Properties

Label 27.1.356...663.1
Degree $27$
Signature $[1, 13]$
Discriminant $-3.568\times 10^{48}$
Root discriminant $62.84$
Ramified primes $7, 367$
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 - 10*x^26 + 80*x^25 - 447*x^24 + 1863*x^23 - 6240*x^22 + 15201*x^21 - 25896*x^20 + 26931*x^19 + 29922*x^18 - 2157*x^17 + 228*x^16 + 1150141*x^15 - 1553359*x^14 + 3361193*x^13 - 357518*x^12 - 4399792*x^11 - 5659177*x^10 - 7694789*x^9 - 6381628*x^8 + 25148690*x^7 + 37280366*x^6 + 10154644*x^5 - 4374482*x^4 + 5689374*x^3 + 3195477*x^2 - 2094606*x + 247941)
 
gp: K = bnfinit(x^27 - 10*x^26 + 80*x^25 - 447*x^24 + 1863*x^23 - 6240*x^22 + 15201*x^21 - 25896*x^20 + 26931*x^19 + 29922*x^18 - 2157*x^17 + 228*x^16 + 1150141*x^15 - 1553359*x^14 + 3361193*x^13 - 357518*x^12 - 4399792*x^11 - 5659177*x^10 - 7694789*x^9 - 6381628*x^8 + 25148690*x^7 + 37280366*x^6 + 10154644*x^5 - 4374482*x^4 + 5689374*x^3 + 3195477*x^2 - 2094606*x + 247941, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![247941, -2094606, 3195477, 5689374, -4374482, 10154644, 37280366, 25148690, -6381628, -7694789, -5659177, -4399792, -357518, 3361193, -1553359, 1150141, 228, -2157, 29922, 26931, -25896, 15201, -6240, 1863, -447, 80, -10, 1]);
 

\( x^{27} - 10 x^{26} + 80 x^{25} - 447 x^{24} + 1863 x^{23} - 6240 x^{22} + 15201 x^{21} - 25896 x^{20} + 26931 x^{19} + 29922 x^{18} - 2157 x^{17} + 228 x^{16} + 1150141 x^{15} - 1553359 x^{14} + 3361193 x^{13} - 357518 x^{12} - 4399792 x^{11} - 5659177 x^{10} - 7694789 x^{9} - 6381628 x^{8} + 25148690 x^{7} + 37280366 x^{6} + 10154644 x^{5} - 4374482 x^{4} + 5689374 x^{3} + 3195477 x^{2} - 2094606 x + 247941 \)

