Properties

Label 27.1.100...375.1
Degree $27$
Signature $[1, 13]$
Discriminant $-1.004\times 10^{50}$
Root discriminant $71.11$
Ramified primes $3, 5, 67$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $D_{27}$ (as 27T8)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 82*x^24 - 9*x^23 + 195*x^22 + 2387*x^21 + 810*x^20 - 11814*x^19 - 33683*x^18 + 2520*x^17 + 230379*x^16 + 441880*x^15 - 1170738*x^14 - 529926*x^13 - 6335185*x^12 + 26111763*x^11 - 28861455*x^10 + 54201969*x^9 - 172123947*x^8 + 231730104*x^7 - 172770969*x^6 + 271759050*x^5 - 338641350*x^4 + 41402575*x^3 + 177368400*x^2 - 96664125*x + 14269375)
 
gp: K = bnfinit(x^27 - 82*x^24 - 9*x^23 + 195*x^22 + 2387*x^21 + 810*x^20 - 11814*x^19 - 33683*x^18 + 2520*x^17 + 230379*x^16 + 441880*x^15 - 1170738*x^14 - 529926*x^13 - 6335185*x^12 + 26111763*x^11 - 28861455*x^10 + 54201969*x^9 - 172123947*x^8 + 231730104*x^7 - 172770969*x^6 + 271759050*x^5 - 338641350*x^4 + 41402575*x^3 + 177368400*x^2 - 96664125*x + 14269375, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14269375, -96664125, 177368400, 41402575, -338641350, 271759050, -172770969, 231730104, -172123947, 54201969, -28861455, 26111763, -6335185, -529926, -1170738, 441880, 230379, 2520, -33683, -11814, 810, 2387, 195, -9, -82, 0, 0, 1]);
 

\( x^{27} - 82 x^{24} - 9 x^{23} + 195 x^{22} + 2387 x^{21} + 810 x^{20} - 11814 x^{19} - 33683 x^{18} + 2520 x^{17} + 230379 x^{16} + 441880 x^{15} - 1170738 x^{14} - 529926 x^{13} - 6335185 x^{12} + 26111763 x^{11} - 28861455 x^{10} + 54201969 x^{9} - 172123947 x^{8} + 231730104 x^{7} - 172770969 x^{6} + 271759050 x^{5} - 338641350 x^{4} + 41402575 x^{3} + 177368400 x^{2} - 96664125 x + 14269375 \)

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:  \(-100449472719512733423421001345835601036409912109375\)\(\medspace = -\,3^{36}\cdot 5^{13}\cdot 67^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $71.11$
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:  $3, 5, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5} a^{9} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{6} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{7} + \frac{1}{5} a^{3}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{8} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{13} - \frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{14} + \frac{2}{5} a^{6} + \frac{2}{5} a^{2}$, $\frac{1}{5} a^{15} + \frac{2}{5} a^{7} + \frac{2}{5} a^{3}$, $\frac{1}{5} a^{16} + \frac{2}{5} a^{8} + \frac{2}{5} a^{4}$, $\frac{1}{25} a^{17} + \frac{2}{25} a^{15} - \frac{1}{25} a^{14} + \frac{1}{25} a^{12} + \frac{1}{25} a^{11} + \frac{2}{25} a^{10} + \frac{2}{25} a^{9} - \frac{7}{25} a^{8} - \frac{8}{25} a^{7} + \frac{9}{25} a^{6} - \frac{3}{25} a^{5} + \frac{6}{25} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2}$, $\frac{1}{25} a^{18} + \frac{2}{25} a^{16} - \frac{1}{25} a^{15} + \frac{1}{25} a^{13} + \frac{1}{25} a^{12} + \frac{2}{25} a^{11} + \frac{2}{25} a^{10} - \frac{2}{25} a^{9} + \frac{2}{25} a^{8} + \frac{4}{25} a^{7} + \frac{12}{25} a^{6} + \frac{1}{25} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{25} a^{19} - \frac{1}{25} a^{16} + \frac{1}{25} a^{15} - \frac{2}{25} a^{14} + \frac{1}{25} a^{13} - \frac{1}{25} a^{10} - \frac{2}{25} a^{9} - \frac{7}{25} a^{8} - \frac{12}{25} a^{7} - \frac{12}{25} a^{6} + \frac{1}{25} a^{5} + \frac{8}{25} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2}$, $\frac{1}{25} a^{20} + \frac{1}{25} a^{16} + \frac{1}{25} a^{12} - \frac{9}{25} a^{8} + \frac{6}{25} a^{4}$, $\frac{1}{125} a^{21} - \frac{1}{125} a^{20} + \frac{1}{125} a^{19} - \frac{2}{125} a^{18} - \frac{1}{125} a^{17} - \frac{11}{125} a^{16} + \frac{4}{125} a^{15} + \frac{1}{25} a^{14} - \frac{2}{25} a^{13} - \frac{2}{25} a^{12} - \frac{11}{125} a^{11} - \frac{4}{125} a^{10} - \frac{6}{125} a^{9} + \frac{7}{125} a^{8} - \frac{44}{125} a^{7} - \frac{24}{125} a^{6} + \frac{41}{125} a^{5} + \frac{8}{25} a^{4} + \frac{1}{5} a$, $\frac{1}{1375} a^{22} - \frac{4}{1375} a^{21} - \frac{21}{1375} a^{20} - \frac{1}{275} a^{19} - \frac{1}{275} a^{18} + \frac{27}{1375} a^{17} + \frac{17}{1375} a^{16} - \frac{102}{1375} a^{15} + \frac{13}{275} a^{14} + \frac{17}{275} a^{13} - \frac{56}{1375} a^{12} - \frac{81}{1375} a^{11} + \frac{81}{1375} a^{10} + \frac{3}{275} a^{9} + \frac{39}{275} a^{8} - \frac{112}{1375} a^{7} + \frac{658}{1375} a^{6} + \frac{327}{1375} a^{5} + \frac{98}{275} a^{4} + \frac{27}{55} a^{3} + \frac{18}{55} a^{2} + \frac{12}{55} a - \frac{2}{11}$, $\frac{1}{3367375} a^{23} + \frac{199}{673475} a^{22} - \frac{8747}{3367375} a^{21} - \frac{20159}{3367375} a^{20} - \frac{1704}{673475} a^{19} + \frac{18517}{3367375} a^{18} + \frac{12174}{673475} a^{17} + \frac{76721}{3367375} a^{16} + \frac{39407}{3367375} a^{15} - \frac{10421}{134695} a^{14} - \frac{59956}{3367375} a^{13} - \frac{14241}{673475} a^{12} + \frac{5857}{108625} a^{11} + \frac{1844}{3367375} a^{10} - \frac{1592}{61225} a^{9} - \frac{766542}{3367375} a^{8} + \frac{107059}{673475} a^{7} + \frac{53769}{3367375} a^{6} + \frac{1244068}{3367375} a^{5} - \frac{142863}{673475} a^{4} - \frac{11599}{26939} a^{3} - \frac{34234}{134695} a^{2} - \frac{3642}{134695} a - \frac{86}{341}$, $\frac{1}{57245375} a^{24} + \frac{3}{57245375} a^{23} - \frac{3942}{57245375} a^{22} - \frac{213413}{57245375} a^{21} - \frac{920354}{57245375} a^{20} + \frac{5287}{3367375} a^{19} - \frac{800093}{57245375} a^{18} + \frac{67264}{5204125} a^{17} + \frac{2133643}{57245375} a^{16} + \frac{5273409}{57245375} a^{15} + \frac{4701179}{57245375} a^{14} + \frac{4841427}{57245375} a^{13} - \frac{54773}{1846625} a^{12} - \frac{975617}{57245375} a^{11} - \frac{3256411}{57245375} a^{10} + \frac{542061}{57245375} a^{9} - \frac{1556901}{3367375} a^{8} + \frac{28277626}{57245375} a^{7} + \frac{6076317}{57245375} a^{6} + \frac{22077846}{57245375} a^{5} - \frac{119522}{1040825} a^{4} - \frac{23451}{208165} a^{3} + \frac{44723}{134695} a^{2} - \frac{1005634}{2289815} a + \frac{1}{11}$, $\frac{1}{19177200625} a^{25} + \frac{52}{19177200625} a^{24} + \frac{353}{19177200625} a^{23} - \frac{3315288}{19177200625} a^{22} + \frac{315268}{242749375} a^{21} - \frac{215348538}{19177200625} a^{20} - \frac{87233771}{19177200625} a^{19} - \frac{260331279}{19177200625} a^{18} + \frac{342710784}{19177200625} a^{17} + \frac{1175372974}{19177200625} a^{16} - \frac{33652726}{19177200625} a^{15} - \frac{1669167407}{19177200625} a^{14} - \frac{840205073}{19177200625} a^{13} + \frac{1321964933}{19177200625} a^{12} - \frac{84493997}{19177200625} a^{11} - \frac{50500427}{1743381875} a^{10} - \frac{1406095124}{19177200625} a^{9} - \frac{3194292796}{19177200625} a^{8} + \frac{8650925641}{19177200625} a^{7} + \frac{50324413}{1128070625} a^{6} + \frac{692816873}{3835440125} a^{5} - \frac{58657842}{153417605} a^{4} - \frac{64760404}{153417605} a^{3} + \frac{307078821}{767088025} a^{2} - \frac{175665}{457963} a - \frac{151}{2077}$, $\frac{1}{3096246747813418256280379993527333516580089372343749982879430698311941646875} a^{26} + \frac{48374877579435475536883969904982045397002671376065929066652022471}{3096246747813418256280379993527333516580089372343749982879430698311941646875} a^{25} + \frac{4644476981148034208878505527188497925425820560473086401462214735941}{3096246747813418256280379993527333516580089372343749982879430698311941646875} a^{24} - \frac{372496166416755731753593744941450524764381752760310290071796251965221}{3096246747813418256280379993527333516580089372343749982879430698311941646875} a^{23} - \frac{35302683230026498785145978387799476733851367750853367142825555658416649}{123849869912536730251215199741093340663203574893749999315177227932477665875} a^{22} + \frac{877714909058968241439739984080671667118077888928081732536229267514459344}{619249349562683651256075998705466703316017874468749996575886139662388329375} a^{21} - \frac{25964514671038050747028312486103654153146476076200473488467830827721089518}{3096246747813418256280379993527333516580089372343749982879430698311941646875} a^{20} + \frac{17206459057714449267703449386486226865547191118836714723644635196468859632}{3096246747813418256280379993527333516580089372343749982879430698311941646875} a^{19} - \frac{17519128157984530368799793042600150082372989012191843637115734556697425067}{3096246747813418256280379993527333516580089372343749982879430698311941646875} a^{18} + \frac{12221466782459008374789542093961611570124755624468290446862880701803828367}{619249349562683651256075998705466703316017874468749996575886139662388329375} a^{17} + \frac{51032024371689363353186591969329227871783544402180594635008899666990009581}{619249349562683651256075998705466703316017874468749996575886139662388329375} a^{16} - \frac{134305898075412362412658775699911532194499257616969450386094023210085985866}{3096246747813418256280379993527333516580089372343749982879430698311941646875} a^{15} - \frac{68622069929379986580189517647899772148772621987427215633331048372270760881}{3096246747813418256280379993527333516580089372343749982879430698311941646875} a^{14} - \frac{259137985264417714192629919888770676943163495984263654580161664184982465964}{3096246747813418256280379993527333516580089372343749982879430698311941646875} a^{13} + \frac{49260239924535005276683757146261752733341449731780746526353198706740884631}{619249349562683651256075998705466703316017874468749996575886139662388329375} a^{12} + \frac{882923461662570918899036747799549158123307272581290768827049593030207343}{26923884763594941358959826030672465361565994542119565068516788680973405625} a^{11} - \frac{170464268399964696909109191120670736537878985511306207378103237528599124742}{3096246747813418256280379993527333516580089372343749982879430698311941646875} a^{10} - \frac{194188906847382377018029430310522709397943529554532598477658559928551007737}{3096246747813418256280379993527333516580089372343749982879430698311941646875} a^{9} - \frac{484351046042632583200360890070789178246553363293163768067341510666516610233}{3096246747813418256280379993527333516580089372343749982879430698311941646875} a^{8} + \frac{94385995148289496409238179950064244153600407534236408815013035044640934422}{619249349562683651256075998