Properties

Label 30.0.893...671.1
Degree $30$
Signature $[0, 15]$
Discriminant $-8.931\times 10^{47}$
Root discriminant $39.66$
Ramified primes $3, 7, 11$
Class number $44$ (GRH)
Class group $[2, 22]$ (GRH)
Galois group $C_5\times S_3$ (as 30T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 3*x^29 - 5*x^28 + 23*x^27 - 7*x^26 - 43*x^25 + 380*x^24 - 1004*x^23 - 505*x^22 + 4147*x^21 - 3960*x^20 + 145*x^19 + 33797*x^18 - 104422*x^17 + 76694*x^16 + 164546*x^15 - 470105*x^14 + 125016*x^13 + 1685293*x^12 - 1720079*x^11 - 1513677*x^10 - 5549753*x^9 + 23270864*x^8 - 21911244*x^7 + 5984424*x^6 - 15339818*x^5 + 45492605*x^4 - 53109297*x^3 + 35921397*x^2 - 13035410*x + 2348809)
 
gp: K = bnfinit(x^30 - 3*x^29 - 5*x^28 + 23*x^27 - 7*x^26 - 43*x^25 + 380*x^24 - 1004*x^23 - 505*x^22 + 4147*x^21 - 3960*x^20 + 145*x^19 + 33797*x^18 - 104422*x^17 + 76694*x^16 + 164546*x^15 - 470105*x^14 + 125016*x^13 + 1685293*x^12 - 1720079*x^11 - 1513677*x^10 - 5549753*x^9 + 23270864*x^8 - 21911244*x^7 + 5984424*x^6 - 15339818*x^5 + 45492605*x^4 - 53109297*x^3 + 35921397*x^2 - 13035410*x + 2348809, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2348809, -13035410, 35921397, -53109297, 45492605, -15339818, 5984424, -21911244, 23270864, -5549753, -1513677, -1720079, 1685293, 125016, -470105, 164546, 76694, -104422, 33797, 145, -3960, 4147, -505, -1004, 380, -43, -7, 23, -5, -3, 1]);
 

\( x^{30} - 3 x^{29} - 5 x^{28} + 23 x^{27} - 7 x^{26} - 43 x^{25} + 380 x^{24} - 1004 x^{23} - 505 x^{22} + 4147 x^{21} - 3960 x^{20} + 145 x^{19} + 33797 x^{18} - 104422 x^{17} + 76694 x^{16} + 164546 x^{15} - 470105 x^{14} + 125016 x^{13} + 1685293 x^{12} - 1720079 x^{11} - 1513677 x^{10} - 5549753 x^{9} + 23270864 x^{8} - 21911244 x^{7} + 5984424 x^{6} - 15339818 x^{5} + 45492605 x^{4} - 53109297 x^{3} + 35921397 x^{2} - 13035410 x + 2348809 \)

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:  \(-893083200934004322334040290940494739136293593671\)\(\medspace = -\,3^{15}\cdot 7^{15}\cdot 11^{27}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $39.66$
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)));
 
