Properties

Label 30.2.114...125.1
Degree $30$
Signature $[2, 14]$
Discriminant $1.145\times 10^{50}$
Root discriminant $46.63$
Ramified primes $5, 11, 61$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $D_{30}$ (as 30T14)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 3*x^29 + 9*x^28 + 14*x^27 - 113*x^26 + 370*x^25 - 643*x^24 - 15*x^23 + 3880*x^22 - 13302*x^21 + 28770*x^20 - 44390*x^19 + 53641*x^18 - 46319*x^17 + 35868*x^16 - 26390*x^15 + 12095*x^14 + 43878*x^13 + 45313*x^12 - 6914*x^11 - 59831*x^10 - 94496*x^9 - 64150*x^8 - 2932*x^7 + 54993*x^6 + 69070*x^5 + 52989*x^4 + 26576*x^3 + 8569*x^2 + 484*x - 121)
 
gp: K = bnfinit(x^30 - 3*x^29 + 9*x^28 + 14*x^27 - 113*x^26 + 370*x^25 - 643*x^24 - 15*x^23 + 3880*x^22 - 13302*x^21 + 28770*x^20 - 44390*x^19 + 53641*x^18 - 46319*x^17 + 35868*x^16 - 26390*x^15 + 12095*x^14 + 43878*x^13 + 45313*x^12 - 6914*x^11 - 59831*x^10 - 94496*x^9 - 64150*x^8 - 2932*x^7 + 54993*x^6 + 69070*x^5 + 52989*x^4 + 26576*x^3 + 8569*x^2 + 484*x - 121, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-121, 484, 8569, 26576, 52989, 69070, 54993, -2932, -64150, -94496, -59831, -6914, 45313, 43878, 12095, -26390, 35868, -46319, 53641, -44390, 28770, -13302, 3880, -15, -643, 370, -113, 14, 9, -3, 1]);
 

\( x^{30} - 3 x^{29} + 9 x^{28} + 14 x^{27} - 113 x^{26} + 370 x^{25} - 643 x^{24} - 15 x^{23} + 3880 x^{22} - 13302 x^{21} + 28770 x^{20} - 44390 x^{19} + 53641 x^{18} - 46319 x^{17} + 35868 x^{16} - 26390 x^{15} + 12095 x^{14} + 43878 x^{13} + 45313 x^{12} - 6914 x^{11} - 59831 x^{10} - 94496 x^{9} - 64150 x^{8} - 2932 x^{7} + 54993 x^{6} + 69070 x^{5} + 52989 x^{4} + 26576 x^{3} + 8569 x^{2} + 484 x - 121 \)

