Properties

Label 27.1.929...947.1
Degree $27$
Signature $[1, 13]$
Discriminant $-9.298\times 10^{50}$
Root discriminant $77.22$
Ramified primes $7, 563$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group $D_{27}$ (as 27T8)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 11*x^26 + 7*x^25 + 393*x^24 - 1777*x^23 - 1339*x^22 + 33149*x^21 - 122161*x^20 + 301221*x^19 - 901165*x^18 + 2771309*x^17 - 5696089*x^16 + 7240134*x^15 - 11573472*x^14 + 45471960*x^13 - 153604016*x^12 + 357338624*x^11 - 615897728*x^10 + 876134656*x^9 - 1154442240*x^8 + 1414654976*x^7 - 1520924672*x^6 + 1526870016*x^5 - 1494863872*x^4 + 1216438272*x^3 - 684785664*x^2 + 248741888*x - 73596928)
 
gp: K = bnfinit(x^27 - 11*x^26 + 7*x^25 + 393*x^24 - 1777*x^23 - 1339*x^22 + 33149*x^21 - 122161*x^20 + 301221*x^19 - 901165*x^18 + 2771309*x^17 - 5696089*x^16 + 7240134*x^15 - 11573472*x^14 + 45471960*x^13 - 153604016*x^12 + 357338624*x^11 - 615897728*x^10 + 876134656*x^9 - 1154442240*x^8 + 1414654976*x^7 - 1520924672*x^6 + 1526870016*x^5 - 1494863872*x^4 + 1216438272*x^3 - 684785664*x^2 + 248741888*x - 73596928, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-73596928, 248741888, -684785664, 1216438272, -1494863872, 1526870016, -1520924672, 1414654976, -1154442240, 876134656, -615897728, 357338624, -153604016, 45471960, -11573472, 7240134, -5696089, 2771309, -901165, 301221, -122161, 33149, -1339, -1777, 393, 7, -11, 1]);
 

\( x^{27} - 11 x^{26} + 7 x^{25} + 393 x^{24} - 1777 x^{23} - 1339 x^{22} + 33149 x^{21} - 122161 x^{20} + 301221 x^{19} - 901165 x^{18} + 2771309 x^{17} - 5696089 x^{16} + 7240134 x^{15} - 11573472 x^{14} + 45471960 x^{13} - 153604016 x^{12} + 357338624 x^{11} - 615897728 x^{10} + 876134656 x^{9} - 1154442240 x^{8} + 1414654976 x^{7} - 1520924672 x^{6} + 1526870016 x^{5} - 1494863872 x^{4} + 1216438272 x^{3} - 684785664 x^{2} + 248741888 x - 73596928 \)

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:  \(-929766412363569678535445062868135961189021577764947\)\(\medspace = -\,7^{18}\cdot 563^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $77.22$
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:  $7, 563$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{16} a^{10} + \frac{1}{16} a^{9} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{3}{16} a^{4} - \frac{3}{16} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{6} + \frac{1}{16} a^{5} + \frac{7}{16} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{9} - \frac{1}{16} a^{6} - \frac{1}{4} a^{5} + \frac{1}{16} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{32} a^{13} - \frac{1}{32} a^{10} - \frac{1}{8} a^{9} + \frac{3}{32} a^{7} + \frac{5}{32} a^{4} - \frac{1}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{64} a^{14} - \frac{1}{64} a^{11} + \frac{1}{16} a^{9} + \frac{3}{64} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{11}{64} a^{5} - \frac{1}{4} a^{4} + \frac{7}{16} a^{3} - \frac{3}{8} a^{2} - \frac{1}{2}$, $\frac{1}{256} a^{15} - \frac{5}{256} a^{12} - \frac{1}{32} a^{11} - \frac{1}{32} a^{10} + \frac{19}{256} a^{9} - \frac{3}{32} a^{8} + \frac{1}{256} a^{6} + \frac{5}{32} a^{5} - \frac{3}{16} a^{4} + \frac{1}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{512} a^{16} - \frac{5}{512} a^{13} + \frac{1}{64} a^{12} - \frac{1}{64} a^{11} + \frac{3}{512} a^{10} + \frac{1}{64} a^{9} + \frac{33}{512} a^{7} - \frac{1}{64} a^{6} - \frac{3}{32} a^{5} - \frac{3}{32} a^{4} + \frac{1}{8} a^{3} - \frac{1}{2} a^{2} + \frac{1}{8} a$, $\frac{1}{1024} a^{17} - \frac{5}{1024} a^{14} + \frac{1}{128} a^{13} - \frac{1}{128} a^{12} + \frac{3}{1024} a^{11} + \frac{1}{128} a^{10} + \frac{33}{1024} a^{8} + \frac{15}{128} a^{7} + \frac{5}{64} a^{6} + \frac{13}{64} a^{5} + \frac{3}{16} a^{4} + \frac{3}{8} a^{3} + \frac{5}{16} a^{2}$, $\frac{1}{1024} a^{18} - \frac{1}{1024} a^{15} - \frac{1}{128} a^{14} - \frac{1}{128} a^{13} - \frac{17}{1024} a^{12} - \frac{1}{128} a^{11} - \frac{1}{32} a^{10} + \frac{45}{1024} a^{9} - \frac{3}{128} a^{8} - \frac{3}{64} a^{7} + \frac{21}{256} a^{6} + \frac{1}{64} a^{5} + \frac{3}{16} a^{4} - \frac{1}{4} a^{3} + \frac{3}{8} a^{2} + \frac{1}{4}$, $\frac{1}{1024} a^{19} - \frac{1}{1024} a^{16} - \frac{1}{128} a^{14} + \frac{15}{1024} a^{13} + \frac{1}{64} a^{12} - \frac{1}{32} a^{11} + \frac{13}{1024} a^{10} - \frac{1}{16} a^{9} - \frac{7}{64} a^{8} + \frac{13}{256} a^{7} - \frac{5}{128} a^{6} + \frac{1}{16} a^{5} - \frac{5}{32} a^{4} + \frac{1}{16} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{20480} a^{20} + \frac{7}{20480} a^{19} - \frac{7}{20480} a^{18} + \frac{3}{20480} a^{17} - \frac{3}{20480} a^{16} - \frac{1}{4096} a^{15} + \frac{43}{20480} a^{14} + \frac{157}{20480} a^{13} + \frac{603}{20480} a^{12} - \frac{319}{20480} a^{11} + \frac{59}{4096} a^{10} - \frac{407}{20480} a^{9} - \frac{23}{320} a^{8} - \frac{237}{2560} a^{7} + \frac{239}{2560} a^{6} - \frac{1}{8} a^{5} + \frac{63}{320} a^{4} + \frac{159}{320} a^{3} - \frac{19}{40} a^{2} - \frac{17}{40} a - \frac{7}{40}$, $\frac{1}{81920} a^{21} + \frac{1}{81920} a^{20} - \frac{29}{81920} a^{19} - \frac{7}{16384} a^{18} - \frac{21}{81920} a^{17} - \frac{7}{81920} a^{16} + \frac{153}{81920} a^{15} + \frac{379}{81920} a^{14} - \frac{39}{81920} a^{13} - \frac{977}{81920} a^{12} + \frac{2209}{81920} a^{11} + \frac{1283}{81920} a^{10} + \frac{761}{8192} a^{9} + \frac{187}{10240} a^{8} - \frac{529}{10240} a^{7} + \frac{493}{5120} a^{6} + \frac{223}{1280} a^{5} - \frac{79}{1280} a^{4} - \frac{193}{640} a^{3} - \frac{3}{160} a^{2} + \frac{9}{32} a - \frac{29}{80}$, $\frac{1}{163840} a^{22} - \frac{1}{163840} a^{21} + \frac{1}{163840} a^{20} - \frac{73}{163840} a^{19} + \frac{13}{32768} a^{18} - \frac{29}{163840} a^{17} + \frac{71}{163840} a^{16} - \frac{7}{163840} a^{15} + \frac{739}{163840} a^{14} - \frac{455}{32768} a^{13} + \frac{1139}{163840} a^{12} + \frac{897}{163840} a^{11} + \frac{741}{40960} a^{10} - \frac{1827}{40960} a^{9} - \frac{1291}{20480} a^{8} + \frac{1}{1024} a^{7} + \frac{5}{1024} a^{6} + \frac{79}{512} a^{5} - \frac{153}{640} a^{4} - \frac{121}{640} a^{3} + \frac{143}{320} a^{2} + \frac{37}{80} a - \frac{27}{80}$, $\frac{1}{655360} a^{23} - \frac{1}{655360} a^{22} - \frac{3}{655360} a^{21} + \frac{3}{655360} a^{20} - \frac{219}{655360} a^{19} - \frac{289}{655360} a^{18} + \frac{15}{131072} a^{17} + \frac{101}{655360} a^{16} + \frac{207}{655360} a^{15} + \frac{5089}{655360} a^{14} - \frac{237}{131072} a^{13} + \frac{1745}{131072} a^{12} + \frac{629}{20480} a^{11} - \frac{401}{16384} a^{10} - \frac{493}{40960} a^{9} + \frac{271}{2560} a^{8} - \frac{403}{10240} a^{7} - \frac{51}{1280} a^{6} - \frac{37}{320} a^{5} + \frac{49}{1280} a^{4} - \frac{67}{640} a^{3} - \frac{9}{32} a^{2} + \frac{3}{20} a + \frac{59}{160}$, $\frac{1}{56360960} a^{24} - \frac{13}{56360960} a^{23} - \frac{47}{56360960} a^{22} - \frac{337}{56360960} a^{21} - \frac{1319}{56360960} a^{20} + \frac{17483}{56360960} a^{19} - \frac{1821}{11272192} a^{18} + \frac{1769}{56360960} a^{17} - \frac{229}{56360960} a^{16} + \frac{106181}{56360960} a^{15} - \frac{116773}{56360960} a^{14} - \frac{10279}{56360960} a^{13} - \frac{80353}{2818048} a^{12} + \frac{70923}{3522560} a^{11} - \frac{22369}{1761280} a^{10} - \frac{97827}{880640} a^{9} + \frac{24789}{220160} a^{8} + \frac{37299}{440320} a^{7} - \frac{185}{5504} a^{6} - \frac{7051}{55040} a^{5} - \frac{577}{27520} a^{4} + \frac{257}{6880} a^{3} - \frac{727}{13760} a^{2} + \frac{529}{13760} a - \frac{1019}{3440}$, $\frac{1}{141466009600} a^{25} + \frac{423}{141466009600} a^{24} - \frac{60927}{141466009600} a^{23} + \frac{366687}{141466009600} a^{22} - \frac{663047}{141466009600} a^{21} - \frac{1768309}{141466009600} a^{20} + \frac{6932819}{28293201920} a^{19} - \frac{21271727}{141466009600} a^{18} + \frac{20167323}{141466009600} a^{17} + \frac{130105589}{141466009600} a^{16} + \frac{83873147}{141466009600} a^{15} - \frac{6938719}{5658640384} a^{14} + \frac{72618983}{35366502400} a^{13} + \frac{15509509}{35366502400} a^{12} - \frac{180241603}{8841625600} a^{11} + \frac{30115279}{2210406400} a^{10} - \frac{247488621}{2210406400} a^{9} - \frac{12324203}{110520320} a^{8} - \frac{71869}{1381504} a^{7} + \frac{1007317}{27630080} a^{6} + \frac{5235779}{138150400} a^{5} - \frac{1911407}{8634400} a^{4} + \frac{792099}{17268800} a^{3} - \frac{16807717}{34537600} a^{2} + \frac{3946809}{8634400} a - \frac{1250577}{8634400}$, $\frac{1}{4979345461732465077885898611318475456555041294260199541964800} a^{26} + \frac{11756009269330471519733084296884465864324858897619}{4979345461732465077885898611318475456555041294260199541964800} a^{25} - \frac{23829446651020766677116185028853378642747558482396679}{4979345461732465077885898611318475456555041294260199541964800} a^{24} - \frac{669007824991844770769162328653890713433685031637573229}{995869092346493015577179722263695091311008258852039908392960} a^{23} + \frac{62496003734858923597227138079618577449753688619744113}{199173818469298603115435944452739018262201651770407981678592} a^{22} + \frac{25567097781630190447390427794555505247726516376242622459}{4979345461732465077885898611318475456555041294260199541964800} a^{21} - \frac{118700760926372038113156739255248818424378132128141425409}{4979345461732465077885898611318475456555041294260199541964800} a^{20} + \frac{833398405479186091221824617928285859079185247580697818873}{4979345461732465077885898611318475456555041294260199541964800} a^{19} + \frac{2352792828520977668611128403990842413530996109374567120571}{4979345461732465077885898611318475456555041294260199541964800} a^{18} - \frac{1557692176862068798291252878520218931155304555977503359083}{4979345461732465077885898611318475456555041294260199541964800} a^{17} - \frac{143083490153258967350622546927805297617696470830511499659}{160624047152660163802770922945757272792098106266458049740800} a^{16} - \frac{3660407624669475542281696580517641098230940451296654348183}{4979345461732465077885898611318475456555041294260199541964800} a^{15} - \frac{6582339359749634080034492366754242795960689068538446742217}{1244836365433116269471474652829618864138760323565049885491200} a^{14} + \frac{644684309923195465273161453605599590353281233471227483337}{77802272839569766841967165801851179008672520222815617843200} a^{13} - \frac{7968925567489152390349500533450372018248767989405407739}{20078005894082520475346365368219659099012263283307256217600} a^{12} + \frac{1856463088977654677703024160082882175352825238618056926047}{77802272839569766841967165801851179008672520222815617843200} a^{11} - \frac{466043004071759490513434739351829970108196939524334635751}{38901136419784883420983582900925589504336260111407808921600} a^{10} - \frac{2513151138941567819367967296839838398957023463708446061993}{38901136419784883420983582900925589504336260111