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Properties

Label 30.2.515...625.1
Degree $30$
Signature $[2, 14]$
Discriminant $5.159\times 10^{49}$
Root discriminant $45.40$
Ramified primes $3, 5, 47$
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 - 10*x^29 + 36*x^28 - 34*x^27 - 129*x^26 + 673*x^25 - 2345*x^24 + 3400*x^23 + 8195*x^22 - 23985*x^21 - 13829*x^20 + 36240*x^19 + 94321*x^18 + 102411*x^17 + 62946*x^16 + 325659*x^15 + 1299805*x^14 + 3826146*x^13 + 7335516*x^12 + 9580506*x^11 + 8499204*x^10 + 5247290*x^9 + 1937111*x^8 + 388946*x^7 - 47819*x^6 - 74039*x^5 - 43685*x^4 - 5057*x^3 + 3493*x^2 + 123*x + 169)
 
gp: K = bnfinit(x^30 - 10*x^29 + 36*x^28 - 34*x^27 - 129*x^26 + 673*x^25 - 2345*x^24 + 3400*x^23 + 8195*x^22 - 23985*x^21 - 13829*x^20 + 36240*x^19 + 94321*x^18 + 102411*x^17 + 62946*x^16 + 325659*x^15 + 1299805*x^14 + 3826146*x^13 + 7335516*x^12 + 9580506*x^11 + 8499204*x^10 + 5247290*x^9 + 1937111*x^8 + 388946*x^7 - 47819*x^6 - 74039*x^5 - 43685*x^4 - 5057*x^3 + 3493*x^2 + 123*x + 169, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![169, 123, 3493, -5057, -43685, -74039, -47819, 388946, 1937111, 5247290, 8499204, 9580506, 7335516, 3826146, 1299805, 325659, 62946, 102411, 94321, 36240, -13829, -23985, 8195, 3400, -2345, 673, -129, -34, 36, -10, 1]);
 

\( x^{30} - 10 x^{29} + 36 x^{28} - 34 x^{27} - 129 x^{26} + 673 x^{25} - 2345 x^{24} + 3400 x^{23} + 8195 x^{22} - 23985 x^{21} - 13829 x^{20} + 36240 x^{19} + 94321 x^{18} + 102411 x^{17} + 62946 x^{16} + 325659 x^{15} + 1299805 x^{14} + 3826146 x^{13} + 7335516 x^{12} + 9580506 x^{11} + 8499204 x^{10} + 5247290 x^{9} + 1937111 x^{8} + 388946 x^{7} - 47819 x^{6} - 74039 x^{5} - 43685 x^{4} - 5057 x^{3} + 3493 x^{2} + 123 x + 169 \)

