Properties

Label 30.0.380...059.1
Degree $30$
Signature $[0, 15]$
Discriminant $-3.807\times 10^{49}$
Root discriminant $44.95$
Ramified primes $11, 19$
Class number $11$ (GRH)
Class group $[11]$ (GRH)
Galois group $C_6\times D_5$ (as 30T5)

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Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 6*x^29 + 25*x^28 - 72*x^27 + 234*x^26 - 520*x^25 + 1111*x^24 - 2189*x^23 + 3903*x^22 - 4957*x^21 + 4623*x^20 - 10333*x^19 + 8702*x^18 + 8683*x^17 - 6137*x^16 + 9595*x^15 - 28557*x^14 + 23199*x^13 - 20007*x^12 + 4195*x^11 + 12906*x^10 - 27137*x^9 + 36711*x^8 - 24629*x^7 + 24367*x^6 - 23558*x^5 + 11264*x^4 + 26568*x^3 - 14144*x^2 - 4128*x + 4928)
 
gp: K = bnfinit(x^30 - 6*x^29 + 25*x^28 - 72*x^27 + 234*x^26 - 520*x^25 + 1111*x^24 - 2189*x^23 + 3903*x^22 - 4957*x^21 + 4623*x^20 - 10333*x^19 + 8702*x^18 + 8683*x^17 - 6137*x^16 + 9595*x^15 - 28557*x^14 + 23199*x^13 - 20007*x^12 + 4195*x^11 + 12906*x^10 - 27137*x^9 + 36711*x^8 - 24629*x^7 + 24367*x^6 - 23558*x^5 + 11264*x^4 + 26568*x^3 - 14144*x^2 - 4128*x + 4928, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4928, -4128, -14144, 26568, 11264, -23558, 24367, -24629, 36711, -27137, 12906, 4195, -20007, 23199, -28557, 9595, -6137, 8683, 8702, -10333, 4623, -4957, 3903, -2189, 1111, -520, 234, -72, 25, -6, 1]);
 

\( x^{30} - 6 x^{29} + 25 x^{28} - 72 x^{27} + 234 x^{26} - 520 x^{25} + 1111 x^{24} - 2189 x^{23} + 3903 x^{22} - 4957 x^{21} + 4623 x^{20} - 10333 x^{19} + 8702 x^{18} + 8683 x^{17} - 6137 x^{16} + 9595 x^{15} - 28557 x^{14} + 23199 x^{13} - 20007 x^{12} + 4195 x^{11} + 12906 x^{10} - 27137 x^{9} + 36711 x^{8} - 24629 x^{7} + 24367 x^{6} - 23558 x^{5} + 11264 x^{4} + 26568 x^{3} - 14144 x^{2} - 4128 x + 4928 \)