705466703316017874468749996575886139662388329375} a^{7} - \frac{121588561384568462460150780001363677430049720199445103963250096561017374686}{3096246747813418256280379993527333516580089372343749982879430698311941646875} a^{6} + \frac{31179947266405183673198172369186129982213942633346269732437092243389806147}{123849869912536730251215199741093340663203574893749999315177227932477665875} a^{5} + \frac{10860313579697435708469924736479104196942514013473905035443956591016015273}{24769973982507346050243039948218668132640714978749999863035445586495533175} a^{4} - \frac{2124420091080165587555608052983941890125621609515787717875570403607118284}{11259079082957884568292290885553940060291234081249999937743384357497969625} a^{3} + \frac{38105890513039209436838915311260424032624613808433810023450064591972934974}{123849869912536730251215199741093340663203574893749999315177227932477665875} a^{2} + \frac{335002189200070615005306275517339196534492330775043236880321342583721088}{2251815816591576913658458177110788012058246816249999987548676871499593925} a + \frac{1810820203602218907309616802610824186648669754576169403646976592734908}{3688752640730803581570072963249243206647909900037230061509373877363445}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 716753350335435.2 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{13}\cdot 716753350335435.2 \cdot 3}{2\sqrt{100449472719512733423421001345835601036409912109375}}\approx 5.10335105787073$ (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

3.1.335.1, 9.1.12594450625.1

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 $27$ R R $27$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ $27$ $27$

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
3Data not computed
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
$67$$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.335.2t1.a.a$1$ $ 5 \cdot 67 $ $x^{2} - x + 84$ $C_2$ (as 2T1) $1$ $-1$
* 2.335.3t2.a.a$2$ $ 5 \cdot 67 $ $x^{3} - x^{2} + 4 x + 1$ $S_3$ (as 3T2) $1$ $0$
* 2.335.9t3.a.c$2$ $ 5 \cdot 67 $ $x^{9} - 3 x^{8} + x^{7} + 4 x^{6} - 2 x^{5} - 6 x^{4} - 2 x^{3} + 7 x^{2} - 5$ $D_{9}$ (as 9T3) $1$ $0$
* 2.335.9t3.a.a$2$ $ 5 \cdot 67 $ $x^{9} - 3 x^{8} + x^{7} + 4 x^{6} - 2 x^{5} - 6 x^{4} - 2 x^{3} + 7 x^{2} - 5$ $D_{9}$ (as 9T3) $1$ $0$
* 2.335.9t3.a.b$2$ $ 5 \cdot 67 $ $x^{9} - 3 x^{8} + x^{7} + 4 x^{6} - 2 x^{5} - 6 x^{4} - 2 x^{3} + 7 x^{2} - 5$ $D_{9}$ (as 9T3) $1$ $0$
* 2.27135.27t8.a.f$2$ $ 3^{4} \cdot 5 \cdot 67 $ $x^{27} - 82 x^{24} - 9 x^{23} + 195 x^{22} + 2387 x^{21} + 810 x^{20} - 11814 x^{19} - 33683 x^{18} + 2520 x^{17} + 230379 x^{16} + 441880 x^{15} - 1170738 x^{14} - 529926 x^{13} - 6335185 x^{12} + 26111763 x^{11} - 28861455 x^{10} + 54201969 x^{9} - 172123947 x^{8} + 231730104 x^{7} - 172770969 x^{6} + 271759050 x^{5} - 338641350 x^{4} + 41402575 x^{3} + 177368400 x^{2} - 96664125 x + 14269375$ $D_{27}$ (as 27T8) $1$ $0$
* 2.27135.27t8.a.d$2$ $ 3^{4} \cdot 5 \cdot 67 $ $x^{27} - 82 x^{24} - 9 x^{23} + 195 x^{22} + 2387 x^{21} + 810 x^{20} - 11814 x^{19} - 33683 x^{18} + 2520 x^{17} + 230379 x^{16} + 441880 x^{15} - 1170738 x^{14} - 529926 x^{13} - 6335185 x^{12} + 26111763 x^{11} - 28861455 x^{10} + 54201969 x^{9} - 172123947 x^{8} + 231730104 x^{7} - 172770969 x^{6} + 271759050 x^{5} - 338641350 x^{4} + 41402575 x^{3} + 177368400 x^{2} - 96664125 x + 14269375$ $D_{27}$ (as 27T8) $1$ $0$