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:  \(51586587308205615747892932945519983768463134765625\)\(\medspace = 3^{15}\cdot 5^{25}\cdot 47^{15}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $45.40$
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:  $3, 5, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5} a^{18} + \frac{1}{5} a^{17} - \frac{1}{5} a^{15} + \frac{1}{5} a^{13} + \frac{1}{5} a^{12} - \frac{1}{5} a^{10} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5}$, $\frac{1}{5} a^{19} - \frac{1}{5} a^{17} - \frac{1}{5} a^{16} + \frac{1}{5} a^{15} + \frac{1}{5} a^{14} - \frac{1}{5} a^{12} - \frac{1}{5} a^{11} + \frac{1}{5} a^{10} - \frac{1}{5} a^{4} + \frac{1}{5} a^{2} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{20} + \frac{1}{5} a^{16} + \frac{1}{5} a^{11} - \frac{1}{5} a^{10} - \frac{1}{5} a^{5} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{21} + \frac{1}{5} a^{17} + \frac{1}{5} a^{12} - \frac{1}{5} a^{11} - \frac{1}{5} a^{6} - \frac{1}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{5} a^{22} - \frac{1}{5} a^{17} + \frac{1}{5} a^{15} - \frac{2}{5} a^{12} + \frac{1}{5} a^{10} - \frac{1}{5} a^{7} + \frac{2}{5} a^{2} - \frac{1}{5}$, $\frac{1}{5} a^{23} + \frac{1}{5} a^{17} + \frac{1}{5} a^{16} - \frac{1}{5} a^{15} - \frac{1}{5} a^{13} + \frac{1}{5} a^{12} + \frac{1}{5} a^{11} - \frac{1}{5} a^{10} - \frac{1}{5} a^{8} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{24} - \frac{1}{5} a^{16} + \frac{1}{5} a^{15} - \frac{1}{5} a^{14} - \frac{1}{5} a^{11} + \frac{1}{5} a^{10} - \frac{1}{5} a^{9} + \frac{1}{5} a^{4} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{25} - \frac{1}{5} a^{17} + \frac{1}{5} a^{16} - \frac{1}{5} a^{15} - \frac{1}{5} a^{12} + \frac{1}{5} a^{11} - \frac{1}{5} a^{10} + \frac{1}{5} a^{5} + \frac{1}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{55} a^{26} - \frac{1}{11} a^{25} + \frac{1}{55} a^{24} - \frac{1}{11} a^{23} + \frac{4}{55} a^{22} - \frac{1}{55} a^{21} + \frac{3}{55} a^{19} - \frac{1}{11} a^{18} - \frac{21}{55} a^{17} - \frac{5}{11} a^{16} + \frac{27}{55} a^{15} + \frac{2}{5} a^{14} - \frac{2}{11} a^{13} - \frac{3}{11} a^{12} + \frac{21}{55} a^{11} - \frac{23}{55} a^{10} + \frac{9}{55} a^{9} - \frac{3}{11} a^{8} + \frac{1}{5} a^{7} - \frac{13}{55} a^{6} - \frac{2}{5} a^{4} + \frac{1}{11} a^{3} - \frac{4}{11} a^{2} - \frac{27}{55} a + \frac{8}{55}$, $\frac{1}{715} a^{27} + \frac{1}{715} a^{26} - \frac{62}{715} a^{25} - \frac{43}{715} a^{24} - \frac{4}{715} a^{23} - \frac{21}{715} a^{22} + \frac{49}{715} a^{21} + \frac{58}{715} a^{20} + \frac{57}{715} a^{19} - \frac{7}{715} a^{18} - \frac{217}{715} a^{17} + \frac{306}{715} a^{16} - \frac{3}{715} a^{15} + \frac{42}{143} a^{14} - \frac{53}{715} a^{13} + \frac{294}{715} a^{12} - \frac{293}{715} a^{11} + \frac{179}{715} a^{10} + \frac{193}{715} a^{9} + \frac{174}{715} a^{8} + \frac{97}{715} a^{7} - \frac{188}{715} a^{6} + \frac{4}{13} a^{5} + \frac{45}{143} a^{4} + \frac{318}{715} a^{3} + \frac{8}{143} a^{2} - \frac{18}{65} a - \frac{9}{55}$, $\frac{1}{3711571435} a^{28} + \frac{1250478}{3711571435} a^{27} + \frac{670028}{337415585} a^{26} + \frac{314417471}{3711571435} a^{25} + \frac{358197821}{3711571435} a^{24} + \frac{292495518}{3711571435} a^{23} - \frac{38469906}{3711571435} a^{22} + \frac{2182259}{337415585} a^{21} - \frac{68672293}{742314287} a^{20} - \frac{256912342}{3711571435} a^{19} + \frac{7593473}{742314287} a^{18} - \frac{1393376612}{3711571435} a^{17} - \frac{1422414064}{3711571435} a^{16} + \frac{44526619}{3711571435} a^{15} + \frac{121817954}{285505495} a^{14} + \frac{41292808}{195345865} a^{13} - \frac{859901201}{3711571435} a^{12} - \frac{136491216}{285505495} a^{11} - \frac{1087297026}{3711571435} a^{10} - \frac{1187511321}{3711571435} a^{9} - \frac{1094589312}{3711571435} a^{8} + \frac{954108204}{3711571435} a^{7} + \frac{1397744474}{3711571435} a^{6} + \frac{1752054496}{3711571435} a^{5} + \frac{428883593}{3711571435} a^{4} - \frac{1382026278}{3711571435} a^{3} - \frac{585108052}{3711571435} a^{2} - \frac{4080556}{742314287} a + \frac{20238029}{57101099}$, $\frac{1}{345982410948808332150415595448998349903581127488116958740138800764641523762825203909745255} a^{29} - \frac{1611661551079523891298608215840441708495760201261635618696578329935651838543924}{69196482189761666430083119089799669980716225497623391748027760152928304752565040781949051} a^{28} - \frac{185569338771105955440179474222771822318311635564041251932689236813270856944640031195782}{345982410948808332150415595448998349903581127488116958740138800764641523762825203909745255} a^{27} + \frac{290410778987096108509012241069571411942567799540864086297217180244426010907228217011682}{345982410948808332150415595448998349903581127488116958740138800764641523762825203909745255} a^{26} + \frac{201099867796168003085392888208123187509824324353286597496839995678755216553367773806971}{3008542703902681149134048656078246520900705456418408336870