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:  \(-38068577552393333958638260901553477592868939543059\)\(\medspace = -\,11^{12}\cdot 19^{29}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $44.95$
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:  $11, 19$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $6$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{11} a^{17} - \frac{1}{11} a^{16} - \frac{4}{11} a^{15} - \frac{3}{11} a^{14} - \frac{1}{11} a^{13} + \frac{1}{11} a^{12} - \frac{3}{11} a^{11} - \frac{3}{11} a^{10} - \frac{2}{11} a^{9} - \frac{5}{11} a^{8} + \frac{3}{11} a^{7} - \frac{4}{11} a^{5} + \frac{5}{11} a^{4} - \frac{5}{11} a^{3} - \frac{5}{11} a^{2} + \frac{4}{11} a$, $\frac{1}{11} a^{18} - \frac{5}{11} a^{16} + \frac{4}{11} a^{15} - \frac{4}{11} a^{14} - \frac{2}{11} a^{12} + \frac{5}{11} a^{11} - \frac{5}{11} a^{10} + \frac{4}{11} a^{9} - \frac{2}{11} a^{8} + \frac{3}{11} a^{7} - \frac{4}{11} a^{6} + \frac{1}{11} a^{5} + \frac{1}{11} a^{3} - \frac{1}{11} a^{2} + \frac{4}{11} a$, $\frac{1}{22} a^{19} - \frac{1}{22} a^{17} - \frac{1}{2} a^{16} + \frac{1}{11} a^{15} - \frac{1}{22} a^{14} + \frac{5}{22} a^{13} + \frac{9}{22} a^{12} + \frac{5}{22} a^{11} + \frac{3}{22} a^{10} - \frac{5}{11} a^{9} + \frac{5}{22} a^{8} + \frac{4}{11} a^{7} - \frac{5}{11} a^{6} - \frac{5}{22} a^{5} + \frac{5}{11} a^{4} + \frac{1}{22} a^{3} - \frac{5}{22} a^{2} + \frac{5}{22} a$, $\frac{1}{22} a^{20} - \frac{1}{22} a^{18} - \frac{1}{22} a^{17} - \frac{4}{11} a^{16} + \frac{3}{22} a^{15} - \frac{3}{22} a^{14} - \frac{1}{22} a^{13} - \frac{7}{22} a^{12} - \frac{5}{22} a^{11} + \frac{2}{11} a^{10} + \frac{7}{22} a^{9} + \frac{1}{11} a^{8} - \frac{1}{11} a^{7} - \frac{5}{22} a^{6} - \frac{4}{11} a^{5} + \frac{7}{22} a^{4} - \frac{1}{2} a^{3} - \frac{1}{22} a^{2} - \frac{2}{11} a$, $\frac{1}{22} a^{21} - \frac{1}{22} a^{18} - \frac{1}{22} a^{17} + \frac{3}{11} a^{16} - \frac{1}{2} a^{15} - \frac{2}{11} a^{14} - \frac{5}{11} a^{13} - \frac{5}{11} a^{12} + \frac{7}{22} a^{11} + \frac{4}{11} a^{10} - \frac{1}{11} a^{9} + \frac{7}{22} a^{8} + \frac{5}{22} a^{7} + \frac{2}{11} a^{6} - \frac{4}{11} a^{5} - \frac{5}{22} a^{4} + \frac{2}{11} a^{3} - \frac{5}{22} a^{2} - \frac{7}{22} a$, $\frac{1}{22} a^{22} - \frac{1}{22} a^{18} - \frac{1}{22} a^{17} + \frac{3}{11} a^{16} + \frac{7}{22} a^{14} + \frac{1}{22} a^{13} + \frac{5}{11} a^{12} + \frac{9}{22} a^{11} - \frac{3}{22} a^{10} + \frac{9}{22} a^{9} - \frac{2}{11} a^{8} - \frac{3}{11} a^{7} + \frac{2}{11} a^{6} - \frac{4}{11} a^{5} + \frac{3}{11} a^{4} + \frac{2}{11} a^{3} - \frac{2}{11} a^{2} + \frac{3}{22} a$, $\frac{1}{22} a^{23} - \frac{1}{22} a^{18} - \frac{1}{22} a^{17} - \frac{5}{22} a^{16} - \frac{1}{2} a^{15} - \frac{2}{11} a^{14} - \frac{1}{22} a^{13} - \frac{5}{11} a^{12} - \frac{1}{11} a^{11} + \frac{4}{11} a^{10} - \frac{1}{11} a^{9} + \frac{7}{22} a^{8} - \frac{3}{11} a^{7} + \frac{2}{11} a^{6} + \frac{3}{22} a^{5} + \frac{3}{11} a^{4} + \frac{5}{22} a^{3} + \frac{3}{11} a^{2} + \frac{3}{22} a$, $\frac{1}{22} a^{24} - \frac{1}{22} a^{18} - \frac{3}{11} a^{16} - \frac{2}{11} a^{15} + \frac{1}{11} a^{14} - \frac{1}{2} a^{13} - \frac{9}{22} a^{12} - \frac{5}{22} a^{11} + \frac{5}{22} a^{10} + \frac{7}{22} a^{9} - \frac{9}{22} a^{8} + \frac{4}{11} a^{7} - \frac{7}{22} a^{6} - \frac{1}{22} a^{5} + \frac{1}{22} a^{4} - \frac{1}{22} a^{3} - \frac{5}{11} a^{2} + \frac{7}{22} a$, $\frac{1}{22} a^