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:  \(-3567989400462155147048339898163452848532229105663\)\(\medspace = -\,7^{18}\cdot 367^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $62.84$
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:  $7, 367$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{11} + \frac{1}{9} a^{10} + \frac{1}{9} a^{8} + \frac{1}{9} a^{7} - \frac{4}{9} a^{6} - \frac{1}{9} a^{5} + \frac{4}{9} a^{4} + \frac{4}{9} a^{2} - \frac{1}{9} a + \frac{1}{3}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{9} a^{7} + \frac{1}{3} a^{6} - \frac{1}{9} a^{5} - \frac{4}{9} a^{4} + \frac{1}{9} a^{3} + \frac{4}{9} a^{2} - \frac{2}{9} a - \frac{1}{3}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{10} - \frac{1}{9} a^{8} + \frac{1}{9} a^{7} - \frac{2}{9} a^{6} - \frac{2}{9} a^{5} + \frac{2}{9} a^{4} + \frac{1}{9} a^{3} - \frac{1}{9} a^{2} - \frac{1}{9} a + \frac{1}{3}$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{1}{9} a^{8} - \frac{1}{9} a^{7} - \frac{1}{3} a^{6} + \frac{1}{9} a^{5} + \frac{2}{9} a^{4} - \frac{1}{9} a^{3} - \frac{1}{3} a^{2} + \frac{2}{9} a + \frac{1}{3}$, $\frac{1}{9} a^{15} + \frac{1}{9} a^{10} - \frac{1}{9} a^{9} + \frac{1}{9} a^{8} - \frac{1}{9} a^{7} + \frac{2}{9} a^{6} - \frac{2}{9} a^{4} + \frac{1}{9} a^{2} + \frac{1}{9} a - \frac{1}{3}$, $\frac{1}{27} a^{16} - \frac{1}{27} a^{15} - \frac{1}{27} a^{14} + \frac{1}{27} a^{13} - \frac{1}{27} a^{12} - \frac{1}{27} a^{11} + \frac{1}{27} a^{10} + \frac{2}{27} a^{9} - \frac{1}{27} a^{8} - \frac{1}{27} a^{7} + \frac{1}{27} a^{6} + \frac{1}{27} a^{5} + \frac{10}{27} a^{4} + \frac{8}{27} a^{3} + \frac{8}{27} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{135} a^{17} + \frac{2}{135} a^{16} + \frac{1}{27} a^{15} - \frac{2}{135} a^{14} + \frac{1}{27} a^{13} + \frac{2}{135} a^{12} - \frac{1}{27} a^{11} + \frac{11}{135} a^{10} - \frac{7}{135} a^{9} - \frac{2}{27} a^{8} - \frac{1}{27} a^{7} + \frac{1}{135} a^{6} - \frac{23}{135} a^{5} + \frac{8}{135} a^{4} - \frac{8}{27} a^{3} + \frac{11}{45} a^{2} - \frac{4}{45} a + \frac{7}{15}$, $\frac{1}{135} a^{18} + \frac{1}{135} a^{16} + \frac{1}{45} a^{15} - \frac{2}{45} a^{14} + \frac{7}{135} a^{13} + \frac{2}{45} a^{12} + \frac{2}{45} a^{11} + \frac{16}{135} a^{10} + \frac{19}{135} a^{9} + \frac{1}{9} a^{8} - \frac{19}{135} a^{7} - \frac{2}{27} a^{6} + \frac{2}{5} a^{5} + \frac{64}{135} a^{4} - \frac{22}{135} a^{3} + \frac{4}{45} a^{2} - \frac{16}{45} a + \frac{1}{15}$, $\frac{1}{405} a^{19} + \frac{1}{405} a^{18} + \frac{7}{405} a^{16} + \frac{17}{405} a^{15} - \frac{17}{405} a^{14} + \frac{13}{405} a^{13} + \frac{1}{81} a^{12} - \frac{8}{405} a^{11} + \frac{14}{405} a^{10} + \frac{4}{45} a^{9} + \frac{61}{405} a^{8} - \frac{29}{405} a^{7} - \frac{59}{135} a^{6} + \frac{116}{405} a^{5} + \frac{7}{15} a^{4} + \frac{26}{81} a^{3} - \frac{119}{405} a^{2} + \frac{61}{135} a + \frac{19}{45}$, $\frac{1}{1215} a^{20} + \frac{1}{1215} a^{19} - \frac{1}{405} a^{18} - \frac{2}{1215} a^{17} + \frac{11}{1215} a^{16} + \frac{49}{1215} a^{15} - \frac{11}{1215} a^{14} - \frac{1}{1215} a^{13} - \frac{59}{1215} a^{12} + \frac{26}{1215} a^{11} + \frac{28}{405} a^{10} - \frac{128}{1215} a^{9} + \frac{1}{1215} a^{8} - \frac{1}{9} a^{7} - \frac{208}{1215} a^{6} - \frac{97}{405} a^{5} + \frac{421}{1215} a^{4} - \frac{293}{1215} a^{3} + \frac{13}{81} a^{2} - \frac{43}{135} a - \frac{2}{45}$, $\frac{1}{13365} a^{21} + \frac{2}{13365} a^{20} - \frac{8}{13365} a^{19} + \frac{5}{2673} a^{18} - \frac{2}{495} a^{16} + \frac{89}{13365} a^{15} - \frac{17}{1485} a^{14} - \frac{17}{405} a^{13} + \frac{5}{297} a^{12} - \frac{436}{13365} a^{11} + \frac{2149}{13365} a^{10} - \frac{856}{13365} a^{9} + \frac{314}{2673} a^{8} - \frac{853}{13365} a^{7} + \frac{4964}{13365} a^{6} + \frac{920}{2673} a^{5} + \frac{6536}{13365} a^{4} - \frac{730}{2673} a^{3} + \frac{443}{1485} a^{2} + \frac{76}{165} a + \frac{56}{495}$, $\frac{1}{13365} a^{22} - \frac{1}{13365} a^{20} - \frac{14}{13365} a^{19} + \frac{49}{13365} a^{18} + \frac{23}{13365} a^{17} - \frac{1}{55} a^{16} + \frac{134}{2673} a^{15} - \frac{29}{2673} a^{14} + \frac{379}{13365} a^{13} + \frac{16}{13365} a^{12} + \frac{568}{13365} a^{11} + \frac{31}{4455} a^{10} + \frac{1577}{13365} a^{9} + \frac{37}{1215} a^{8} + \frac{1852}{13365} a^{7} + \frac{41}{2673} a^{6} - \frac{877}{4455} a^{5} - \frac{1201}{13365} a^{4} + \frac{1181}{4455} a^{3} - \frac{719}{1485} a^{2} - \frac{191}{495} a - \frac{2}{11}$, $\frac{1}{761805} a^{23} - \frac{4}{253935} a^{22} - \frac{1}{761805} a^{21} + \frac{14}{253935} a^{20} + \frac{13}{50787} a^{19} + \frac{1613}{761805} a^{18} + \frac{28}{8019} a^{17} - \frac{212}{69255} a^{16} + \frac{8887}{761805} a^{15} + \frac{9829}{253935} a^{14} + \frac{2233}{69255} a^{13} - \frac{764}{50787} a^{12} + \frac{5405}{152361} a^{11} - \frac{40129}{761805} a^{10} + \frac{41}{152361} a^{9} + \frac{41222}{253935} a^{8} - \frac{20699}{761805} a^{7} - \frac{321539}{761805} a^{6} + \frac{5464}{69255} a^{5} + \frac{64498}{152361} a^{4} - \frac{94516}{761805} a^{3} - \frac{41797}{84645} a^{2} + \frac{11893}{28215} a + \frac{12619}{28215}$, $\frac{1}{3809025} a^{24} + \frac{2}{3809025} a^{23} + \frac{2}{3809025} a^{22} + \frac{142}{3809025} a^{21} - \frac{46}{423225} a^{20} - \frac{3979}{3809025} a^{19} + \frac{4633}{1269675} a^{18} + \frac{398}{115425} a^{17} + \frac{7024}{761805} a^{16} + \frac{28387}{761805} a^{15} + \frac{15809}{3809025} a^{14} - \frac{126314}{3809025} a^{13} - \frac{25771}{761805} a^{12} - \frac{1246}{761805} a^{11} - \frac{565933}{3809025} a^{10} - \frac{133384}{3809025} a^{9} - \frac{331229}{3809025} a^{8} - \frac{712}{6075} a^{7} + \frac{43616}{761805} a^{6} + \frac{24202}{141075} a^{5} + \frac{561334}{1269675} a^{4} - \frac{570398}{3809025} a^{3} - \frac{12403}{84645} a^{2} - \frac{5431}{15675} a - \frac{39877}{141075}$, $\frac{1}{11427075} a^{25} + \frac{1}{11427075} a^{24} - \frac{1}{2285415} a^{23} + \frac{8}{457083} a^{22} + \frac{16}{601425} a^{21} + \frac{841}{2285415} a^{20} - \frac{11882}{11427075} a^{19} + \frac{878}{457083} a^{18} - \frac{32069}{11427075} a^{17} - \frac{39461}{2285415} a^{16} - \frac{520826}{11427075} a^{15} - \frac{573988}{11427075} a^{14} + \frac{36769}{1038825} a^{13} + \frac{18968}{761805} a^{12} - \frac{481363}{11427075} a^{11} - \frac{3614}{54675} a^{10} + \frac{1576}{50787} a^{9} - \frac{51914}{2285415} a^{8} + \frac{48071}{601425} a^{7} + \frac{415741}{1269675} a^{6} + \frac{581258}{11427075} a^{5} - \frac{45959}{761805} a^{4} + \frac{3737548}{11427075} a^{3} + \frac{200078}{1269675} a^{2} - \frac{20633}{423225} a - \frac{122}{2025}$, $\frac{1}{5502512240037965492035778968748927816089121282421593028768759268271737275} a^{26} + \frac{2385349607303947709301351710904199851358272061582539618819889204}{289605907370419236422935735197311990320480067495873317303618908856407225} a^{25} + \frac{548681358981200839796926441238114554286091856166440968470589778117}{5502512240037965492035778968748927816089121282421593028768759268271737275} a^{24} + \frac{3004346885409987365799061383923241238022553390189987919024748478704}{5502512240037965492035778968748927816089121282421593028768759268271737275} a^{23} + \frac{25286080946250490640866332368369078303528480942353451176758250025593}{5502512240037965492035778968748927816089121282421593028768759268271737275} a^{22} - \frac{37119213385728389151481975116098180103901616920160374563047375836291}{5502512240037965492035778968748927816089121282421593028768759268271737275} a^{21} - \frac{209098211095639409378559442030400951555343269655787266925568580477646}{1100502448007593098407155793749785563217824256484318605753751853654347455} a^{20} - \frac{3470682431516328101743477381210351766424442094005219891615928286159703}{5502512240037965492035778968748927816089121282421593028768759268271737275} a^{19} + \frac{9361888980052547208796745975053176783949366475161249332085487948202514}{5502512240037965492035778968748927816089121282421593028768759268271737275} a^{18} + \frac{7221929200616952678923666140443151900587892221748710481883907035042798}{5502512240037965492035778968748927816089121282421593028768759268271737275} a^{17} - \frac{79492651296406012814971870509294367011208942753794991723377653719720586}{5502512240037965492035778968748927816089121282421593028768759268271737275} a^{16} + \frac{152017254945268798604166608091518039063923636094335164613346843831958017}{5502512240037965492035778968748927816089121282421593028768759268271737275} a^{15} + \frac{249176317620759527029314509598695319790066449201597488973128200794104732}{5502512240037965492035778968748927816089121282421593028768759268271737275} a^{14} - \frac{25871473815282054809371021222218349052819970496623559600126358746553831}{1834170746679321830678592989582975938696373760807197676256253089423912425} a^{13} - \frac{284584690199090637333892189058337164379315398532093030424150973119472713}{5502512240037965492035778968748927816089121282421593028768759268271737275} a^{12} + \frac{239764832674113510967550294713361361598380881809596768090060775185292049}{5502512240037965492035778968748927816089121282421593028768759268271737275} a^{11} + \frac{24311320181402962563150666786518761978518144376147688963333727652615798}{166742795152665620970781180871179630790579432800654334205113917220355675} a^{10} + \frac{93568983715398082770385640217962591303395733919156246895590569099246867}{5502512240037965492035778968748927816089121282421593028768759268271737275} a^{9} + \frac{26542401702990592704452917414358965155765864210155926496105679211103356}{500228385457996862912343542613538892371738298401963002615341751661067025} a^{8} - \frac{1756666616010247050157138694287207596057268352261002176143600188403498}{15158435922969601906434652806470875526416312072786757655010356110941425} a^{7} + \frac{1326066992642010091717426925356895506856767059218478319210663165087580768}{5502512240037965492035778968748927816089121282421593028768759268271737275} a^{6} - \frac{296427058997263497427431077458096838947050367687694624176022815860098893}{611390248893107276892864329860991979565457920269065892085417696474637475} a^{5} - \frac{1047582490258798714517549031150711495118229032834087809238970683575931868}{5502512240037965492035778968748927816089121282421593028768759268271737275} a^{4} + \frac{792336218136749326139107997694320507843831572770509806533121987298981377}{1834170746679321830678592989582975938696373760807197676256253089423912425} a^{3} + \frac{5506090243531249401156571706932197514379967745793846816495065669371874}{67932249877011919654762703317887997729495324474340654676157521830515275} a^{2} + \frac{89309789704891288536541405432032194183880550827728000766676226136362474}{203796749631035758964288109953663993188485973423021964028472565491545825} a - \frac{8228774837981888436217805700485084697917846746753769487117270171092293}{67932249877011919654762703317887997729495324474340654676157521830515275}$

