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Properties

Label 27.1.108...375.1
Degree $27$
Signature $[1, 13]$
Discriminant $-1.090\times 10^{48}$
Root discriminant $60.14$
Ramified primes $5, 7, 67$
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 - 8*x^26 + 11*x^25 + 79*x^24 - 318*x^23 + 272*x^22 + 1817*x^21 - 11487*x^20 + 27792*x^19 - 5214*x^18 - 88324*x^17 + 101476*x^16 + 384139*x^15 - 2894444*x^14 + 8285833*x^13 - 7065937*x^12 - 6208737*x^11 - 6594753*x^10 + 51530498*x^9 - 40765921*x^8 + 72314518*x^7 - 318855243*x^6 + 340132420*x^5 + 140246505*x^4 - 148040525*x^3 - 594053575*x^2 + 805097125*x - 295863625)
 
gp: K = bnfinit(x^27 - 8*x^26 + 11*x^25 + 79*x^24 - 318*x^23 + 272*x^22 + 1817*x^21 - 11487*x^20 + 27792*x^19 - 5214*x^18 - 88324*x^17 + 101476*x^16 + 384139*x^15 - 2894444*x^14 + 8285833*x^13 - 7065937*x^12 - 6208737*x^11 - 6594753*x^10 + 51530498*x^9 - 40765921*x^8 + 72314518*x^7 - 318855243*x^6 + 340132420*x^5 + 140246505*x^4 - 148040525*x^3 - 594053575*x^2 + 805097125*x - 295863625, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-295863625, 805097125, -594053575, -148040525, 140246505, 340132420, -318855243, 72314518, -40765921, 51530498, -6594753, -6208737, -7065937, 8285833, -2894444, 384139, 101476, -88324, -5214, 27792, -11487, 1817, 272, -318, 79, 11, -8, 1]);
 

\( x^{27} - 8 x^{26} + 11 x^{25} + 79 x^{24} - 318 x^{23} + 272 x^{22} + 1817 x^{21} - 11487 x^{20} + 27792 x^{19} - 5214 x^{18} - 88324 x^{17} + 101476 x^{16} + 384139 x^{15} - 2894444 x^{14} + 8285833 x^{13} - 7065937 x^{12} - 6208737 x^{11} - 6594753 x^{10} + 51530498 x^{9} - 40765921 x^{8} + 72314518 x^{7} - 318855243 x^{6} + 340132420 x^{5} + 140246505 x^{4} - 148040525 x^{3} - 594053575 x^{2} + 805097125 x - 295863625 \)

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:  \(-1089801024238603052304820653163822019730224609375\)\(\medspace = -\,5^{13}\cdot 7^{18}\cdot 67^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $60.14$
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:  $5, 7, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{6} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{7} + \frac{1}{5} a^{3}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{8} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{9} + \frac{1}{5} a^{5}$, $\frac{1}{5} a^{14} + \frac{2}{5} a^{6} + \frac{2}{5} a^{2}$, $\frac{1}{5} a^{15} + \frac{2}{5} a^{7} + \frac{2}{5} a^{3}$, $\frac{1}{5} a^{16} + \frac{2}{5} a^{8} + \frac{2}{5} a^{4}$, $\frac{1}{5} a^{17} + \frac{2}{5} a^{9} + \frac{2}{5} a^{5}$, $\frac{1}{25} a^{18} + \frac{1}{25} a^{17} + \frac{2}{25} a^{16} - \frac{2}{25} a^{15} + \frac{1}{25} a^{14} - \frac{1}{25} a^{13} - \frac{1}{25} a^{11} + \frac{9}{25} a^{9} - \frac{11}{25} a^{8} - \frac{12}{25} a^{7} - \frac{7}{25} a^{6} + \frac{11}{25} a^{5} - \frac{6}{25} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{25} a^{19} + \frac{1}{25} a^{17} + \frac{1}{25} a^{16} - \frac{2}{25} a^{15} - \frac{2}{25} a^{14} + \frac{1}{25} a^{13} - \frac{1}{25} a^{12} + \frac{1}{25} a^{11} - \frac{1}{25} a^{10} + \frac{1}{5} a^{9} + \frac{9}{25} a^{8} - \frac{1}{5} a^{7} - \frac{12}{25} a^{6} + \frac{8}{25} a^{5} + \frac{11}{25} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{25} a^{20} + \frac{1}{25} a^{16} + \frac{1}{25} a^{12} - \frac{9}{25} a^{8} + \frac{6}{25} a^{4}$, $\frac{1}{25} a^{21} + \frac{1}{25} a^{17} + \frac{1}{25} a^{13} - \frac{9}{25} a^{9} + \frac{6}{25} a^{5}$, $\frac{1}{2125} a^{22} - \frac{36}{2125} a^{21} - \frac{36}{2125} a^{20} + \frac{3}{425} a^{19} + \frac{43}{2125} a^{17} - \frac{108}{2125} a^{16} - \frac{138}{2125} a^{15} - \frac{2}{25} a^{14} - \frac{41}{425} a^{13} - \frac{46}{2125} a^{12} - \frac{94}{2125} a^{11} - \frac{149}{2125} a^{10} - \frac{137}{425} a^{9} + \frac{18}{425} a^{8} - \frac{863}{2125} a^{7} + \frac{1028}{2125} a^{6} - \frac{287}{2125} a^{5} + \frac{103}{425} a^{4} + \frac{22}{85} a^{3} - \frac{3}{17} a^{2} - \frac{2}{85} a + \frac{1}{17}$, $\frac{1}{1566125} a^{23} + \frac{1}{92125} a^{22} + \frac{27211}{1566125} a^{21} + \frac{28112}{1566125} a^{20} - \frac{52}{28475} a^{19} - \frac{10157}{1566125} a^{18} + \frac{52321}{1566125} a^{17} - \frac{75987}{1566125} a^{16} - \frac{54149}{1566125} a^{15} - \frac{31321}{313225} a^{14} + \frac{129764}{1566125} a^{13} + \frac{71503}{1566125} a^{12} - \frac{131611}{1566125} a^{11} - \frac{107352}{1566125} a^{10} - \frac{14587}{313225} a^{9} - \frac{494108}{1566125} a^{8} + \frac{610639}{1566125} a^{7} - \frac{669068}{1566125} a^{6} - \frac{269781}{1566125} a^{5} + \frac{154744}{313225} a^{4} + \frac{26787}{62645} a^{3} - \frac{17763}{62645} a^{2} + \frac{67}{935} a + \frac{49}{187}$, $\frac{1}{1566125} a^{24} - \frac{347}{1566125} a^{22} - \frac{16596}{1566125} a^{21} - \frac{48}{313225} a^{20} + \frac{5298}{1566125} a^{19} - \frac{5118}{313225} a^{18} - \frac{70726}{1566125} a^{17} + \frac{110757}{1566125} a^{16} - \frac{1976}{28475} a^{15} - \frac{152266}{1566125} a^{14} + \frac{14566}{313225} a^{13} - \frac{30143}{1566125} a^{12} - \frac{5054}{1566125} a^{11} + \frac{22887}{313225} a^{10} + \frac{5102}{1566125} a^{9} + \frac{120998}{313225} a^{8} - \frac{171074}{1566125} a^{7} - \frac{264587}{1566125} a^{6} + \frac{26288}{62645} a^{5} - \frac{18182}{313225} a^{4} + \frac{16963}{62645} a^{3} + \frac{1261}{5695} a^{2} - \frac{16}{187} a + \frac{69}{187}$, $\frac{1}{4019358014375} a^{25} - \frac{1092866}{4019358014375} a^{24} + \frac{476863}{4019358014375} a^{23} + \frac{733574131}{4019358014375} a^{22} + \frac{50923054101}{4019358014375} a^{21} - \frac{70866659987}{4019358014375} a^{20} - \frac{62376008723}{4019358014375} a^{19} + \frac{79492017619}{4019358014375} a^{18} - \frac{146356384442}{4019358014375} a^{17} + \frac{299501636793}{4019358014375} a^{16} - \frac{246968380411}{4019358014375} a^{15} + \frac{220948660616}{4019358014375} a^{14} - \frac{18324496293}{365396183125} a^{13} + \frac{51219224684}{4019358014375} a^{12} + \frac{281579707479}{4019358014375} a^{11} + \frac{16433562326}{236432824375} a^{10} - \frac{54779437001}{236432824375} a^{9} + \frac{114291735318}{236432824375} a^{8} - \frac{2289056378}{4019358014375} a^{7} + \frac{1574165690332}{4019358014375} a^{6} - \frac{71883675883}{160774320575} a^{5} + \frac{205226591736}{803871602875} a^{4} + \frac{2050424108}{14615847325} a^{3} + \frac{737604766}{3420730225} a^{2} + \frac{210878394}{479923345} a + \frac{112787458}{479923345}$, $\frac{1}{45376936732691348104761371734040311316554582938624950431462281528972520064444375} a^{26} - \frac{4513461660528888427991569493073674937760750255369714909153918022216}{45376936732691348104761371734040311316554582938624950431462281528972520064444375} a^{25} - \frac{410064483574045635555427618625359424317225932280089245690832798530276202}{45376936732691348104761371734040311316554582938624950431462281528972520064444375} a^{24} - \frac{13399847417108082311458417218431923098515592820899120678144928042639777364}{45376936732691348104761371734040311316554582938624950431462281528972520064444375} a^{23} - \frac{1841221176941680778636198050770345319651060199700292381177405602543606729749}{45376936732691348104761371734040311316554582938624950431462281528972520064444375} a^{22} - \frac{17950174076955799042632699913326695712812087451697353449869584155253777398224}{1972910292725710787163537901480013535502373171244563062237490501259674785410625} a^{21} - \frac{19888163621745446467741137189724647951559118230504029123502116433131122657451}{1972910292725710787163537901480013535502373171244563062237490501259674785410625} a^{20} + \frac{285621995123179985667257913057598047202089246293643884362254464342467803861949}{45376936732691348104761371734040311316554582938624950431462281528972520064444375} a^{19} + \frac{875561051395356442433580334904471846214900114514859876069603143486459204334948}{45376936732691348104761371734040311316554582938624950431462281528972520064444375} a^{18} - \frac{1933871007334067137722801421689650942924358227827162321600339491646442260535532}{45376936732691348104761371734040311316554582938624950431462281528972520064444375} a^{17} + \frac{2681929221981697407735213552767208180482240386252730548529618925636160544600169}{45376936732691348104761371734040311316554582938624950431462281528972520064444375} a^{16} + \frac{13614054681626008184042756617831329419018143550590812487394400738183663517943}{242657415682841433715301453123210220944142154752005082521188671277927914783125} a^{15} + \frac{657530197455033409434525410087951227499096546176220049161426305757021214399217}{45376936732691348104761371734040311316554582938624950431462281528972520064444375} a^{14} + \frac{1878086909965726154659734064671969072590048924463289061905855476077344701381054}{45376936732691348104761371734040311316554582938624950431462281528972520064444375} a^{13} + \frac{23135520372344323827708015652964702344909476369570164842156418663447652605562}{677267712428229076190468234836422556963501237889924633305407186999589851708125} a^{12} + \frac{573490749154096480385066661992177684256667115975829272681566937629485745927207}{45376936732691348104761371734040311316554582938624950431462281528972520064444375} a^{11} - \frac{42039817147728753323905301386643640373596870861296217645334134412358150564947}{743884208732645050897727405476070677320566933420081154614135762770041312531875} a^{10} - \frac{475859845084757160670604200505493372200828357686191699743619046310552105810424}{45376936732691348104761371734040311316554582938624950431462281528972520064444375} a^{9} + \frac{4486225755923723680824641095264153027638278496176094321053284068367888827340257}{45376936732691348104761371734040311316554582938624950431462281528972520064444375} a^{8} - \frac{12351244124047860835248035145842892509396271576389452552846027906886050504947943}{45376936732691348104761371734040311316554582938624950431462281528972520064444375} a^{7} - \frac{3687042986413462726163358486449831941687521586399218687824025864528562479317191}{9075387346538269620952274346808062263310916587724990086292456305794504012888875} a^{6} - \frac{3271419639977570543474932133456034215381419123251935882715551751305866733124269}{9075387346538269620952274346808062263310916587724990086292456305794504