{25} - \frac{1}{22} a^{17} + \frac{1}{22} a^{16} + \frac{1}{11} a^{15} - \frac{4}{11} a^{14} - \frac{5}{11} a^{13} + \frac{5}{11} a^{12} - \frac{4}{11} a^{11} - \frac{4}{11} a^{10} - \frac{9}{22} a^{9} + \frac{5}{22} a^{8} - \frac{3}{22} a^{7} - \frac{1}{2} a^{6} - \frac{3}{11} a^{5} - \frac{5}{22} a^{4} + \frac{5}{22} a^{3} - \frac{3}{11} a^{2} + \frac{7}{22} a$, $\frac{1}{1936} a^{26} + \frac{15}{968} a^{25} + \frac{21}{1936} a^{24} - \frac{7}{484} a^{23} + \frac{1}{88} a^{22} + \frac{3}{242} a^{21} - \frac{3}{176} a^{20} + \frac{39}{1936} a^{19} - \frac{25}{1936} a^{18} + \frac{19}{1936} a^{17} - \frac{529}{1936} a^{16} + \frac{643}{1936} a^{15} - \frac{237}{968} a^{14} - \frac{401}{1936} a^{13} - \frac{157}{1936} a^{12} - \frac{757}{1936} a^{11} - \frac{229}{1936} a^{10} - \frac{1}{1936} a^{9} - \frac{287}{1936} a^{8} + \frac{587}{1936} a^{7} + \frac{477}{968} a^{6} - \frac{741}{1936} a^{5} - \frac{685}{1936} a^{4} - \frac{645}{1936} a^{3} + \frac{343}{1936} a^{2} - \frac{171}{968} a + \frac{19}{44}$, $\frac{1}{27104} a^{27} + \frac{1}{13552} a^{26} + \frac{61}{27104} a^{25} + \frac{3}{308} a^{24} - \frac{257}{13552} a^{23} + \frac{3}{3388} a^{22} - \frac{529}{27104} a^{21} + \frac{347}{27104} a^{20} + \frac{115}{27104} a^{19} - \frac{249}{27104} a^{18} + \frac{1139}{27104} a^{17} + \frac{117}{2464} a^{16} - \frac{377}{1936} a^{15} + \frac{12519}{27104} a^{14} + \frac{9927}{27104} a^{13} - \frac{11057}{27104} a^{12} - \frac{1033}{27104} a^{11} + \frac{8083}{27104} a^{10} + \frac{6957}{27104} a^{9} - \frac{265}{27104} a^{8} - \frac{6157}{13552} a^{7} - \frac{12933}{27104} a^{6} + \frac{3959}{27104} a^{5} + \frac{39}{224} a^{4} - \frac{199}{2464} a^{3} - \frac{503}{1936} a^{2} - \frac{1709}{6776} a + \frac{3}{22}$, $\frac{1}{2385152} a^{28} + \frac{3}{596288} a^{27} - \frac{227}{2385152} a^{26} + \frac{10601}{1192576} a^{25} + \frac{3989}{1192576} a^{24} + \frac{6113}{596288} a^{23} + \frac{6487}{2385152} a^{22} - \frac{38207}{2385152} a^{21} - \frac{49083}{2385152} a^{20} - \frac{44375}{2385152} a^{19} + \frac{10763}{340736} a^{18} + \frac{12767}{340736} a^{17} + \frac{150431}{596288} a^{16} + \frac{935791}{2385152} a^{15} + \frac{385829}{2385152} a^{14} - \frac{584151}{2385152} a^{13} - \frac{1073487}{2385152} a^{12} + \frac{60091}{340736} a^{11} - \frac{148103}{340736} a^{10} - \frac{433043}{2385152} a^{9} - \frac{1755}{298144} a^{8} + \frac{53469}{340736} a^{7} - \frac{284915}{2385152} a^{6} - \frac{161127}{2385152} a^{5} - \frac{27353}{216832} a^{4} + \frac{50879}{149072} a^{3} + \frac{118439}{596288} a^{2} - \frac{255}{9317} a + \frac{877}{1936}$, $\frac{1}{1465591793553231278994577319010110762426867691794969828745298432} a^{29} - \frac{41436511344138686248446418696804450641345415386715208371}{366397948388307819748644329752527690606716922948742457186324608} a^{28} - \frac{2440709993122541998719662914069089176968632460406006768757}{209370256221890182713511045572872966060981098827852832677899776} a^{27} + \frac{185778158119520927333903333789894397758671507792357447049853}{732795896776615639497288659505055381213433845897484914372649216} a^{26} + \frac{710544992704297668581158538453788359646081092453089262023685}{732795896776615639497288659505055381213433845897484914372649216} a^{25} + \frac{8281494590390003473518591553261434189553525885350275651488645}{366397948388307819748644329752527690606716922948742457186324608} a^{24} - \frac{18998273443575443