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:  \(114463058803512608608063235872919561806671142578125\)\(\medspace = 5^{15}\cdot 11^{14}\cdot 61^{14}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $46.63$
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:  $5, 11, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{11} a^{19} + \frac{3}{11} a^{18} - \frac{2}{11} a^{17} + \frac{1}{11} a^{16} + \frac{2}{11} a^{15} + \frac{3}{11} a^{14} + \frac{5}{11} a^{13} - \frac{4}{11} a^{12} - \frac{2}{11} a^{11} - \frac{4}{11} a^{10} + \frac{2}{11} a^{9} + \frac{3}{11} a^{8} + \frac{3}{11} a^{7} + \frac{1}{11} a^{6} + \frac{2}{11} a^{5} + \frac{5}{11} a^{4} + \frac{3}{11} a^{2}$, $\frac{1}{11} a^{20} - \frac{4}{11} a^{17} - \frac{1}{11} a^{16} - \frac{3}{11} a^{15} - \frac{4}{11} a^{14} + \frac{3}{11} a^{13} - \frac{1}{11} a^{12} + \frac{2}{11} a^{11} + \frac{3}{11} a^{10} - \frac{3}{11} a^{9} + \frac{5}{11} a^{8} + \frac{3}{11} a^{7} - \frac{1}{11} a^{6} - \frac{1}{11} a^{5} - \frac{4}{11} a^{4} + \frac{3}{11} a^{3} + \frac{2}{11} a^{2}$, $\frac{1}{11} a^{21} - \frac{4}{11} a^{18} - \frac{1}{11} a^{17} - \frac{3}{11} a^{16} - \frac{4}{11} a^{15} + \frac{3}{11} a^{14} - \frac{1}{11} a^{13} + \frac{2}{11} a^{12} + \frac{3}{11} a^{11} - \frac{3}{11} a^{10} + \frac{5}{11} a^{9} + \frac{3}{11} a^{8} - \frac{1}{11} a^{7} - \frac{1}{11} a^{6} - \frac{4}{11} a^{5} + \frac{3}{11} a^{4} + \frac{2}{11} a^{3}$, $\frac{1}{11} a^{22} - \frac{2}{11} a^{12} + \frac{1}{11} a^{2}$, $\frac{1}{11} a^{23} - \frac{2}{11} a^{13} + \frac{1}{11} a^{3}$, $\frac{1}{11} a^{24} - \frac{2}{11} a^{14} + \frac{1}{11} a^{4}$, $\frac{1}{11} a^{25} - \frac{2}{11} a^{15} + \frac{1}{11} a^{5}$, $\frac{1}{121} a^{26} + \frac{1}{121} a^{25} - \frac{5}{121} a^{24} + \frac{4}{121} a^{23} + \frac{5}{121} a^{22} - \frac{3}{121} a^{21} - \frac{3}{121} a^{20} + \frac{4}{121} a^{19} + \frac{24}{121} a^{18} + \frac{7}{121} a^{17} + \frac{3}{121} a^{16} - \frac{28}{121} a^{15} - \frac{41}{121} a^{14} - \frac{60}{121} a^{13} - \frac{40}{121} a^{12} + \frac{43}{121} a^{11} - \frac{49}{121} a^{10} + \frac{46}{121} a^{9} + \frac{54}{121} a^{8} - \frac{60}{121} a^{7} + \frac{13}{121} a^{5} - \frac{59}{121} a^{4} + \frac{2}{11} a^{3} + \frac{2}{11} a^{2}$, $\frac{1}{299959} a^{27} + \frac{204}{299959} a^{26} - \frac{108}{2479} a^{25} + \frac{1035}{299959} a^{24} + \frac{6559}{299959} a^{23} + \frac{1175}{27269} a^{22} + \frac{6802}{299959} a^{21} + \frac{412}{27269} a^{20} + \frac{1145}{27269} a^{19} + \frac{62772}{299959} a^{18} + \frac{69734}{299959} a^{17} + \frac{127741}{299959} a^{16} + \frac{101503}{299959} a^{15} + \frac{88505}{299959} a^{14} - \frac{14992}{299959} a^{13} + \frac{115156}{299959} a^{12} + \frac{108549}{299959} a^{11} - \frac{139030}{299959} a^{10} + \frac{85182}{299959} a^{9} - \frac{23990}{299959} a^{8} + \frac{4837}{299959} a^{7} - \frac{35990}{299959} a^{6} - \frac{7683}{299959} a^{5} - \frac{89901}{299959} a^{4} + \frac{345}{737} a^{3} + \frac{13627}{27269} a^{2} + \frac{798}{2479} a + \frac{288}{2479}$, $\frac{1}{130821683492797} a^{28} - \frac{175586907}{130821683492797} a^{27} - \frac{30086998630}{11892880317527} a^{26} + \frac{1770560150226}{130821683492797} a^{25} - \frac{2444924461061}{130821683492797} a^{24} - \frac{1184756940473}{130821683492797} a^{23} - \frac{1465845665477}{130821683492797} a^{22} - \frac{210156322812}{11892880317527} a^{21} - \frac{1193575272598}{130821683492797} a^{20} - \frac{1549060590035}{130821683492797} a^{19} - \frac{62606628895258}{130821683492797} a^{18} - \frac{986012447287}{3535721175481} a^{17} - \frac{128079399533}{130821683492797} a^{16} + \frac{25678222895063}{130821683492797} a^{15} + \frac{31986578084771}{130821683492797} a^{14} - \frac{1328759311520}{11892880317527} a^{13} - \frac{26037714705604}{130821683492797} a^{12} - \frac{17877882414926}{130821683492797} a^{11} + \frac{10242591762715}{130821683492797} a^{10} + \frac{304938592996}{1081170937957} a^{9} - \frac{32076537984774}{130821683492797} a^{8} + \frac{1395903965088}{11892880317527} a^{7} - \frac{54000319054690}{130821683492797} a^{6} + \frac{870478855678}{11892880317527} a^{5} - \frac{17578677391237}{130821683492797} a^{4} - \frac{4190059167901}{11892880317527} a^{3} + \frac{4776014271441}{11892880317527} a^{2} + \frac{278547752728}{1081170937957} a - \frac{97432074220}{1081170937957}$, $\frac{1}{292006327359192399436863518058612942784463902474613526740743} a^{29} + \frac{126005453173180612188454809220204616102615944}{292006327359192399436863518058612942784463902474613526740743} a^{28} + \frac{376373793093353354197599213506313580295496403919283864}{292006327359192399436863518058612942784463902474613526740743} a^{27} - \frac{51457052977090114883099460431924556867105611280205157860}{292006327359192399436863518058612942784463902474613526740743} a^{26} - \frac{7102682936997524412004273270528169372431816380722549390743}{292006327359192399436863518058612942784463902474613526740743} a^{25} - \frac{13046716631194230902227457788487485902623902633712024188086}{292006327359192399436863518058612942784463902474613526740743} a^{24} - \frac{6956041283988356571594010629673250466713520141388505610122}{292006327359192399436863518058612942784463902474613526740743} a^{23} + \frac{8445864539775305554629470486396098281040024087650968121170}{292006327359192399436863518058612942784463902474613526740743} a^{22} + \frac{10018438917551142115265879122463827873764099944897215140}{7122105545346156083825939464844218116694241523771061627823} a^{21} + \frac{1194057674674597851665391844463508457198320082241003916926}{26546029759926581766987592550782994798587627497692138794613} a^{20} - \frac{1083690262917713174826274723459379633615778537902022159161}{292006327359192399436863518058612942784463902474613526740743} a^{19} + \frac{30864928510758896273240588136544415494966952687191625750218}{292006327359192399436863518058612942784463902474613526740743} a^{18} + \frac{23198837571528583494330113768013261093950278339460621793559}{292006327359192399436863518058612942784463902474613526740743} a^{17} - \frac{1134076684148632851960193789732848437547711262851323529749}{7892062901599794579374689677259809264444970337151716938939} a^{16} + \frac{1870562319409391172353419070585312387559632453697583498951}{7122105545346156083825939464844218116694241523771061627823} a^{15} + \frac{135732365793399133547269184680395601384665953632517190231380}{292006327359192399436863518058612942784463902474613526740743} a^{14} - \frac{98418032712308572863482075229173676052471189528576360068876}{292006327359192399436863518058612942784463902474613526740743} a^{13} + \frac{115688153010387608984784885045069470273815229589057725874343}{292006327359192399436863518058612942784463902474613526740743} a^{12} + \frac{118571497989137154192151899640944763574943824419145353919730}{292006327359192399436863518058612942784463902474613526740743} a^{11} - \frac{48535718372927576623798219983707310509709144322953708499535}{292006327359192399436863518058612942784463902474613526740743} a^{10} + \frac{20835868328683605076195573149159768823340115139843723022169}{292006327359192399436863518058612942784463902474613526740743} a^{9} + \frac{31963929608640684884567877593069823301810487255225550451802}{292006327359192399436863518058612942784463902474613526740743} a^{8} - \frac{13719965049965218559696944961561782671567294958279687576055}{292006327359192399436863518058612942784463902474613526740743} a^{7} - \frac{32280214243749729483742271799228645383080328925126157237584}{292006327359192399436863518058612942784463902474613526740743} a^{6} - \frac{141091896257597601905292856035428228120377285133785625836477}{292006327359192399436863518058612942784463902474613526740743} a^{5} - \frac{83463952927010781031490257319504999662840659598312537587745}{292006327359192399436863518058612942784463902474613526740743} a^{4} - \frac{4577526070370831646944155292355662625875287225457951092732}{26546029759926581766987592550782994798587627497692138794613} a^{3} + \frac{5286312736589727969465172563984951165501971159697663296650}{26546029759926581766987592550782994798587627497692138794613} a^{2} - \frac{1048492186094320198178136244579022905457213494444122673329}{2413275432720598342453417504616635890780693408881103526783} a + \frac{1037857669233767302148446295021292277110326267202813128760}{2413275432720598342453417504616635890780693408881103526783}$