$|\Gal(K/\Q)|$:  $30$
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{11} a^{10} - \frac{1}{11} a^{9} + \frac{1}{11} a^{8} - \frac{1}{11} a^{7} + \frac{1}{11} a^{6} - \frac{1}{11} a^{5} + \frac{1}{11} a^{4} - \frac{1}{11} a^{3} + \frac{1}{11} a^{2} - \frac{1}{11} a + \frac{1}{11}$, $\frac{1}{11} a^{11} + \frac{1}{11}$, $\frac{1}{11} a^{12} + \frac{1}{11} a$, $\frac{1}{11} a^{13} + \frac{1}{11} a^{2}$, $\frac{1}{11} a^{14} + \frac{1}{11} a^{3}$, $\frac{1}{11} a^{15} + \frac{1}{11} a^{4}$, $\frac{1}{11} a^{16} + \frac{1}{11} a^{5}$, $\frac{1}{11} a^{17} + \frac{1}{11} a^{6}$, $\frac{1}{11} a^{18} + \frac{1}{11} a^{7}$, $\frac{1}{121} a^{19} - \frac{3}{121} a^{18} - \frac{5}{121} a^{17} + \frac{1}{121} a^{16} + \frac{4}{121} a^{15} + \frac{1}{121} a^{14} - \frac{5}{121} a^{13} - \frac{3}{121} a^{12} + \frac{1}{121} a^{11} - \frac{2}{11} a^{9} + \frac{45}{121} a^{8} + \frac{52}{121} a^{7} - \frac{38}{121} a^{6} + \frac{12}{121} a^{5} + \frac{15}{121} a^{4} - \frac{32}{121} a^{3} + \frac{50}{121} a^{2} + \frac{41}{121} a - \frac{21}{121}$, $\frac{1}{121} a^{20} - \frac{3}{121} a^{18} - \frac{3}{121} a^{17} - \frac{4}{121} a^{16} + \frac{2}{121} a^{15} - \frac{2}{121} a^{14} + \frac{4}{121} a^{13} + \frac{3}{121} a^{12} + \frac{3}{121} a^{11} - \frac{43}{121} a^{9} - \frac{3}{11} a^{8} - \frac{14}{121} a^{7} + \frac{52}{121} a^{6} + \frac{18}{121} a^{5} + \frac{24}{121} a^{4} + \frac{53}{121} a^{3} - \frac{7}{121} a^{2} - \frac{30}{121} a - \frac{41}{121}$, $\frac{1}{121} a^{21} - \frac{1}{121} a^{18} + \frac{3}{121} a^{17} + \frac{5}{121} a^{16} - \frac{1}{121} a^{15} - \frac{4}{121} a^{14} - \frac{1}{121} a^{13} + \frac{5}{121} a^{12} + \frac{3}{121} a^{11} + \frac{1}{121} a^{10} - \frac{2}{11} a^{9} + \frac{4}{11} a^{8} + \frac{54}{121} a^{7} - \frac{30}{121} a^{6} + \frac{16}{121} a^{5} + \frac{10}{121} a^{4} - \frac{37}{121} a^{3} + \frac{54}{121} a^{2} + \frac{49}{121} a - \frac{19}{121}$, $\frac{1}{121} a^{22} + \frac{2}{121} a^{11} + \frac{1}{121}$, $\frac{1}{121} a^{23} + \frac{2}{121} a^{12} + \frac{1}{121} a$, $\frac{1}{1331} a^{24} + \frac{2}{1331} a^{23} + \frac{1}{1331} a^{22} + \frac{1}{121} a^{17} + \frac{5}{121} a^{16} + \frac{5}{121} a^{15} - \frac{3}{121} a^{14} - \frac{42}{1331} a^{13} + \frac{26}{1331} a^{12} + \frac{24}{1331} a^{11} + \frac{1}{11} a^{9} + \frac{2}{11} a^{8} + \frac{56}{121} a^{6} + \frac{49}{121} a^{5} + \frac{16}{121} a^{4} + \frac{41}{121} a^{3} - \frac{648}{1331} a^{2} - \frac{97}{1331} a + \frac{628}{1331}$, $\frac{1}{3993} a^{25} - \frac{1}{3993} a^{24} + \frac{2}{1331} a^{23} - \frac{14}{3993} a^{22} - \frac{1}{363} a^{21} - \frac{1}{363} a^{20} + \frac{16}{363} a^{18} + \frac{13}{363} a^{17} + \frac{14}{363} a^{15} + \frac{41}{1331} a^{14} - \frac{2}{3993} a^{13} + \frac{122}{3993} a^{12} + \frac{82}{3993} a^{11} + \frac{10}{363} a^{10} + \frac{18}{121} a^{9} + \frac{4}{33} a^{8} - \frac{94}{363} a^{7} - \frac{3}{121} a^{6} + \frac{8}{33} a^{5} - \frac{8}{363} a^{4} + \frac{1816}{3993} a^{3} + \frac{403}{1331} a^{2} - \frac{368}{3993} a + \frac{1669}{3993}$, $\frac{1}{59621954667} a^{26} - \frac{6902629}{59621954667} a^{25} + \frac{2429246}{19873984889} a^{24} + \frac{141808192}{59621954667} a^{23} + \frac{65446792}{59621954667} a^{22} - \frac{12883714}{5420177697} a^{21} + \frac{7312964}{1806725899} a^{20} + \frac{20807641}{5420177697} a^{19} - \frac{240673475}{5420177697} a^{18} + \frac{29002280}{1806725899} a^{17} - \frac{173803255}{5420177697} a^{16} - \frac{781708403}{19873984889} a^{15} + \frac{2676435757}{59621954667} a^{14} + \frac{1624889258}{59621954667} a^{13} - \frac{1405816163}{59621954667} a^{12} - \frac{2198444218}{59621954667} a^{11} - \frac{57631727}{1806725899} a^{10} + \frac{2553487571}{5420177697} a^{9} - \frac{2066164114}{5420177697} a^{8} - \frac{712442283}{1806725899} a^{7} - \frac{2034667649}{5420177697} a^{6} + \frac{2128778389}{5420177697} a^{5} - \frac{10919268095}{59621954667} a^{4} - \frac{1991375759}{19873984889} a^{3} - \frac{5945406317}{59621954667} a^{2} - \frac{6369566315}{59621954667} a + \frac{3101436343}{19873984889}$, $\frac{1}{59621954667} a^{27} - \frac{6201568}{59621954667} a^{25} - \frac{20529320}{59621954667} a^{24} - \frac{151259893}{59621954667} a^{23} - \frac{230278261}{59621954667} a^{22} - \frac{22368829}{5420177697} a^{21} + \frac{6073660}{5420177697} a^{20} + \frac{2080190}{5420177697} a^{19} + \frac{5227009}{5420177697} a^{18} + \frac{95845283}{5420177697} a^{17} - \frac{2227311821}{59621954667} a^{16} + \frac{13344245}{5420177697} a^{15} - \frac{648775416}{19873984889} a^{14} - \frac{3604959}{19873984889} a^{13} + \frac{278930598}{19873984889} a^{12} - \frac{1189585768}{59621954667} a^{11} - \frac{80317291}{5420177697} a^{10} - \frac{2235603140}{5420177697} a^{9} + \frac{2372550578}{5420177697} a^{8} - \frac{1953151061}{5420177697} a^{7} - \frac{2476034038}{5420177697} a^{6} + \frac{3228814307}{19873984889} a^{5} - \frac{1116893809}{5420177697} a^{4} + \frac{24334959133}{59621954667} a^{3} - \frac{16260433792}{59621954667} a^{2} + \frac{743245576}{59621954667} a - \frac{4165159938}{19873984889}$, $\frac{1}{59621954667} a^{28} + \frac{1295509}{59621954667} a^{25} + \frac{286346}{5420177697} a^{24} + \frac{11822477}{19873984889} a^{23} + \frac{32461104}{19873984889} a^{22} + \frac{5263679}{5420177697} a^{21} + \frac{8975413}{5420177697} a^{20} - \frac{1213519}{5420177697} a^{19} + \frac{43960711}{5420177697} a^{18} + \frac{172433104}{19873984889} a^{17} - \frac{6691877}{5420177697} a^{16} + \frac{109468982}{5420177697} a^{15} - \frac{19707929}{59621954667} a^{14} - \frac{26043500}{1806725899} a^{13} + \frac{1710510668}{59621954667} a^{12} - \frac{2100466811}{59621954667} a^{11} + \frac{96247484}{5420177697} a^{10} - \frac{2248869737}{5420177697} a^{9} - \frac{1153044793}{5420177697} a^{8} - \frac{462856259}{5420177697} a^{7} - \frac{3158078593}{59621954667} a^{6} - \frac{481003088}{5420177697} a^{5} - \frac{1272001009}{5420177697} a^{4} - \frac{2135684208}{19873984889} a^{3} - \frac{2127177197}{5420177697} a^{2} + \frac{22646381966}{59621954667} a - \frac{26436440048}{59621954667}$, $\frac{1}{91409034650310260542863866164981827738731095611252002598801002511} a^{29} + \frac{485005753077985049467741742486573114366637662910860360}{91409034650310260542863866164981827738731095611252002598801002511} a^{28} + \frac{107034225754206635242542137583775282926162707400176926}{30469678216770086847621288721660609246243698537084000866267000837} a^{27} - \frac{253453000890069547784250677369301965387570995869672020}{30469678216770086847621288721660609246243698537084000866267000837} a^{26} + \frac{8028607433231133204351386826643847379692183935211935918666820}{91409034650310260542863866164981827738731095611252002598801002511} a^{25} - \frac{7501888702183355374740635163511395566852803775976766994169971}{91409034650310260542863866164981827738731095611252002598801002511} a^{24} + \frac{331581692266473170613500090749630909071442149650596932846871406}{91409034650310260542863866164981827738731095611252002598801002511} a^{23} + \frac{154097644647701701238304969853569948331311189490169479628386645}{91409034650310260542863866164981827738731095611252002598801002511} a^{22} - \frac{16188579529593019211021255163757014869599758065910305526232882}{8309912240937296412987624196816529794430099601022909327163727501} a^{21} + \frac{7784171014912413478538879827070277139810716612265980777033953}{2769970746979098804329208065605509931476699867007636442387909167} a^{20} + \frac{16990741772964171549769334835392786047979770333316149754199232}{8309912240937296412987624196816529794430099601022909327163727501} a^{19} - \frac{3281738350450874707365127233253229718825059173595252588805520952}{91409034650310260542863866164981827738731095611252002598801002511} a^{18} + \frac{3022114805969568560393022362214069588266393384776073899769862907}{91409034650310260542863866164981827738731095611252002598801002511} a^{17} + \frac{2790594574335523497309695171215407721489393750232635368509029438}{91409034650310260542863866164981827738731095611252002598801002511} a^{16} + \frac{1430329282601852778072210992256709869291590845124143064475745690}{91409034650310260542863866164981827738731095611252002598801002511} a^{15} - \frac{3412120521260075764584677988166704876842098710794400165590889332}{91409034650310260542863866164981827738731095611252002598801002511} a^{14} + \frac{1689852040829810091935299817969371768970963014879523667835741891}{91409034650310260542863866164981827738731095611252002598801002511} a^{13} - \frac{928206598295223653797605194672280374292235898304973360611215854}{30469678216770086847621288721660609246243698537084000866267000837} a^{12} - \frac{489256437511244714778356724732486405779959834283247956343006127}{91409034650310260542863866164981827738731095611252002598801002511} a^{11} + \frac{21765293029965881285547564063199995442931235490050668654619598}{2769970746979098804329208065605509931476699867007636442387909167} a^{10} + \frac{1319144139743794449966486544288936141286563393279087891858085360}{8309912240937296412987624196816529794430099601022909327163727501} a^{9} + \frac{2227557004849368407416859238346002440675590716883722097793262593}{8309912240937296412987624196816529794430099601022909327163727501} a^{8} - \frac{12153307590368138595209696771287273072255349322691121854298210804}{91409034650310260542863866164981827738731095611252002598801002511} a^{7} - \frac{40578396593337548124087278363131622246192744609833947810346294982}{91409034650310260542863866164981827738731095611252002598801002511} a^{6} + \frac{25185479725896285807419372181667239618477931390975991174357650781}{91409034650310260542863866164981827738731095611252002598801002511} a^{5} + \frac{11509832453915744539081594349990801596639676103584154993623975254}{30469678216770086847621288721660609246243698537084000866267000837} a^{4} + \frac{15026546135279224630020894895342049140132785204004467480874279986}{30469678216770086847621288721660609246243698537084000866267000837} a^{3} + \frac{31667800222061126547030517446642255384230988817665587613235983356}{91409034650310260542863866164981827738731095611252002598801002511} a^{2} - \frac{25631992245028695061985805899919842378524729582659982984734129313}{91409034650310260542863866164981827738731095611252002598801002511} a + \frac{13095041641551442627517432431971334781957293931181358741288436718}{30469678216770086847621288721660609246243698537084000866267000837}$