* 2.27135.27t8.a.c$2$ $ 3^{4} \cdot 5 \cdot 67 $ $x^{27} - 82 x^{24} - 9 x^{23} + 195 x^{22} + 2387 x^{21} + 810 x^{20} - 11814 x^{19} - 33683 x^{18} + 2520 x^{17} + 230379 x^{16} + 441880 x^{15} - 1170738 x^{14} - 529926 x^{13} - 6335185 x^{12} + 26111763 x^{11} - 28861455 x^{10} + 54201969 x^{9} - 172123947 x^{8} + 231730104 x^{7} - 172770969 x^{6} + 271759050 x^{5} - 338641350 x^{4} + 41402575 x^{3} + 177368400 x^{2} - 96664125 x + 14269375$ $D_{27}$ (as 27T8) $1$ $0$
* 2.27135.27t8.a.a$2$ $ 3^{4} \cdot 5 \cdot 67 $ $x^{27} - 82 x^{24} - 9 x^{23} + 195 x^{22} + 2387 x^{21} + 810 x^{20} - 11814 x^{19} - 33683 x^{18} + 2520 x^{17} + 230379 x^{16} + 441880 x^{15} - 1170738 x^{14} - 529926 x^{13} - 6335185 x^{12} + 26111763 x^{11} - 28861455 x^{10} + 54201969 x^{9} - 172123947 x^{8} + 231730104 x^{7} - 172770969 x^{6} + 271759050 x^{5} - 338641350 x^{4} + 41402575 x^{3} + 177368400 x^{2} - 96664125 x + 14269375$ $D_{27}$ (as 27T8) $1$ $0$
* 2.27135.27t8.a.b$2$ $ 3^{4} \cdot 5 \cdot 67 $ $x^{27} - 82 x^{24} - 9 x^{23} + 195 x^{22} + 2387 x^{21} + 810 x^{20} - 11814 x^{19} - 33683 x^{18} + 2520 x^{17} + 230379 x^{16} + 441880 x^{15} - 1170738 x^{14} - 529926 x^{13} - 6335185 x^{12} + 26111763 x^{11} - 28861455 x^{10} + 54201969 x^{9} - 172123947 x^{8} + 231730104 x^{7} - 172770969 x^{6} + 271759050 x^{5} - 338641350 x^{4} + 41402575 x^{3} + 177368400 x^{2} - 96664125 x + 14269375$ $D_{27}$ (as 27T8) $1$ $0$
* 2.27135.27t8.a.e$2$ $ 3^{4} \cdot 5 \cdot 67 $ $x^{27} - 82 x^{24} - 9 x^{23} + 195 x^{22} + 2387 x^{21} + 810 x^{20} - 11814 x^{19} - 33683 x^{18} + 2520 x^{17} + 230379 x^{16} + 441880 x^{15} - 1170738 x^{14} - 529926 x^{13} - 6335185 x^{12} + 26111763 x^{11} - 28861455 x^{10} + 54201969 x^{9} - 172123947 x^{8} + 231730104 x^{7} - 172770969 x^{6} + 271759050 x^{5} - 338641350 x^{4} + 41402575 x^{3} + 177368400 x^{2} - 96664125 x + 14269375$ $D_{27}$ (as 27T8) $1$ $0$
* 2.27135.27t8.a.i$2$ $ 3^{4} \cdot 5 \cdot 67 $ $x^{27} - 82 x^{24} - 9 x^{23} + 195 x^{22} + 2387 x^{21} + 810 x^{20} - 11814 x^{19} - 33683 x^{18} + 2520 x^{17} + 230379 x^{16} + 441880 x^{15} - 1170738 x^{14} - 529926 x^{13} - 6335185 x^{12} + 26111763 x^{11} - 28861455 x^{10} + 54201969 x^{9} - 172123947 x^{8} + 231730104 x^{7} - 172770969 x^{6} + 271759050 x^{5} - 338641350 x^{4} + 41402575 x^{3} + 177368400 x^{2} - 96664125 x + 14269375$ $D_{27}$ (as 27T8) $1$ $0$
* 2.27135.27t8.a.h$2$ $ 3^{4} \cdot 5 \cdot 67 $ $x^{27} - 82 x^{24} - 9 x^{23} + 195 x^{22} + 2387 x^{21} + 810 x^{20} - 11814 x^{19} - 33683 x^{18} + 2520 x^{17} + 230379 x^{16} + 441880 x^{15} - 1170738 x^{14} - 529926 x^{13} - 6335185 x^{12} + 26111763 x^{11} - 28861455 x^{10} + 54201969 x^{9} - 172123947 x^{8} + 231730104 x^{7} - 172770969 x^{6} + 271759050 x^{5} - 338641350 x^{4} + 41402575 x^{3} + 177368400 x^{2} - 96664125 x + 14269375$ $D_{27}$ (as 27T8) $1$ $0$
* 2.27135.27t8.a.g$2$ $ 3^{4} \cdot 5 \cdot 67 $ $x^{27} - 82 x^{24} - 9 x^{23} + 195 x^{22} + 2387 x^{21} + 810 x^{20} - 11814 x^{19} - 33683 x^{18} + 2520 x^{17} + 230379 x^{16} + 441880 x^{15} - 1170738 x^{14} - 529926 x^{13} - 6335185 x^{12} + 26111763 x^{11} - 28861455 x^{10} + 54201969 x^{9} - 172123947 x^{8} + 231730104 x^{7} - 172770969 x^{6} + 271759050 x^{5} - 338641350 x^{4} + 41402575 x^{3} + 177368400 x^{2} - 96664125 x + 14269375$ $D_{27}$ (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 *.