Properties

Label 30.2.110...000.1
Degree $30$
Signature $[2, 14]$
Discriminant $1.109\times 10^{48}$
Root discriminant $39.95$
Ramified primes $2, 5, 179$
Class number $9$ (GRH)
Class group $[3, 3]$ (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 + 21*x^28 - 42*x^27 + 225*x^26 - 359*x^25 + 651*x^24 - 2808*x^23 - 1822*x^22 - 11536*x^21 - 10414*x^20 - 26496*x^19 - 25330*x^18 - 33066*x^17 - 28312*x^16 - 19700*x^15 - 18200*x^14 + 6446*x^13 - 8210*x^12 + 13412*x^11 - 5750*x^10 + 6200*x^9 - 4422*x^8 + 1276*x^7 - 711*x^6 + 161*x^5 - 113*x^4 - 42*x^3 + 3*x^2 - 3*x - 1)
 
gp: K = bnfinit(x^30 - 3*x^29 + 21*x^28 - 42*x^27 + 225*x^26 - 359*x^25 + 651*x^24 - 2808*x^23 - 1822*x^22 - 11536*x^21 - 10414*x^20 - 26496*x^19 - 25330*x^18 - 33066*x^17 - 28312*x^16 - 19700*x^15 - 18200*x^14 + 6446*x^13 - 8210*x^12 + 13412*x^11 - 5750*x^10 + 6200*x^9 - 4422*x^8 + 1276*x^7 - 711*x^6 + 161*x^5 - 113*x^4 - 42*x^3 + 3*x^2 - 3*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -3, 3, -42, -113, 161, -711, 1276, -4422, 6200, -5750, 13412, -8210, 6446, -18200, -19700, -28312, -33066, -25330, -26496, -10414, -11536, -1822, -2808, 651, -359, 225, -42, 21, -3, 1]);
 

\( x^{30} - 3 x^{29} + 21 x^{28} - 42 x^{27} + 225 x^{26} - 359 x^{25} + 651 x^{24} - 2808 x^{23} - 1822 x^{22} - 11536 x^{21} - 10414 x^{20} - 26496 x^{19} - 25330 x^{18} - 33066 x^{17} - 28312 x^{16} - 19700 x^{15} - 18200 x^{14} + 6446 x^{13} - 8210 x^{12} + 13412 x^{11} - 5750 x^{10} + 6200 x^{9} - 4422 x^{8} + 1276 x^{7} - 711 x^{6} + 161 x^{5} - 113 x^{4} - 42 x^{3} + 3 x^{2} - 3 x - 1 \)