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

Class group and class number

$C_{2}$, which has order $2$ (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 $2$)
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:  \( 16637349953072.254 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{14}\cdot 16637349953072.254 \cdot 2}{2\sqrt{114463058803512608608063235872919561806671142578125}}\approx 0.929671733202842$ (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

\(\Q(\sqrt{5}) \), 3.1.671.1, 5.1.450241.1, 6.2.56280125.1, 10.2.633490494003125.1, 15.1.61243167054566186591.1

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 $30$ $30$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{5}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{15}$ $30$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{3}$ $30$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
$61$$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.3355.2t1.a.a$1$ $ 5 \cdot 11 \cdot 61 $ \(\Q(\sqrt{-3355}) \) $C_2$ (as 2T1) $1$ $-1$
1.671.2t1.a.a$1$ $ 11 \cdot 61 $ \(\Q(\sqrt{-671}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
* 2.16775.6t3.d.a$2$ $ 5^{2} \cdot 11 \cdot 61 $ 6.0.37763963875.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.671.3t2.a.a$2$ $ 11 \cdot 61 $ 3.1.671.1 $S_3$ (as 3T2) $1$ $0$
* 2.671.5t2.a.a$2$ $ 11 \cdot 61 $ 5.1.450241.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.671.5t2.a.b$2$ $ 11 \cdot 61 $ 5.1.450241.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.16775.10t3.a.a$2$ $ 5^{2} \cdot 11 \cdot 61 $ 10.0.425072121476096875.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.16775.10t3.a.b$2$ $ 5^{2} \cdot 11 \cdot 61 $ 10.0.425072121476096875.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.16775.30t14.a.d$2$ $ 5^{2} \cdot 11 \cdot 61 $ 30.2.114463058803512608608063235872919561806671142578125.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.671.15t2.a.d$2$ $ 11 \cdot 61 $ 15.1.61243167054566186591.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.671.15t2.a.b$2$ $ 11 \cdot 61 $ 15.1.61243167054566186591.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.16775.30t14.a.b$2$ $ 5^{2} \cdot 11 \cdot 61 $ 30.2.114463058803512608608063235872919561806671142578125.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.671.15t2.a.a$2$ $ 11 \cdot 61 $ 15.1.61243167054566186591.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.16775.30t14.a.c$2$ $ 5^{2} \cdot 11 \cdot 61 $ 30.2.114463058803512608608063235872919561806671142578125.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.671.15t2.a.c$2$ $ 11 \cdot 61 $ 15.1.61243167054566186591.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.16775.30t14.a.a$2$ $ 5^{2} \cdot 11 \cdot 61 $ 30.2.114463058803512608608063235872919561806671142578125.1 $D_{30}$ (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 *.