Properties

Label 27.1.119...311.1
Degree $27$
Signature $[1, 13]$
Discriminant $-1.196\times 10^{50}$
Root discriminant $71.57$
Ramified primes $19, 31$
Class number $9$ (GRH)
Class group $[9]$ (GRH)
Galois group $D_{27}$ (as 27T8)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 9*x^26 + 56*x^25 - 273*x^24 + 1290*x^23 - 5604*x^22 + 24003*x^21 - 98346*x^20 + 381168*x^19 - 1383408*x^18 + 4700958*x^17 - 15265944*x^16 + 47759673*x^15 - 142609974*x^14 + 403676163*x^13 - 1071234237*x^12 + 2616260829*x^11 - 5766916275*x^10 + 11330239941*x^9 - 19648861461*x^8 + 29452821249*x^7 - 36679431165*x^6 + 35809916697*x^5 - 25389897726*x^4 + 11811942110*x^3 - 3328128234*x^2 + 1063555192*x - 565863243)
 
gp: K = bnfinit(x^27 - 9*x^26 + 56*x^25 - 273*x^24 + 1290*x^23 - 5604*x^22 + 24003*x^21 - 98346*x^20 + 381168*x^19 - 1383408*x^18 + 4700958*x^17 - 15265944*x^16 + 47759673*x^15 - 142609974*x^14 + 403676163*x^13 - 1071234237*x^12 + 2616260829*x^11 - 5766916275*x^10 + 11330239941*x^9 - 19648861461*x^8 + 29452821249*x^7 - 36679431165*x^6 + 35809916697*x^5 - 25389897726*x^4 + 11811942110*x^3 - 3328128234*x^2 + 1063555192*x - 565863243, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-565863243, 1063555192, -3328128234, 11811942110, -25389897726, 35809916697, -36679431165, 29452821249, -19648861461, 11330239941, -5766916275, 2616260829, -1071234237, 403676163, -142609974, 47759673, -15265944, 4700958, -1383408, 381168, -98346, 24003, -5604, 1290, -273, 56, -9, 1]);
 