772180562100206633262642693437} a^{25} - \frac{14763684988092696753926667192151418789344079337200753853862306696416883785184516107725824}{345982410948808332150415595448998349903581127488116958740138800764641523762825203909745255} a^{24} + \frac{162762657812141576237666386124016540865164489485989264581066802730065469203210007028477}{7361327892527836854264161605297837231991087818896105505109336186481734548145217104462665} a^{23} - \frac{27240532135585986256222164393237454835456171286171576217301813412431065696493312364212137}{345982410948808332150415595448998349903581127488116958740138800764641523762825203909745255} a^{22} - \frac{2850872669553092053433359581508838579625963973789187539053722701380963337976463076801804}{69196482189761666430083119089799669980716225497623391748027760152928304752565040781949051} a^{21} - \frac{3732630802122309942963489472669934932419043939245084683247583472770735661781060674161877}{69196482189761666430083119089799669980716225497623391748027760152928304752565040781949051} a^{20} - \frac{101801962090953371625008701046176626132153643196354771724385514287553352393494014989956}{5322806322289358956160239929984589998516632730586414749848289242532946519428080060149927} a^{19} + \frac{21625102393473188877856099488891198695634980606991094192932777089883431575051811770944194}{345982410948808332150415595448998349903581127488116958740138800764641523762825203909745255} a^{18} + \frac{1680785413426522105184760458677538993771186595183152461396030142757162407434289062925240}{6290589289978333311825738099072697270974202317965762886184341832084391341142276434722641} a^{17} - \frac{634030411078910743853434531188126715429706804930328269671201565874376454612010884752128}{1472265578505567370852832321059567446398217563779221101021867237296346909629043420892533} a^{16} + \frac{9692225491565395189736292535802271172607177166978229547000461776915951519404572503170952}{31452946449891666559128690495363486354871011589828814430921709160421956705711382173613205} a^{15} - \frac{117631814070607467484199618465125003241534921529285912389478390085722123477316802802727891}{345982410948808332150415595448998349903581127488116958740138800764641523762825203909745255} a^{14} - \frac{24398598097485526836176367459014822954511790300392707531724274324980636007166201267164144}{69196482189761666430083119089799669980716225497623391748027760152928304752565040781949051} a^{13} + \frac{42413859788899944803903374671443642296864274345449168422757310874117917018386236577441682}{345982410948808332150415595448998349903581127488116958740138800764641523762825203909745255} a^{12} - \frac{28682529953675433017774405904237459948725979527606864937792384042944190102498709406804494}{69196482189761666430083119089799669980716225497623391748027760152928304752565040781949051} a^{11} - \frac{88902157777724011580975477136641714194478747750825816602592453419354681614130681883727323}{345982410948808332150415595448998349903581127488116958740138800764641523762825203909745255} a^{10} + \frac{24631086509245765699952530427052403328825486350349212435262051287170079254106095307968553}{69196482189761666430083119089799669980716225497623391748027760152928304752565040781949051} a^{9} + \frac{138482149976694655634753374150778587995632761601444026030921574553323574861864143040814746}{345982410948808332150415595448998349903581127488116958740138800764641523762825203909745255} a^{8} + \frac{6697599555180521114465842410639919574976294332335678626915689414199691848204725655535068}{69196482189761666430083119089799669980716225497623391748027760152928304752565040781949051} a^{7} + \frac{52019262921541275974075134744283837364173324238537738551624631526504323504216318984726707}{345982410948808332150415595448998349903581127488116958740138800764641523762825203909745255} a^{6} - \frac{2686789802326225011903071419751830324499440071152900706393359843568975181034502296269404}{69196482189761666430083119089799669980716225497623391748027760152928304752565040781949051} a^{5} - \frac{68346486244155066560253520108638325953693977936566838519514719651591835677739197182519}{3008542703902681149134048656078246520900705456418408336870772180562100206633262642693437} a^{4} + \frac{4586969214203233853838865345038579064996270210832748539082899197653466843110980038994958}{69196482189761666430083119089799669980716225497623391748027760152928304752565040781949051} a^{3} + \frac{22973893761490877366897243762241873161596312515580304885079495770771993981547670627124447}{69196482189761666430083119089799669980716225497623391748027760152928304752565040781949051} a^{2} - \frac{142485687940192571114344364886680777114768066189054472366555485945623840131317876319594762}{345982410948808332150415595448998349903581127488116958740138800764641523762825203909745255} a - \frac{10794942288233549541686885546228951875694175452525119345849412641639792695306379244304129}{26614031611446794780801199649922949992583163652932073749241446212664732597140400300749635}$