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

Class group and class number

$C_{2}\times C_{22}$, which has order $44$ (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:  \( -1 \) (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:  \( 9800307802.95059 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{15}\cdot 9800307802.95059 \cdot 44}{2\sqrt{893083200934004322334040290940494739136293593671}}\approx 0.214246858238136$ (assuming GRH)

Galois group

$C_5\times S_3$ (as 30T2):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 30
The 15 conjugacy class representatives for $C_5\times S_3$
Character table for $C_5\times S_3$

Intermediate fields

\(\Q(\sqrt{-231}) \), 3.1.231.1 x3, \(\Q(\zeta_{11})^+\), 6.0.12326391.1, 10.0.9630096522760791.1, 15.5.140994243189740741031.1 x3

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

Sibling fields

Degree 15 sibling: 15.5.140994243189740741031.1

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 $15^{2}$ R R $15^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{3}$ $15^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{15}$ $15^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{3}$ $15^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{15}$ $15^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{3}$ $15^{2}$

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}$$[\ ]_{2}^{5}$
$7$7.10.5.1$x^{10} - 98 x^{6} + 2401 x^{2} - 268912$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
7.10.5.1$x^{10} - 98 x^{6} + 2401 x^{2} - 268912$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
7.10.5.1$x^{10} - 98 x^{6} + 2401 x^{2} - 268912$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.231.2t1.a.a$1$ $ 3 \cdot 7 \cdot 11 $ \(\Q(\sqrt{-231}) \) $C_2$ (as 2T1) $1$ $-1$
* 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) $0$ $-1$
* 1.11.5t1.a.a$1$ $ 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.b$1$ $ 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) $0$ $-1$
* 1.11.5t1.a.c$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
* 1.11.5t1.a.d$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
*2 2.231.3t2.a.a$2$ $ 3 \cdot 7 \cdot 11 $ 3.1.231.1 $S_3$ (as 3T2) $1$ $0$
*2 2.2541.15t4.a.a$2$ $ 3 \cdot 7 \cdot 11^{2}$ 30.0.893083200934004322334040290940494739136293593671.1 $C_5\times S_3$ (as 30T2) $0$ $0$
*2 2.2541.15t4.a.b$2$ $ 3 \cdot 7 \cdot 11^{2}$ 30.0.893083200934004322334040290940494739136293593671.1 $C_5\times S_3$ (as 30T2) $0$ $0$
*2 2.2541.15t4.a.c$2$ $ 3 \cdot 7 \cdot 11^{2}$ 30.0.893083200934004322334040290940494739136293593671.1 $C_5\times S_3$ (as 30T2) $0$ $0$
*2 2.2541.15t4.a.d$2$ $ 3 \cdot 7 \cdot 11^{2}$ 30.0.893083200934004322334040290940494739136293593671.1 $C_5\times S_3$ (as 30T2) $0$ $0$

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 *.