\( x^{27} - 9 x^{26} + 56 x^{25} - 273 x^{24} + 1290 x^{23} - 5604 x^{22} + 24003 x^{21} - 98346 x^{20} + 381168 x^{19} - 1383408 x^{18} + 4700958 x^{17} - 15265944 x^{16} + 47759673 x^{15} - 142609974 x^{14} + 403676163 x^{13} - 1071234237 x^{12} + 2616260829 x^{11} - 5766916275 x^{10} + 11330239941 x^{9} - 19648861461 x^{8} + 29452821249 x^{7} - 36679431165 x^{6} + 35809916697 x^{5} - 25389897726 x^{4} + 11811942110 x^{3} - 3328128234 x^{2} + 1063555192 x - 565863243 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $27$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[1, 13]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-119615770666944050013402329147269064161944026933311\)\(\medspace = -\,19^{24}\cdot 31^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $71.57$
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:  $19, 31$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{6}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{7}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{8}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{9} - \frac{1}{3} a^{8} - \frac{2}{9} a + \frac{1}{3}$, $\frac{1}{9} a^{18} + \frac{1}{9} a^{10} - \frac{2}{9} a^{2}$, $\frac{1}{279} a^{19} - \frac{7}{279} a^{18} + \frac{5}{279} a^{17} - \frac{11}{93} a^{16} - \frac{5}{31} a^{15} + \frac{11}{93} a^{14} + \frac{13}{93} a^{13} - \frac{13}{93} a^{12} + \frac{25}{279} a^{11} - \frac{43}{279} a^{10} - \frac{7}{279} a^{9} - \frac{7}{31} a^{7} + \frac{4}{93} a^{6} + \frac{20}{93} a^{5} - \frac{29}{93} a^{4} - \frac{35}{279} a^{3} + \frac{68}{279} a^{2} - \frac{4}{9} a - \frac{1}{3}$, $\frac{1}{279} a^{20} - \frac{13}{279} a^{18} + \frac{2}{279} a^{17} + \frac{1}{93} a^{16} - \frac{1}{93} a^{15} - \frac{1}{31} a^{14} - \frac{5}{31} a^{13} + \frac{1}{9} a^{12} + \frac{13}{93} a^{11} + \frac{2}{279} a^{10} + \frac{44}{279} a^{9} - \frac{7}{31} a^{8} + \frac{43}{93} a^{7} - \frac{15}{31} a^{6} + \frac{6}{31} a^{5} - \frac{86}{279} a^{4} - \frac{28}{93} a^{3} + \frac{11}{279} a^{2} + \frac{2}{9} a - \frac{1}{3}$, $\frac{1}{3069} a^{21} + \frac{4}{3069} a^{20} + \frac{76}{3069} a^{18} + \frac{14}{3069} a^{17} + \frac{7}{93} a^{16} + \frac{170}{1023} a^{15} + \frac{49}{341} a^{14} - \frac{14}{3069} a^{13} - \frac{65}{3069} a^{12} + \frac{3}{31} a^{11} - \frac{476}{3069} a^{10} + \frac{53}{3069} a^{9} + \frac{269}{1023} a^{8} + \frac{40}{1023} a^{7} + \frac{46}{341} a^{6} + \frac{1003}{3069} a^{5} + \frac{1231}{3069} a^{4} - \frac{4}{341} a^{3} + \frac{742}{3069} a^{2} - \frac{19}{99} a + \frac{1}{3}$, $\frac{1}{3069} a^{22} - \frac{5}{3069} a^{20} - \frac{1}{3069} a^{19} + \frac{106}{3069} a^{18} + \frac{17}{341} a^{17} + \frac{38}{1023} a^{16} - \frac{71}{1023} a^{15} - \frac{326}{3069} a^{14} - \frac{146}{1023} a^{13} - \frac{191}{3069} a^{12} - \frac{91}{3069} a^{11} + \frac{175}{3069} a^{10} - \frac{29}{1023} a^{9} + \frac{97}{1023} a^{8} - \frac{29}{93} a^{7} - \frac{1016}{3069} a^{6} - \frac{223}{1023} a^{5} - \frac{1253}{3069} a^{4} - \frac{412}{3069} a^{3} - \frac{488}{3069} a^{2} - \frac{4}{33} a$, $\frac{1}{9207} a^{23} - \frac{1}{9207} a^{22} - \frac{1}{9207} a^{21} + \frac{1}{1023} a^{20} + \frac{8}{9207} a^{19} + \frac{505}{9207} a^{18} - \frac{500}{9207} a^{17} - \frac{428}{3069} a^{16} + \frac{1300}{9207} a^{15} - \frac{1516}{9207} a^{14} - \frac{1129}{9207} a^{13} + \frac{146}{1023} a^{12} - \frac{427}{9207} a^{11} + \frac{364}{9207} a^{10} + \frac{799}{9207} a^{9} - \frac{136}{279} a^{8} - \frac{62}{297} a^{7} - \frac{3838}{9207} a^{6} + \frac{4055}{9207} a^{5} - \frac{116}{1023} a^{4} + \frac{286}{837} a^{3} - \frac{4451}{9207} a^{2} + \frac{112}{297} a - \frac{4}{9}$, $\frac{1}{285417} a^{24} - \frac{4}{285417} a^{23} + \frac{29}{285417} a^{22} - \frac{1}{10571} a^{21} + \frac{350}{285417} a^{20} + \frac{157}{285417} a^{19} + \frac{85}{9207} a^{18} - \frac{1516}{95139} a^{17} - \frac{16619}{285417} a^{16} - \frac{28879}{285417} a^{15} + \frac{26126}{285417} a^{14} + \frac{46}{10571} a^{13} - \frac{17914}{285417} a^{12} + \frac{18196}{285417} a^{11} - \frac{43730}{285417} a^{10} + \frac{15131}{95139} a^{9} + \frac{53761}{285417} a^{8} - \frac{26017}{285417} a^{7} + \frac{127916}{285417} a^{6} + \frac{1298}{2883} a^{5} + \frac{932}{285417} a^{4} - \frac{117758}{285417} a^{3} - \frac{12217}{25947} a^{2} + \frac{1523}{3069} a + \frac{35}{93}$, $\frac{1}{20451569691267} a^{25} + \frac{46577}{44556796713} a^{24} + \frac{32263211}{757465544121} a^{23} - \frac{177116}{17892886869} a^{22} + \frac{490675814}{6817189897089} a^{21} - \frac{7551434366}{6817189897089} a^{20} - \frac{1004281537}{757465544121} a^{19} + \frac{373744860578}{6817189897089} a^{18} + \frac{251688597671}{6817189897089} a^{17} + \frac{659389316180}{6817189897089} a^{16} + \frac{8310598388}{2272396632363} a^{15} - \frac{71565813340}{757465544121} a^{14} + \frac{28150584290}{6817189897089} a^{13} + \frac{508487317213}{6817189897089} a^{12} + \frac{71694210319}{2272396632363} a^{11} + \frac{466531688990}{6817189897089} a^{10} + \frac{752926571533}{6817189897089} a^{9} + \frac{713599758140}{6817189897089} a^{8} - \frac{31433311405}{133670390139} a^{7} - \frac{5194084522}{12151853649} a^{6} + \frac{31683630787}{401011170417} a^{5} - \frac{145744333133}{6817189897089} a^{4} + \frac{4733781311}{2272396632363} a^{3} + \frac{2759911192223}{6817189897089} a^{2} + \frac{307758307772}{659728054557} a + \frac{7533418987}{19991759229}$, $\frac{1}{301220765238143377886721192064023177693905849096910245855830856907532039523621687382751026171} a^{26} + \frac{1050040209314978484205189824186703361426503104747468890440683472662401995766335}{301220765238143377886721192064023177693905849096910245855830856907532039523621687382751026171} a^{25} - \frac{56273469315580812547692145045673031641002014184807294520795750279750370823529374976144}{33468973915349264209635688007113686410433983232990027317314539656392448835957965264750114019} a^{24} - \frac{104540137070700290470455199593154545898108273073237474205601824560040874808439503253727}{100406921746047792628907064021341059231301949698970081951943618969177346507873895794250342057} a^{23} + \frac{10718049446422275989507760546251546924415407575112055744775879654658639725950502228874}{3718774879483251578848409778568187378937109248110003035257171072932494315106440584972234891} a^{22} + \frac{11689955188709430423917602508809262903595139859975501978560761735397214066580649978943092}{100406921746047792628907064021341059231301949698970081951943618969177346507873895794250342057} a^{21} + \frac{35164995002348697893311340101925353649780962685216595353764374090762035667294640457937922}{100406921746047792628907064021341059231301949698970081951943618969177346507873895794250342057} a^{20} - \frac{23625289852267582414875123403492295462532253720962055495079260165808288459361920777884127}{33468973915349264209635688007113686410433983232990027317314539656392448835957965264750114019} a^{19} - \frac{105877826951811755065407199879746215025629164856725041161302317751735472636525357321698255}{1968763171491133188802099294536099200613763719587648665724384685670144049173997956750006707} a^{18} + \frac{835497862243088013234921222636170694595706853615543995545699523272609110607953983679230369}{33468973915349264209635688007113686410433983232990027317314539656392448835957965264750114019} a^{17} - \frac{49091802390703996771424683547511538523373830956484219149369910124546794738354114885826815}{454329962651799966646638298739099815526253166058688153628704158231571703655537990019232317} a^{16} + \frac{4170897791936427925779894152615967914927823122620621601450428587458286067609679577853108637}{100406921746047792628907064021341059231301949698970081951943618969177346507873895794250342057} a^{15} - \frac{40298262551970018255440822428194809588064703187087080199716836699175624524506904209466977}{384700849601715680570525149507053866786597508425172727785224593751637342942045577755748437} a^{14} + \frac{5975209390224680707419641459198726924253610114857674235905224974904814692136944213446971677}{100406921746047792628907064021341059231301949698970081951943618969177346507873895794250342057} a^{13} + \frac{7892674871853438037825469952437990032820887341834443246714218458595075777103261084827192567}{100406921746047792628907064021341059231301949698970081951943618969177346507873895794250342057} a^{12} + \frac{93939491718001326948447381305775656219740898001768813492200687615555406922004152528409779}{778348230589517772317109023421248521172883330999768077146849759450987182231580587552328233} a^{11} + \frac{431072655325160622973527244631338214405106812252921920278379576969341867049257059909611187}{3238932959549928794480873032946485781654901603192583288772374805457333758318512767556462647} a^{10} + \frac{498023844007958881419394856551607010888505336717806790548030089307441793538714405678130733}{3462307646415441125134726345563484801079377575826554550067021343764736086478410199801735933} a^{9} + \frac{104976787659119865051895244427052209709433455663419636726698711993909302973309159758347784}{3462307646415441125134726345563484801079377575826554550067021343764736086478410199801735933} a^{8} + \frac{568270650120972895335139251680714541643490221788296201031453366125460262789071628169600278}{5906289514473399566406297883608297601841291158762945997173154057010432147521993870250020121} a^{7} - \frac{191645442892234194521322974297954140979474892900902177240233371488530080136130585087507489}{656254390497044396267366431512033066871254573195882888574794895223381349724665985583335569} a^{6} - \frac{13389345957383136170649641515805811936334898730694835445287834639520718277305042853514825722}{100406921746047792628907064021341059231301949698970081951943618969177346507873895794250342057} a^{5} - \frac{19073015076037597891160489896366714930400901082921434741226982285329345879077598552931416073}{100406921746047792628907064021341059231301949698970081951943618969177346507873895794250342057} a^{4} + \frac{186822063971123216084006306386734994373092537351676432909953732381820164979732712039363208}{2574536455026866477664283692854898954648767940999232870562656896645572987381381943442316463} a^{3} - \frac{7274976039275209484392603909556884246152980338744835366697787468042156834594790395444192096}{23170828095241798298978553235694090591838911468993095835063912069810156886432437490980848167} a^{2} - \frac{3452347861884718992591024074221035979803648088041578048813921074721954691978112773408189199}{9716798878649786383442619098839457344964704809577749866317124416372001274955538302669387941} a + \frac{106857015954402715379527924606182004206554908469369449153308155023114409234834129382343846}{294448450868175344952806639358771434695900145744780298979306800496121250756228433414223877}$