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

\(\Q(\sqrt{705}) \), 3.1.1175.1, 5.1.2209.1, 6.2.8760065625.1, 10.2.174158864690625.1, 15.1.4947491410771484375.1

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

Sibling fields

Degree 30 sibling: 30.0.1097586964004374803146658147777020931243896484375.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $15^{2}$ R R $30$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ $15^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/53.5.0.1}{5} }^{6}$ $30$

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
3Data not computed
$5$5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.12.10.1$x^{12} + 6 x^{11} + 27 x^{10} + 80 x^{9} + 195 x^{8} + 366 x^{7} + 571 x^{6} + 702 x^{5} + 1005 x^{4} + 1140 x^{3} + 357 x^{2} - 138 x + 44$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$
5.12.10.1$x^{12} + 6 x^{11} + 27 x^{10} + 80 x^{9} + 195 x^{8} + 366 x^{7} + 571 x^{6} + 702 x^{5} + 1005 x^{4} + 1140 x^{3} + 357 x^{2} - 138 x + 44$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$
$47$47.2.1.2$x^{2} + 94$$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$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.15.2t1.a.a$1$ $ 3 \cdot 5 $ \(\Q(\sqrt{-15}) \) $C_2$ (as 2T1) $1$ $-1$
1.47.2t1.a.a$1$ $ 47 $ \(\Q(\sqrt{-47}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.705.2t1.a.a$1$ $ 3 \cdot 5 \cdot 47 $ \(\Q(\sqrt{705}) \) $C_2$ (as 2T1) $1$ $1$
* 2.10575.6t3.a.a$2$ $ 3^{2} \cdot 5^{2} \cdot 47 $ 6.0.186384375.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.1175.3t2.a.a$2$ $ 5^{2} \cdot 47 $ 3.1.1175.1 $S_3$ (as 3T2) $1$ $0$
* 2.10575.10t3.a.b$2$ $ 3^{2} \cdot 5^{2} \cdot 47 $ 10.0.3705507759375.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.10575.10t3.a.a$2$ $ 3^{2} \cdot 5^{2} \cdot 47 $ 10.0.3705507759375.1 $D_{10}$ (as 10T3) $1$ $0$
* 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) $1$ $0$
* 2.10575.30t14.a.a$2$ $ 3^{2} \cdot 5^{2} \cdot 47 $ 30.2.51586587308205615747892932945519983768463134765625.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.10575.30t14.a.b$2$ $ 3^{2} \cdot 5^{2} \cdot 47 $ 30.2.51586587308205615747892932945519983768463134765625.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.10575.30t14.a.d$2$ $ 3^{2} \cdot 5^{2} \cdot 47 $ 30.2.51586587308205615747892932945519983768463134765625.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.10575.30t14.a.c$2$ $ 3^{2} \cdot 5^{2} \cdot 47 $ 30.2.51586587308205615747892932945519983768463134765625.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.1175.15t2.a.c$2$ $ 5^{2} \cdot 47 $ 15.1.4947491410771484375.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.1175.15t2.a.d$2$ $ 5^{2} \cdot 47 $ 15.1.4947491410771484375.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.1175.15t2.a.a$2$ $ 5^{2} \cdot 47 $ 15.1.4947491410771484375.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.1175.15t2.a.b$2$ $ 5^{2} \cdot 47 $ 15.1.4947491410771484375.1 $D_{15}$ (as 15T2) $1$ $0$

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 *.