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:  \( 67227716701120.3 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{13}\cdot 67227716701120.3 \cdot 3}{2\sqrt{3567989400462155147048339898163452848532229105663}}\approx 2.53978016346981$ (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

3.1.367.1, 9.1.18141126721.1

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 $27$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{3}$ $27$ $27$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $27$ $27$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$

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
367Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.367.2t1.a.a$1$ $ 367 $ \(\Q(\sqrt{-367}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.367.3t2.a.a$2$ $ 367 $ 3.1.367.1 $S_3$ (as 3T2) $1$ $0$
* 2.367.9t3.a.b$2$ $ 367 $ 9.1.18141126721.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.367.9t3.a.c$2$ $ 367 $ 9.1.18141126721.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.367.9t3.a.a$2$ $ 367 $ 9.1.18141126721.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.17983.27t8.a.b$2$ $ 7^{2} \cdot 367 $ 27.1.3567989400462155147048339898163452848532229105663.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.17983.27t8.a.g$2$ $ 7^{2} \cdot 367 $ 27.1.3567989400462155147048339898163452848532229105663.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.17983.27t8.a.f$2$ $ 7^{2} \cdot 367 $ 27.1.3567989400462155147048339898163452848532229105663.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.17983.27t8.a.e$2$ $ 7^{2} \cdot 367 $ 27.1.3567989400462155147048339898163452848532229105663.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.17983.27t8.a.d$2$ $ 7^{2} \cdot 367 $ 27.1.3567989400462155147048339898163452848532229105663.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.17983.27t8.a.c$2$ $ 7^{2} \cdot 367 $ 27.1.3567989400462155147048339898163452848532229105663.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.17983.27t8.a.a$2$ $ 7^{2} \cdot 367 $ 27.1.3567989400462155147048339898163452848532229105663.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.17983.27t8.a.i$2$ $ 7^{2} \cdot 367 $ 27.1.3567989400462155147048339898163452848532229105663.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.17983.27t8.a.h$2$ $ 7^{2} \cdot 367 $ 27.1.3567989400462155147048339898163452848532229105663.1 $D_{27}$ (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 *.