685215983400346266875380227912694732933513257}{1465591793553231278994577319010110762426867691794969828745298432} a^{23} + \frac{3436853730767626397734311501500464612010937100173088428322137}{1465591793553231278994577319010110762426867691794969828745298432} a^{22} - \frac{22944647501421020296647522299868124647452683881261969059273443}{1465591793553231278994577319010110762426867691794969828745298432} a^{21} + \frac{4262833550028329005164480554853199778832644540292899689332065}{1465591793553231278994577319010110762426867691794969828745298432} a^{20} - \frac{17654197434794968436590821459031079256752327157650622316992859}{1465591793553231278994577319010110762426867691794969828745298432} a^{19} + \frac{120610976985870652040336274232091513617510243514874042450619}{6456351513450358057244834004449827147254923752400748144252416} a^{18} - \frac{8061160372101982284997619569187022326962673963898797334494379}{366397948388307819748644329752527690606716922948742457186324608} a^{17} - \frac{82723087145462528448359468515596580037641772537178491893636535}{209370256221890182713511045572872966060981098827852832677899776} a^{16} - \frac{581036346579649324664039009469136816473443484921687984439010787}{1465591793553231278994577319010110762426867691794969828745298432} a^{15} - \frac{373054649634554763591245977717812400504373736991543262797182303}{1465591793553231278994577319010110762426867691794969828745298432} a^{14} - \frac{203697662005218186592150759610310486658972255570426258356485175}{1465591793553231278994577319010110762426867691794969828745298432} a^{13} - \frac{711147644264380924695408080770835537912586133766207353982135051}{1465591793553231278994577319010110762426867691794969828745298432} a^{12} + \frac{489298715641559725840183286621647361839026814930282947611890823}{1465591793553231278994577319010110762426867691794969828745298432} a^{11} + \frac{566841317103803037415452985350965751704081682575744771024783029}{1465591793553231278994577319010110762426867691794969828745298432} a^{10} - \frac{9977636908875501324687091114723151939825828718443176374083655}{22899871774269238734290270609532980662919807684296403574145288} a^{9} + \frac{555153867511537754601747086295982229782802783918373526689085819}{1465591793553231278994577319010110762426867691794969828745298432} a^{8} + \frac{407860133173355673141587296313545022172134798234731615480492965}{1465591793553231278994577319010110762426867691794969828745298432} a^{7} - \frac{299892959750080739308422929873469780280783288679054191280227695}{1465591793553231278994577319010110762426867691794969828745298432} a^{6} + \frac{24330027119734948787041381305439821475646983405354508230565743}{133235617595748298090416119910010069311533426526815438976845312} a^{5} + \frac{9953654919290477491545382986474665866764952038289255456851157}{183198974194153909874322164876263845303358461474371228593162304} a^{4} - \frac{49419796605525193824636601885588808604709165835324654075886493}{366397948388307819748644329752527690606716922948742457186324608} a^{3} + \frac{2090676569021646699579099995530410584038514882930964345527203}{22899871774269238734290270609532980662919807684296403574145288} a^{2} + \frac{3016868877356380070421218486391646244991028387972379546952847}{8327226099734268630651007494375629331970839157925964936052832} a + \frac{3042412795763668041014384512110320935349151440481269124729}{13518224187880306218589297880479917746705907723905787233852}$

