Properties

Label 27.1.863...271.1
Degree $27$
Signature $[1, 13]$
Discriminant $-8.631\times 10^{49}$
Root discriminant $70.71$
Ramified primes $13, 199$
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 - 9*x^26 + 51*x^25 - 157*x^24 + 381*x^23 + 94*x^22 - 4772*x^21 + 22385*x^20 - 19633*x^19 - 132957*x^18 + 462384*x^17 + 49228*x^16 - 4729835*x^15 + 16673567*x^14 - 15312382*x^13 - 102305328*x^12 + 361800096*x^11 - 196173035*x^10 - 890969729*x^9 + 1569260108*x^8 - 437613721*x^7 - 1278467664*x^6 + 2134193061*x^5 - 1922367953*x^4 + 1085805378*x^3 - 524572351*x^2 + 152402636*x - 24878621)
 
gp: K = bnfinit(x^27 - 9*x^26 + 51*x^25 - 157*x^24 + 381*x^23 + 94*x^22 - 4772*x^21 + 22385*x^20 - 19633*x^19 - 132957*x^18 + 462384*x^17 + 49228*x^16 - 4729835*x^15 + 16673567*x^14 - 15312382*x^13 - 102305328*x^12 + 361800096*x^11 - 196173035*x^10 - 890969729*x^9 + 1569260108*x^8 - 437613721*x^7 - 1278467664*x^6 + 2134193061*x^5 - 1922367953*x^4 + 1085805378*x^3 - 524572351*x^2 + 152402636*x - 24878621, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-24878621, 152402636, -524572351, 1085805378, -1922367953, 2134193061, -1278467664, -437613721, 1569260108, -890969729, -196173035, 361800096, -102305328, -15312382, 16673567, -4729835, 49228, 462384, -132957, -19633, 22385, -4772, 94, 381, -157, 51, -9, 1]);
 

\(x^{27} - 9 x^{26} + 51 x^{25} - 157 x^{24} + 381 x^{23} + 94 x^{22} - 4772 x^{21} + 22385 x^{20} - 19633 x^{19} - 132957 x^{18} + 462384 x^{17} + 49228 x^{16} - 4729835 x^{15} + 16673567 x^{14} - 15312382 x^{13} - 102305328 x^{12} + 361800096 x^{11} - 196173035 x^{10} - 890969729 x^{9} + 1569260108 x^{8} - 437613721 x^{7} - 1278467664 x^{6} + 2134193061 x^{5} - 1922367953 x^{4} + 1085805378 x^{3} - 524572351 x^{2} + 152402636 x - 24878621\)  Toggle raw display