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:  \(1109410857892705938292621064252192000000000000000\)\(\medspace = 2^{20}\cdot 5^{15}\cdot 179^{14}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $39.95$
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:  $2, 5, 179$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{12} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{12} - \frac{1}{2} a^{6} - \frac{1}{4} a^{3} - \frac{1}{4}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{13} - \frac{1}{2} a^{7} - \frac{1}{4} a^{4} - \frac{1}{4} a$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{14} - \frac{1}{2} a^{8} - \frac{1}{4} a^{5} - \frac{1}{4} a^{2}$, $\frac{1}{28} a^{18} + \frac{1}{28} a^{17} + \frac{3}{28} a^{16} - \frac{1}{14} a^{15} + \frac{5}{28} a^{14} + \frac{1}{28} a^{13} + \frac{3}{28} a^{12} + \frac{1}{14} a^{11} - \frac{1}{7} a^{10} - \frac{1}{14} a^{9} + \frac{3}{14} a^{7} + \frac{1}{28} a^{6} + \frac{1}{28} a^{5} - \frac{1}{4} a^{4} - \frac{5}{28} a^{2} + \frac{5}{28} a - \frac{1}{28}$, $\frac{1}{56} a^{19} - \frac{1}{56} a^{18} - \frac{3}{28} a^{17} + \frac{3}{28} a^{16} + \frac{1}{28} a^{15} + \frac{3}{14} a^{14} - \frac{13}{56} a^{13} + \frac{3}{56} a^{12} - \frac{1}{7} a^{11} - \frac{1}{7} a^{10} + \frac{1}{14} a^{9} - \frac{1}{7} a^{8} + \frac{3}{56} a^{7} + \frac{13}{56} a^{6} + \frac{13}{28} a^{5} - \frac{1}{4} a^{4} - \frac{13}{28} a^{3} + \frac{1}{7} a^{2} - \frac{11}{56} a + \frac{9}{56}$, $\frac{1}{56} a^{20} - \frac{1}{56} a^{18} + \frac{3}{28} a^{17} - \frac{1}{28} a^{16} + \frac{1}{28} a^{15} + \frac{1}{56} a^{14} - \frac{1}{14} a^{13} + \frac{13}{56} a^{12} - \frac{1}{14} a^{11} + \frac{3}{14} a^{9} - \frac{5}{56} a^{8} + \frac{3}{7} a^{7} + \frac{17}{56} a^{6} + \frac{9}{28} a^{5} - \frac{13}{28} a^{4} + \frac{5}{28} a^{3} - \frac{5}{56} a^{2} - \frac{25}{56}$, $\frac{1}{56} a^{21} - \frac{1}{56} a^{18} + \frac{1}{14} a^{16} + \frac{1}{56} a^{15} - \frac{1}{7} a^{14} + \frac{1}{7} a^{13} - \frac{5}{56} a^{12} + \frac{1}{7} a^{11} + \frac{11}{56} a^{9} + \frac{2}{7} a^{8} - \frac{2}{7} a^{7} - \frac{3}{56} a^{6} + \frac{1}{7} a^{5} - \frac{1}{14} a^{4} - \frac{17}{56} a^{3} + \frac{3}{7} a^{2} - \frac{3}{7} a - \frac{27}{56}$, $\frac{1}{392} a^{22} + \frac{1}{196} a^{21} + \frac{1}{392} a^{20} + \frac{1}{392} a^{19} - \frac{1}{392} a^{18} + \frac{11}{98} a^{17} - \frac{11}{392} a^{16} + \frac{5}{98} a^{15} + \frac{79}{392} a^{14} + \frac{55}{392} a^{13} + \frac{57}{392} a^{12} - \frac{10}{49} a^{11} - \frac{1}{8} a^{10} - \frac{31}{196} a^{9} - \frac{173}{392} a^{8} - \frac{149}{392} a^{7} + \frac{1}{8} a^{6} - \frac{19}{49} a^{5} - \frac{149}{392} a^{4} + \frac{18}{49} a^{3} + \frac{85}{392} a^{2} + \frac{19}{56} a - \frac{37}{392}$, $\frac{1}{392} a^{23} - \frac{3}{392} a^{21} - \frac{1}{392} a^{20} - \frac{3}{392} a^{19} + \frac{1}{98} a^{18} - \frac{43}{392} a^{17} + \frac{1}{28} a^{16} + \frac{25}{392} a^{15} - \frac{19}{392} a^{14} + \frac{3}{392} a^{13} - \frac{13}{196} a^{12} + \frac{27}{392} a^{11} + \frac{1}{49} a^{10} + \frac{5}{56} a^{9} + \frac{1}{392} a^{8} + \frac{95}{392} a^{7} - \frac{12}{49} a^{6} + \frac{15}{392} a^{5} + \frac{25}{196} a^{4} - \frac{15}{56} a^{3} + \frac{75}{392} a^{2} + \frac{173}{392} a + \frac{9}{196}$, $\frac{1}{392} a^{24} - \frac{1}{196} a^{21} - \frac{1}{98} a^{18} + \frac{5}{98} a^{17} + \frac{3}{196} a^{16} - \frac{9}{98} a^{15} - \frac{5}{49} a^{14} + \frac{3}{196} a^{13} - \frac{12}{49} a^{12} + \frac{5}{98} a^{11} + \frac{1}{14} a^{10} + \frac{23}{196} a^{9} + \frac{27}{98} a^{8} - \frac{11}{49} a^{7} + \frac{15}{49} a^{6} + \frac{3}{7} a^{5} - \frac{17}{196} a^{4} + \frac{3}{49} a^{3} - \frac{33}{98} a^{2} + \frac{9}{196} a - \frac{13}{392}$, $\frac{1}{392} a^{25} - \frac{3}{392} a^{21} + \frac{1}{196} a^{20} - \frac{1}{196} a^{19} - \frac{3}{392} a^{18} - \frac{4}{49} a^{17} + \frac{13}{196} a^{16} - \frac{1}{8} a^{15} - \frac{9}{196} a^{14} - \frac{5}{28} a^{13} - \frac{13}{392} a^{12} - \frac{6}{49} a^{11} + \frac{15}{98} a^{10} - \frac{37}{392} a^{9} - \frac{11}{28} a^{8} + \frac{79}{196} a^{7} + \frac{9}{56} a^{6} - \frac{16}{49} a^{5} + \frac{73}{196} a^{4} - \frac{19}{392} a^{3} - \frac{67}{196} a^{2} - \frac{111}{392} a + \frac{45}{392}$, $\frac{1}{2744} a^{26} + \frac{3}{2744} a^{25} - \frac{1}{2744} a^{23} - \frac{1}{1372} a^{22} - \frac{2}{343} a^{21} - \frac{15}{2744} a^{20} + \frac{2}{343} a^{19} + \frac{3}{343} a^{18} + \frac{17}{2744} a^{17} - \frac{38}{343} a^{16} - \frac{37}{343} a^{15} + \frac{9}{2744} a^{14} - \frac{153}{686} a^{13} - \frac{46}{343} a^{12} - \frac{471}{2744} a^{11} - \frac{153}{1372} a^{10} - \frac{3}{686} a^{9} - \frac{569}{2744} a^{8} - \frac{289}{686} a^{7} + \frac{153}{686} a^{6} + \frac{491}{2744} a^{5} + \frac{279}{686} a^{4} + \frac{277}{686} a^{3} + \frac{557}{1372} a^{2} - \frac{1315}{2744} a - \frac{15}{686}$, $\frac{1}{126224} a^{27} + \frac{5}{31556} a^{26} - \frac{5}{15778} a^{25} + \frac{41}{126224} a^{24} + \frac{11}{31556} a^{23} - \frac{37}{31556} a^{22} - \frac{13}{2254} a^{21} + \frac{271}{31556} a^{20} - \frac{517}{63112} a^{19} + \frac{73}{31556} a^{18} + \frac{1623}{15778} a^{17} + \frac{1809}{15778} a^{16} - \frac{1549}{63112} a^{15} - \frac{2703}{31556} a^{14} - \frac{12883}{63112} a^{13} - \frac{1823}{9016} a^{12} + \frac{7195}{31556} a^{11} - \frac{1293}{31556} a^{10} + \frac{3753}{31556} a^{9} + \frac{1825}{4508} a^{8} - \frac{2493}{9016} a^{7} + \frac{1699}{7889} a^{6} - \frac{123}{686} a^{5} - \frac{1244}{7889} a^{4} - \frac{523}{2576} a^{3} - \frac{5575}{15778} a^{2} - \frac{22845}{63112} a + \frac{6253}{126224}$, $\frac{1}{73588592} a^{28} + \frac{139}{36794296} a^{27} - \frac{2063}{36794296} a^{26} - \frac{3775}{10512656} a^{25} - \frac{24313}{36794296} a^{24} - \frac{2349}{3344936} a^{23} - \frac{288}{418117} a^{22} + \frac{20365}{36794296} a^{21} - \frac{276107}{36794296} a^{20} + \frac{323931}{36794296} a^{19} + \frac{2395}{1599752} a^{18} + \frac{324211}{5256328} a^{17} + \frac{2695639}{36794296} a^{16} - \frac{2644489}{36794296} a^{15} + \frac{964241}{5256328} a^{14} + \frac{384471}{9198574} a^{13} - \frac{7141649}{36794296} a^{12} + \frac{4730843}{36794296} a^{11} - \frac{446037}{18397148} a^{10} - \frac{1327201}{36794296} a^{9} - \frac{18294669}{36794296} a^{8} + \frac{2238895}{36794296} a^{7} - \frac{6295657}{36794296} a^{6} - \frac{6769445}{36794296} a^{5} + \frac{16417925}{73588592} a^{4} + \frac{6939231}{18397148} a^{3} - \frac{2652073}{9198574} a^{2} + \frac{19037589}{73588592} a - \frac{3637471}{18397148}$, $\frac{1}{132965980904100515651103455267744} a^{29} + \frac{36188735073007170380285}{9497570064578608260793103947696} a^{28} - \frac{379700217566415516910771649}{132965980904100515651103455267744} a^{27} - \frac{903176998530643750248947375}{5781129604526109376134932837728} a^{26} - \frac{5360311174915453717267809471}{4748785032289304130396551973848} a^{25} + \frac{53356696242464117338962312933}{132965980904100515651103455267744} a^{24} + \frac{2901285257293617856309237609}{6043908222913659802322884330352} a^{23} + \frac{30144482147926675483147611829}{66482990452050257825551727633872} a^{22} + \frac{118432116223276125356554297019}{16620747613012564456387931908468} a^{21} + \frac{2731772905524553404292746937}{755488527864207475290360541294} a^{20} - \frac{23538384753231694118243421651}{2890564802263054688067466418864} a^{19} - \frac{369443404532242573163460157113}{66482990452050257825551727633872} a^{18} - \frac{3801326048386382060173116043109}{33241495226025128912775863816936} a^{17} - \frac{3046156954568775063620807584853}{66482990452050257825551727633872} a^{16} + \frac{773084996515232446823181054549}{66482990452050257825551727633872} a^{15} - \frac{3696144850746994305475988545599}{66482990452050257825551727633872} a^{14} - \frac{16141477137503042892149629712397}{66482990452050257825551727633872} a^{13} - \frac{257790887402063672147266753101}{3021954111456829901161442165176} a^{12} - \frac{2331808770064783363433612969875}{9497570064578608260793103947696} a^{11} + \frac{912549050675458155326990977505}{9497570064578608260793103947696} a^{10} + \frac{130156583961223604199477382277}{3021954111456829901161442165176} a^{9} + \frac{5375444622677156215217263199237}{33241495226025128912775863816936} a^{8} + \frac{21137990440323514766981548501325}{66482990452050257825551727633872} a^{7} - \frac{439389009841565965225344860219}{6043908222913659802322884330352} a^{6} + \frac{15725209595268868452157819973359}{132965980904100515651103455267744} a^{5} - \frac{433661518439976534687795603185}{3021954111456829901161442165176} a^{4} + \frac{14001900952948143703070133256771}{132965980904100515651103455267744} a^{3} - \frac{18027355445324740739441780136253}{132965980904100515651103455267744} a^{2} - \frac{26518970972092265188802644102645}{66482990452050257825551727633872} a - \frac{41174248340291077061513561413309}{132965980904100515651103455267744}$