012888875} a^{5} + \frac{875979252537103876025278402267718262328618680962785412150265536987709578190937}{1815077469307653924190454869361612452662183317544998017258491261158900802577775} a^{4} + \frac{255528745868813682419974558160511843385927904247496537981509424044482191166587}{1815077469307653924190454869361612452662183317544998017258491261158900802577775} a^{3} + \frac{2751545162321599070427791141703515853346681554526408093301619196258586965047}{15783282341805686297308303211840108284018985369956504497899924010077398283285} a^{2} + \frac{560854573737487709118655268438947970635514241806287823972806669996870157088}{5418141699425832609523745878691380455708009903119397066443257495996718813665} a + \frac{121682527211440882607949895758835686999776433643535435147680465991331204285}{1083628339885166521904749175738276091141601980623879413288651499199343762733}$

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:  \( 71364448730505.47 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{13}\cdot 71364448730505.47 \cdot 3}{2\sqrt{1089801024238603052304820653163822019730224609375}}\approx 4.87829267984210$ (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 $27$ $27$ R R ${\href{/LocalNumberField/11.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ $27$ $27$

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
$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}$
7Data not computed
$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 Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.335.2t1.a.a$1$ $ 5 \cdot 67 $ \(\Q(\sqrt{-335}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.335.3t2.a.a$2$ $ 5 \cdot 67 $ 3.1.335.1 $S_3$ (as 3T2) $1$ $0$
* 2.335.9t3.a.b$2$ $ 5 \cdot 67 $ 9.1.12594450625.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.335.9t3.a.c$2$ $ 5 \cdot 67 $ 9.1.12594450625.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.335.9t3.a.a$2$ $ 5 \cdot 67 $ 9.1.12594450625.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.16415.27t8.a.e$2$ $ 5 \cdot 7^{2} \cdot 67 $ 27.1.1089801024238603052304820653163822019730224609375.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.16415.27t8.a.f$2$ $ 5 \cdot 7^{2} \cdot 67 $ 27.1.1089801024238603052304820653163822019730224609375.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.16415.27t8.a.a$2$ $ 5 \cdot 7^{2} \cdot 67 $ 27.1.1089801024238603052304820653163822019730224609375.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.16415.27t8.a.h$2$ $ 5 \cdot 7^{2} \cdot 67 $ 27.1.1089801024238603052304820653163822019730224609375.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.16415.27t8.a.c$2$ $ 5 \cdot 7^{2} \cdot 67 $ 27.1.1089801024238603052304820653163822019730224609375.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.16415.27t8.a.i$2$ $ 5 \cdot 7^{2} \cdot 67 $ 27.1.1089801024238603052304820653163822019730224609375.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.16415.27t8.a.g$2$ $ 5 \cdot 7^{2} \cdot 67 $ 27.1.1089801024238603052304820653163822019730224609375.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.16415.27t8.a.d$2$ $ 5 \cdot 7^{2} \cdot 67 $ 27.1.1089801024238603052304820653163822019730224609375.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.16415.27t8.a.b$2$ $ 5 \cdot 7^{2} \cdot 67 $ 27.1.1089801024238603052304820653163822019730224609375.1 $D_{27}$ (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 *.