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:  \(-86311832016540901180083101912199499488333854262271\)\(\medspace = -\,13^{18}\cdot 199^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $70.71$
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:  $13, 199$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{6}$, $\frac{1}{9} a^{15} + \frac{1}{9} a^{14} + \frac{1}{9} a^{13} + \frac{1}{9} a^{12} + \frac{1}{9} a^{11} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{9} a^{8} + \frac{2}{9} a^{7} + \frac{2}{9} a^{6} + \frac{2}{9} a^{5} + \frac{2}{9} a^{4} + \frac{2}{9} a^{3} + \frac{2}{9} a^{2} + \frac{2}{9} a + \frac{2}{9}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{8} - \frac{2}{9}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{9} - \frac{2}{9} a$, $\frac{1}{9} a^{18} + \frac{1}{9} a^{10} - \frac{2}{9} a^{2}$, $\frac{1}{9} a^{19} + \frac{1}{9} a^{11} - \frac{2}{9} a^{3}$, $\frac{1}{99} a^{20} + \frac{4}{99} a^{19} + \frac{4}{99} a^{18} - \frac{2}{99} a^{16} - \frac{1}{99} a^{15} - \frac{10}{99} a^{14} + \frac{2}{99} a^{13} - \frac{1}{33} a^{12} - \frac{5}{33} a^{10} + \frac{1}{9} a^{9} - \frac{1}{33} a^{8} - \frac{38}{99} a^{7} - \frac{20}{99} a^{6} + \frac{4}{99} a^{5} + \frac{35}{99} a^{4} + \frac{38}{99} a^{3} + \frac{17}{99} a^{2} - \frac{14}{99} a + \frac{47}{99}$, $\frac{1}{297} a^{21} + \frac{1}{297} a^{20} + \frac{14}{297} a^{19} - \frac{1}{297} a^{18} + \frac{1}{33} a^{17} - \frac{2}{99} a^{16} + \frac{4}{297} a^{15} + \frac{43}{297} a^{14} - \frac{31}{297} a^{13} - \frac{46}{297} a^{12} + \frac{2}{33} a^{11} - \frac{7}{99} a^{10} - \frac{14}{297} a^{9} + \frac{37}{297} a^{8} - \frac{82}{297} a^{7} + \frac{86}{297} a^{6} - \frac{7}{99} a^{5} + \frac{40}{99} a^{4} + \frac{79}{297} a^{3} - \frac{65}{297} a^{2} + \frac{89}{297} a - \frac{64}{297}$, $\frac{1}{297} a^{22} + \frac{1}{297} a^{20} + \frac{1}{99} a^{19} - \frac{5}{297} a^{18} - \frac{5}{99} a^{17} + \frac{1}{297} a^{16} - \frac{5}{99} a^{15} - \frac{20}{297} a^{14} - \frac{2}{99} a^{13} + \frac{34}{297} a^{12} - \frac{13}{99} a^{11} - \frac{4}{27} a^{10} - \frac{16}{99} a^{9} + \frac{16}{297} a^{8} - \frac{34}{99} a^{7} + \frac{1}{297} a^{6} - \frac{46}{99} a^{5} + \frac{1}{297} a^{4} + \frac{1}{11} a^{3} - \frac{50}{297} a^{2} + \frac{3}{11} a + \frac{127}{297}$, $\frac{1}{297} a^{23} - \frac{1}{297} a^{20} + \frac{2}{297} a^{19} + \frac{7}{297} a^{18} - \frac{8}{297} a^{17} - \frac{1}{99} a^{16} + \frac{4}{99} a^{15} + \frac{14}{297} a^{14} - \frac{7}{297} a^{13} + \frac{49}{297} a^{12} + \frac{4}{297} a^{11} - \frac{5}{99} a^{10} + \frac{10}{99} a^{9} + \frac{2}{297} a^{8} - \frac{34}{297} a^{7} - \frac{98}{297} a^{6} - \frac{122}{297} a^{5} - \frac{4}{9} a^{4} + \frac{2}{11} a^{3} - \frac{103}{297} a^{2} + \frac{146}{297} a - \frac{10}{27}$, $\frac{1}{5049} a^{24} - \frac{2}{1683} a^{23} - \frac{2}{1683} a^{22} - \frac{2}{5049} a^{21} - \frac{1}{459} a^{20} + \frac{80}{5049} a^{19} + \frac{65}{5049} a^{18} + \frac{86}{1683} a^{17} - \frac{37}{1683} a^{16} + \frac{73}{5049} a^{15} + \frac{337}{5049} a^{14} - \frac{526}{5049} a^{13} - \frac{280}{5049} a^{12} - \frac{73}{1683} a^{11} - \frac{245}{1683} a^{10} - \frac{239}{5049} a^{9} + \frac{47}{459} a^{8} - \frac{1945}{5049} a^{7} + \frac{1670}{5049} a^{6} + \frac{3}{187} a^{5} + \frac{287}{1683} a^{4} + \frac{1879}{5049} a^{3} - \frac{97}{297} a^{2} + \frac{290}{5049} a + \frac{2269}{5049}$, $\frac{1}{14953709133} a^{25} - \frac{184139}{14953709133} a^{24} + \frac{13298552}{14953709133} a^{23} + \frac{665845}{879629949} a^{22} - \frac{15546227}{14953709133} a^{21} - \frac{43849637}{14953709133} a^{20} + \frac{162067979}{4984569711} a^{19} + \frac{20334461}{453142701} a^{18} + \frac{441235415}{14953709133} a^{17} - \frac{183728869}{14953709133} a^{16} + \frac{53792111}{1359428103} a^{15} + \frac{792680269}{14953709133} a^{14} + \frac{8277404}{553841079} a^{13} + \frac{101175307}{4984569711} a^{12} + \frac{1592927354}{14953709133} a^{11} + \frac{1495508315}{14953709133} a^{10} - \frac{2400051164}{14953709133} a^{9} - \frac{1387539845}{14953709133} a^{8} + \frac{1017524362}{4984569711} a^{7} - \frac{119896763}{553841079} a^{6} + \frac{432359318}{14953709133} a^{5} + \frac{3483323165}{14953709133} a^{4} + \frac{5822836801}{14953709133} a^{3} - \frac{1872215201}{14953709133} a^{2} - \frac{1646808103}{14953709133} a + \frac{2206228043}{14953709133}$, $\frac{1}{264265920239931360217747614001632569964506869861645323740755464316265641327209843396006091023325327664637} a^{26} - \frac{1532449358862185518821915041737608911742410717147680991896003744824869051058460666696365664703}{88088640079977120072582538000544189988168956620548441246918488105421880442403281132002030341108442554879} a^{25} - \frac{14733310108900179741386912039705903959963637725119951419541990299695117187147521788449588110340334779}{264265920239931360217747614001632569964506869861645323740755464316265641327209843396006091023325327664637} a^{24} - \frac{143261638498089190022628748783135478771406441550116942392534038082292046570316341145773520075306829572}{88088640079977120072582538000544189988168956620548441246918488105421880442403281132002030341108442554879} a^{23} + \frac{406289791705705