407808921600} a^{9} + \frac{11852566867450141008963180813379639252123015712225530767}{97252841049462208552458957252313973760840650278519522304} a^{8} + \frac{37678608941190829707835964317707818029262701281823252907}{486264205247311042762294786261569868804203251392597611520} a^{7} - \frac{6549295412298497902110040376925801732857418312335347839}{167677312154245187159411995262610299587656293583654348800} a^{6} - \frac{21427883021986337142908736681572669373362121987652895989}{151957564139784700863217120706740584001313516060186753600} a^{5} - \frac{1105839512653952881747307964249420783389351648965931341}{8383865607712259357970599763130514979382814679182717440} a^{4} + \frac{467681760193625802046795286244974591166058048684506897881}{1215660513118277606905736965653924672010508128481494028800} a^{3} - \frac{110146925973164977941791215482061450776737390450230972359}{303915128279569401726434241413481168002627032120373507200} a^{2} - \frac{1310581185011603542912805169487220167133913711155037971}{37989391034946175215804280176685146000328379015046688400} a - \frac{66835142597591681217857703332855661785428079311693575741}{151957564139784700863217120706740584001313516060186753600}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  \( 5470716283346407.0 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{13}\cdot 5470716283346407.0 \cdot 6}{2\sqrt{929766412363569678535445062868135961189021577764947}}\approx 25.6062985545585$ (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

3.1.563.1, 9.1.100469346961.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ $27$ $27$ $27$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ $27$

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
7Data not computed
563Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.563.2t1.a.a$1$ $ 563 $ \(\Q(\sqrt{-563}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.563.3t2.a.a$2$ $ 563 $ 3.1.563.1 $S_3$ (as 3T2) $1$ $0$
* 2.563.9t3.a.b$2$ $ 563 $ 9.1.100469346961.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.563.9t3.a.c$2$ $ 563 $ 9.1.100469346961.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.563.9t3.a.a$2$ $ 563 $ 9.1.100469346961.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.27587.27t8.a.f$2$ $ 7^{2} \cdot 563 $ 27.1.929766412363569678535445062868135961189021577764947.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.27587.27t8.a.d$2$ $ 7^{2} \cdot 563 $ 27.1.929766412363569678535445062868135961189021577764947.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.27587.27t8.a.c$2$ $ 7^{2} \cdot 563 $ 27.1.929766412363569678535445062868135961189021577764947.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.27587.27t8.a.a$2$ $ 7^{2} \cdot 563 $ 27.1.929766412363569678535445062868135961189021577764947.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.27587.27t8.a.b$2$ $ 7^{2} \cdot 563 $ 27.1.929766412363569678535445062868135961189021577764947.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.27587.27t8.a.e$2$ $ 7^{2} \cdot 563 $ 27.1.929766412363569678535445062868135961189021577764947.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.27587.27t8.a.i$2$ $ 7^{2} \cdot 563 $ 27.1.929766412363569678535445062868135961189021577764947.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.27587.27t8.a.h$2$ $ 7^{2} \cdot 563 $ 27.1.929766412363569678535445062868135961189021577764947.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.27587.27t8.a.g$2$ $ 7^{2} \cdot 563 $ 27.1.929766412363569678535445062868135961189021577764947.1 $D_{27}$ (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 *.