.875463320250981.1 $C_{10}$ (as 10T1) $0$ $1$
1.231.10t1.b.a$1$ $ 3 \cdot 7 \cdot 11 $ 10.0.9630096522760791.1 $C_{10}$ (as 10T1) $0$ $-1$
1.231.10t1.b.b$1$ $ 3 \cdot 7 \cdot 11 $ 10.0.9630096522760791.1 $C_{10}$ (as 10T1) $0$ $-1$
* 1.11.10t1.a.b$1$ $ 11 $ \(\Q(\zeta_{11})\) $C_{10}$ (as 10T1) $0$ $-1$
* 1.11.10t1.a.c$1$ $ 11 $ \(\Q(\zeta_{11})\) $C_{10}$ (as 10T1) $0$ $-1$
1.231.10t1.a.b$1$ $ 3 \cdot 7 \cdot 11 $ 10.10.875463320250981.1 $C_{10}$ (as 10T1) $0$ $1$
* 1.11.10t1.a.d$1$ $ 11 $ \(\Q(\zeta_{11})\) $C_{10}$ (as 10T1) $0$ $-1$
1.231.10t1.a.c$1$ $ 3 \cdot 7 \cdot 11 $ 10.10.875463320250981.1 $C_{10}$ (as 10T1) $0$ $1$
* 1.11.5t1.a.b$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
1.231.10t1.a.d$1$ $ 3 \cdot 7 \cdot 11 $ 10.10.875463320250981.1 $C_{10}$ (as 10T1) $0$ $1$
* 1.11.5t1.a.c$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
1.231.10t1.b.c$1$ $ 3 \cdot 7 \cdot 11 $ 10.0.9630096522760791.1 $C_{10}$ (as 10T1) $0$ $-1$
* 1.11.5t1.a.d$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
1.231.10t1.b.d$1$ $ 3 \cdot 7 \cdot 11 $ 10.0.9630096522760791.1 $C_{10}$ (as 10T1) $0$ $-1$
* 2.231.6t3.b.a$2$ $ 3 \cdot 7 \cdot 11 $ 6.2.1120581.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.231.3t2.a.a$2$ $ 3 \cdot 7 \cdot 11 $ 3.1.231.1 $S_3$ (as 3T2) $1$ $0$
* 2.2541.15t4.b.a$2$ $ 3 \cdot 7 \cdot 11^{2}$ 15.5.140994243189740741031.1 $S_3 \times C_5$ (as 15T4) $0$ $0$
* 2.2541.15t4.b.b$2$ $ 3 \cdot 7 \cdot 11^{2}$ 15.5.140994243189740741031.1 $S_3 \times C_5$ (as 15T4) $0$ $0$
* 2.2541.15t4.b.c$2$ $ 3 \cdot 7 \cdot 11^{2}$ 15.5.140994243189740741031.1 $S_3 \times C_5$ (as 15T4) $0$ $0$
* 2.2541.15t4.b.d$2$ $ 3 \cdot 7 \cdot 11^{2}$ 15.5.140994243189740741031.1 $S_3 \times C_5$ (as 15T4) $0$ $0$
* 2.2541.30t12.b.a$2$ $ 3 \cdot 7 \cdot 11^{2}$ 30.0.218673142739125286650364496602923076372571.1 $C_{10}\times S_3$ (as 30T12) $0$ $0$
* 2.2541.30t12.b.b$2$ $ 3 \cdot 7 \cdot 11^{2}$ 30.0.218673142739125286650364496602923076372571.1 $C_{10}\times S_3$ (as 30T12) $0$ $0$
* 2.2541.30t12.b.c$2$ $ 3 \cdot 7 \cdot 11^{2}$ 30.0.218673142739125286650364496602923076372571.1 $C_{10}\times S_3$ (as 30T12) $0$ $0$
* 2.2541.30t12.b.d$2$ $ 3 \cdot 7 \cdot 11^{2}$ 30.0.218673142739125286650364496602923076372571.1 $C_{10}\times S_3$ (as 30T12) $0$ $0$

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 *.