284353444669737314333674693714404028968498028313104908581309684343179931673636269532789}{264265920239931360217747614001632569964506869861645323740755464316265641327209843396006091023325327664637} a^{22} - \frac{8194477614235858667143739237985765843663989738183986402398979569334043480573343767409060803471412607}{88088640079977120072582538000544189988168956620548441246918488105421880442403281132002030341108442554879} a^{21} + \frac{574240383987682296217623727860190443997577097977541097412643178204101484148949047643303522100903325678}{264265920239931360217747614001632569964506869861645323740755464316265641327209843396006091023325327664637} a^{20} - \frac{3788517095939718937571993616551522408364675618955566573040710175832278270113264965336708362628192976733}{88088640079977120072582538000544189988168956620548441246918488105421880442403281132002030341108442554879} a^{19} + \frac{10974946733799964543087823058223683926935921302935977949595741918727604209379614308866187463179636684122}{264265920239931360217747614001632569964506869861645323740755464316265641327209843396006091023325327664637} a^{18} - \frac{1086081974121148646357639002838007999392318139358616503141144828679063687002972225547809700092341133858}{29362880026659040024194179333514729996056318873516147082306162701807293480801093710667343447036147518293} a^{17} - \frac{5657299848022545517356052654104008694607471209529082588393997180840240173199029551865164102493001497854}{264265920239931360217747614001632569964506869861645323740755464316265641327209843396006091023325327664637} a^{16} + \frac{976041099209163981937693788843406975824505445141643343129157593904031623525731161693832903239654534515}{29362880026659040024194179333514729996056318873516147082306162701807293480801093710667343447036147518293} a^{15} - \frac{37128994156280411367774046508821981479972069598426626099345847170917508135784130933402511860614572175369}{264265920239931360217747614001632569964506869861645323740755464316265641327209843396006091023325327664637} a^{14} + \frac{4583348373913641573269709015993504324762004578890859491773548794076474078679761312540929984095444507510}{88088640079977120072582538000544189988168956620548441246918488105421880442403281132002030341108442554879} a^{13} - \frac{40255116356329629450199742850470626386218978428027995416482954255492851202513036860785297415365508329107}{264265920239931360217747614001632569964506869861645323740755464316265641327209843396006091023325327664637} a^{12} + \frac{11230348499383026308750346858335499198144670128856608721124289950891141990984779025079342267567512496133}{88088640079977120072582538000544189988168956620548441246918488105421880442403281132002030341108442554879} a^{11} + \frac{29309711238956617533329309183681262604207902028406063490819297143530979265509271105204876239064322464137}{264265920239931360217747614001632569964506869861645323740755464316265641327209843396006091023325327664637} a^{10} + \frac{79937046169997213962279802434667517814235672777861481677016327572393310094170522565893750154505715184}{2148503416584807806648354585379126585077291624891425396266304587937119035180567832487854398563620550119} a^{9} - \frac{25515598496570340392360370639271998082739863640048449231971950483219848659540532545999513198896658288310}{264265920239931360217747614001632569964506869861645323740755464316265641327209843396006091023325327664637} a^{8} + \frac{37078331544713554426750495630406499410714236157303188473524921901440744082146710481889807476610999500192}{88088640079977120072582538000544189988168956620548441246918488105421880442403281132002030341108442554879} a^{7} - \frac{67154744073739602040713026278178910604255126006824735320541002282490779791752297974939703352667279496796}{264265920239931360217747614001632569964506869861645323740755464316265641327209843396006091023325327664637} a^{6} + \frac{1051366124848245510927143059923123637342591182257284019696266279443746885001566908021521463477316864351}{3262542225184337780466019925946081110672924319279571898034018077978588164533454856740815938559571946477} a^{5} - \frac{70832550351715158197483720227031512327890720391211898096855788501587546486008733378034027241117511187182}{264265920239931360217747614001632569964506869861645323740755464316265641327209843396006091023325327664637} a^{4} + \frac{8731779640567336073965268168127706487536719682819929742839850875102390151346541291201833227277873248808}{29362880026659040024194179333514729996056318873516147082306162701807293480801093710667343447036147518293} a^{3} - \frac{8343648682418798873094566112410929955208169870206585348069147164575847795881722713578976219436646572610}{24024174567266487292522510363784779087682442714695029430977769483296876484291803945091462820302302514967} a^{2} + \frac{7541130816420362322225578148317510133488456771517707452265721430888061561845738747952230108745350367039}{29362880026659040024194179333514729996056318873516147082306162701807293480801093710667343447036147518293} a - \frac{72566300678803785140089121424245794416525064416974750760554728470623493788228352597115476980684515197869}{264265920239931360217747614001632569964506869861645323740755464316265641327209843396006091023325327664637}$  Toggle raw display