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

Class group and class number

$C_{9}$, 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:  $13$
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:  \( 111879013692337.2 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{13}\cdot 111879013692337.2 \cdot 9}{2\sqrt{119615770666944050013402329147269064161944026933311}}\approx 2.18995507510331$ (assuming GRH)

Galois group

$D_{27}$ (as 27T8):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 54
The 15 conjugacy class representatives for $D_{27}$
Character table for $D_{27}$

Intermediate fields

3.1.31.1, 9.1.43447859067001.1

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

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $27$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ $27$ $27$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ R ${\href{/LocalNumberField/37.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ $27$

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
19Data not computed
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.31.2t1.a.a$1$ $ 31 $ \(\Q(\sqrt{-31}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.31.3t2.b.a$2$ $ 31 $ 3.1.31.1 $S_3$ (as 3T2) $1$ $0$
* 2.11191.9t3.a.c$2$ $ 19^{2} \cdot 31 $ 9.1.43447859067001.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.11191.9t3.a.a$2$ $ 19^{2} \cdot 31 $ 9.1.43447859067001.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.11191.9t3.a.b$2$ $ 19^{2} \cdot 31 $ 9.1.43447859067001.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.11191.27t8.a.c$2$ $ 19^{2} \cdot 31 $ 27.1.119615770666944050013402329147269064161944026933311.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.11191.27t8.a.b$2$ $ 19^{2} \cdot 31 $ 27.1.119615770666944050013402329147269064161944026933311.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.11191.27t8.a.e$2$ $ 19^{2} \cdot 31 $ 27.1.119615770666944050013402329147269064161944026933311.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.11191.27t8.a.i$2$ $ 19^{2} \cdot 31 $ 27.1.119615770666944050013402329147269064161944026933311.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.11191.27t8.a.f$2$ $ 19^{2} \cdot 31 $ 27.1.119615770666944050013402329147269064161944026933311.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.11191.27t8.a.a$2$ $ 19^{2} \cdot 31 $ 27.1.119615770666944050013402329147269064161944026933311.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.11191.27t8.a.h$2$ $ 19^{2} \cdot 31 $ 27.1.119615770666944050013402329147269064161944026933311.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.11191.27t8.a.g$2$ $ 19^{2} \cdot 31 $ 27.1.119615770666944050013402329147269064161944026933311.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.11191.27t8.a.d$2$ $ 19^{2} \cdot 31 $ 27.1.119615770666944050013402329147269064161944026933311.1 $D_{27}$ (as 27T8) $1$ $0$

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 *.