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

Class group and class number

$C_{11}$, which has order $11$ (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:  \( 2589248825821.8447 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{15}\cdot 2589248825821.8447 \cdot 11}{2\sqrt{38068577552393333958638260901553477592868939543059}}\approx 2.16746043250945$ (assuming GRH)

Galois group

$C_6\times D_5$ (as 30T5):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 60
The 24 conjugacy class representatives for $C_6\times D_5$
Character table for $C_6\times D_5$ is not computed

Intermediate fields

\(\Q(\sqrt{-19}) \), 3.3.361.1, 5.1.15768841.1, 6.0.2476099.1, 10.0.4724470583182339.1, 15.3.1415489083272211976282881.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{5}$ $30$ $15^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{6}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{5}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ R $15^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{5}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{5}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{4}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ $15^{2}$ $30$ $30$

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
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
19Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.209.2t1.a.a$1$ $ 11 \cdot 19 $ \(\Q(\sqrt{209}) \) $C_2$ (as 2T1) $1$ $1$
1.11.2t1.a.a$1$ $ 11 $ \(\Q(\sqrt{-11}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.19.2t1.a.a$1$ $ 19 $ \(\Q(\sqrt{-19}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.19.6t1.a.b$1$ $ 19 $ 6.0.2476099.1 $C_6$ (as 6T1) $0$ $-1$
* 1.19.3t1.a.a$1$ $ 19 $ 3.3.361.1 $C_3$ (as 3T1) $0$ $1$
1.209.6t1.b.b$1$ $ 11 \cdot 19 $ 6.0.173457251.1 $C_6$ (as 6T1) $0$ $-1$
1.209.6t1.a.b$1$ $ 11 \cdot 19 $ 6.6.3295687769.1 $C_6$ (as 6T1) $0$ $1$
1.209.6t1.b.a$1$ $ 11 \cdot 19 $ 6.0.173457251.1 $C_6$ (as 6T1) $0$ $-1$
* 1.19.6t1.a.a$1$ $ 19 $ 6.0.2476099.1 $C_6$ (as 6T1) $0$ $-1$
1.209.6t1.a.a$1$ $ 11 \cdot 19 $ 6.6.3295687769.1 $C_6$ (as 6T1) $0$ $1$
* 1.19.3t1.a.b$1$ $ 19 $ 3.3.361.1 $C_3$ (as 3T1) $0$ $1$
* 2.3971.10t3.a.a$2$ $ 11 \cdot 19^{2}$ 10.0.4724470583182339.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.3971.5t2.a.b$2$ $ 11 \cdot 19^{2}$ 5.1.15768841.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.3971.10t3.a.b$2$ $ 11 \cdot 19^{2}$ 10.0.4724470583182339.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.3971.5t2.a.a$2$ $ 11 \cdot 19^{2}$ 5.1.15768841.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.3971.15t3.a.a$2$ $ 11 \cdot 19^{2}$ 15.3.1415489083272211976282881.1 $D_5\times C_3$ (as 15T3) $0$ $0$
* 2.3971.30t5.a.a$2$ $ 11 \cdot 19^{2}$ 30.0.38068577552393333958638260901553477592868939543059.1 $C_6\times D_5$ (as 30T5) $0$ $0$
* 2.3971.30t5.a.c$2$ $ 11 \cdot 19^{2}$ 30.0.38068577552393333958638260901553477592868939543059.1 $C_6\times D_5$ (as 30T5) $0$ $0$
* 2.3971.15t3.a.b$2$ $ 11 \cdot 19^{2}$ 15.3.1415489083272211976282881.1 $D_5\times C_3$ (as 15T3) $0$ $0$
* 2.3971.30t5.a.b$2$ $ 11 \cdot 19^{2}$ 30.0.38068577552393333958638260901553477592868939543059.1 $C_6\times D_5$ (as 30T5) $0$ $0$
* 2.3971.15t3.a.c$2$ $ 11 \cdot 19^{2}$ 15.3.1415489083272211976282881.1 $D_5\times C_3$ (as 15T3) $0$ $0$
* 2.3971.15t3.a.d$2$ $ 11 \cdot 19^{2}$ 15.3.1415489083272211976282881.1 $D_5\times C_3$ (as 15T3) $0$ $0$
* 2.3971.30t5.a.d$2$ $ 11 \cdot 19^{2}$ 30.0.38068577552393333958638260901553477592868939543059.1 $C_6\times D_5$ (as 30T5) $0$ $0$

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 *.