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 $2$)  Toggle raw display
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:  \( 174903583233230.53 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{13}\cdot 174903583233230.53 \cdot 3}{2\sqrt{86311832016540901180083101912199499488333854262271}}\approx 1.34345487275669$ (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

3.1.199.1, 9.1.1568239201.1

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 $27$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ $27$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $27$ $27$ $27$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
13Data not computed
$199$$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
199.2.1.2$x^{2} + 398$$2$$1$$1$$C_2$$[\ ]_{2}$
199.2.1.2$x^{2} + 398$$2$$1$$1$$C_2$$[\ ]_{2}$
199.2.1.2$x^{2} + 398$$2$$1$$1$$C_2$$[\ ]_{2}$
199.2.1.2$x^{2} + 398$$2$$1$$1$$C_2$$[\ ]_{2}$
199.2.1.2$x^{2} + 398$$2$$1$$1$$C_2$$[\ ]_{2}$
199.2.1.2$x^{2} + 398$$2$$1$$1$$C_2$$[\ ]_{2}$
199.2.1.2$x^{2} + 398$$2$$1$$1$$C_2$$[\ ]_{2}$
199.2.1.2$x^{2} + 398$$2$$1$$1$$C_2$$[\ ]_{2}$
199.2.1.2$x^{2} + 398$$2$$1$$1$$C_2$$[\ ]_{2}$
199.2.1.2$x^{2} + 398$$2$$1$$1$$C_2$$[\ ]_{2}$
199.2.1.2$x^{2} + 398$$2$$1$$1$$C_2$$[\ ]_{2}$
199.2.1.2$x^{2} + 398$$2$$1$$1$$C_2$$[\ ]_{2}$
199.2.1.2$x^{2} + 398$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.199.2t1.a.a$1$ $ 199 $ \(\Q(\sqrt{-199}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.199.3t2.a.a$2$ $ 199 $ 3.1.199.1 $S_3$ (as 3T2) $1$ $0$
* 2.199.9t3.a.c$2$ $ 199 $ 9.1.1568239201.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.199.9t3.a.a$2$ $ 199 $ 9.1.1568239201.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.199.9t3.a.b$2$ $ 199 $ 9.1.1568239201.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.33631.27t8.a.c$2$ $ 13^{2} \cdot 199 $ 27.1.86311832016540901180083101912199499488333854262271.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.33631.27t8.a.b$2$ $ 13^{2} \cdot 199 $ 27.1.86311832016540901180083101912199499488333854262271.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.33631.27t8.a.e$2$ $ 13^{2} \cdot 199 $ 27.1.86311832016540901180083101912199499488333854262271.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.33631.27t8.a.i$2$ $ 13^{2} \cdot 199 $ 27.1.86311832016540901180083101912199499488333854262271.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.33631.27t8.a.f$2$ $ 13^{2} \cdot 199 $ 27.1.86311832016540901180083101912199499488333854262271.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.33631.27t8.a.a$2$ $ 13^{2} \cdot 199 $ 27.1.86311832016540901180083101912199499488333854262271.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.33631.27t8.a.h$2$ $ 13^{2} \cdot 199 $ 27.1.86311832016540901180083101912199499488333854262271.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.33631.27t8.a.g$2$ $ 13^{2} \cdot 199 $ 27.1.86311832016540901180083101912199499488333854262271.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.33631.27t8.a.d$2$ $ 13^{2} \cdot 199 $ 27.1.86311832016540901180083101912199499488333854262271.1 $D_{27}$ (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 *.