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

Class group and class number

$C_{3}\times C_{3}$, which has order $9$ (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:  \( 383760855207.4596 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{14}\cdot 383760855207.4596 \cdot 9}{2\sqrt{1109410857892705938292621064252192000000000000000}}\approx 0.980178403517299$ (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

\(\Q(\sqrt{5}) \), 3.1.716.1, 5.1.32041.1, 6.2.64082000.2, 10.2.3208205253125.1, 15.1.6029359418000239616.1

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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{3}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $30$ $30$ $15^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{15}$ $15^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $30$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{5}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{15}$ $15^{2}$

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
$2$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5Data not computed
$179$$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{179}$$x + 3$$1$$1$$0$Trivial$[\ ]$
179.2.1.2$x^{2} + 537$$2$$1$$1$$C_2$$[\ ]_{2}$
179.2.1.2$x^{2} + 537$$2$$1$$1$$C_2$$[\ ]_{2}$
179.2.1.2$x^{2} + 537$$2$$1$$1$$C_2$$[\ ]_{2}$
179.2.1.2$x^{2} + 537$$2$$1$$1$$C_2$$[\ ]_{2}$
179.2.1.2$x^{2} + 537$$2$$1$$1$$C_2$$[\ ]_{2}$
179.2.1.2$x^{2} + 537$$2$$1$$1$$C_2$$[\ ]_{2}$
179.2.1.2$x^{2} + 537$$2$$1$$1$$C_2$$[\ ]_{2}$
179.2.1.2$x^{2} + 537$$2$$1$$1$$C_2$$[\ ]_{2}$
179.2.1.2$x^{2} + 537$$2$$1$$1$$C_2$$[\ ]_{2}$
179.2.1.2$x^{2} + 537$$2$$1$$1$$C_2$$[\ ]_{2}$
179.2.1.2$x^{2} + 537$$2$$1$$1$$C_2$$[\ ]_{2}$
179.2.1.2$x^{2} + 537$$2$$1$$1$$C_2$$[\ ]_{2}$
179.2.1.2$x^{2} + 537$$2$$1$$1$$C_2$$[\ ]_{2}$
179.2.1.2$x^{2} + 537$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.895.2t1.a.a$1$ $ 5 \cdot 179 $ \(\Q(\sqrt{-895}) \) $C_2$ (as 2T1) $1$ $-1$
1.179.2t1.a.a$1$ $ 179 $ \(\Q(\sqrt{-179}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
* 2.17900.6t3.b.a$2$ $ 2^{2} \cdot 5^{2} \cdot 179 $ 6.0.11470678000.2 $D_{6}$ (as 6T3) $1$ $0$
* 2.716.3t2.a.a$2$ $ 2^{2} \cdot 179 $ 3.1.716.1 $S_3$ (as 3T2) $1$ $0$
* 2.179.5t2.a.b$2$ $ 179 $ 5.1.32041.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.179.5t2.a.a$2$ $ 179 $ 5.1.32041.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.4475.10t3.a.b$2$ $ 5^{2} \cdot 179 $ 10.0.574268740309375.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.4475.10t3.a.a$2$ $ 5^{2} \cdot 179 $ 10.0.574268740309375.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.716.15t2.a.c$2$ $ 2^{2} \cdot 179 $ 15.1.6029359418000239616.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.17900.30t14.a.d$2$ $ 2^{2} \cdot 5^{2} \cdot 179 $ 30.2.1109410857892705938292621064252192000000000000000.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.716.15t2.a.d$2$ $ 2^{2} \cdot 179 $ 15.1.6029359418000239616.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.17900.30t14.a.c$2$ $ 2^{2} \cdot 5^{2} \cdot 179 $ 30.2.1109410857892705938292621064252192000000000000000.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.716.15t2.a.b$2$ $ 2^{2} \cdot 179 $ 15.1.6029359418000239616.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.17900.30t14.a.b$2$ $ 2^{2} \cdot 5^{2} \cdot 179 $ 30.2.1109410857892705938292621064252192000000000000000.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.716.15t2.a.a$2$ $ 2^{2} \cdot 179 $ 15.1.6029359418000239616.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.17900.30t14.a.a$2$ $ 2^{2} \cdot 5^{2} \cdot 179 $ 30.2.1109410857892705938292621064252192000000000000000.1 $D_{30}$ (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 *.