Properties

Label 28.0.849...381.1
Degree $28$
Signature $[0, 14]$
Discriminant $8.498\times 10^{43}$
Root discriminant $37.06$
Ramified primes $11, 181$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{28}$ (as 28T10)

Related objects

Downloads

Learn more about

Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 8*x^27 + 39*x^26 - 147*x^25 + 484*x^24 - 1461*x^23 + 4001*x^22 - 10052*x^21 + 23841*x^20 - 52926*x^19 + 109173*x^18 - 208723*x^17 + 369896*x^16 - 608215*x^15 + 913744*x^14 - 1246890*x^13 + 1584122*x^12 - 1829692*x^11 + 1878311*x^10 - 1777347*x^9 + 1518093*x^8 - 1036936*x^7 + 598732*x^6 - 344364*x^5 + 177291*x^4 - 79332*x^3 + 46992*x^2 - 15367*x + 6413)
 
gp: K = bnfinit(x^28 - 8*x^27 + 39*x^26 - 147*x^25 + 484*x^24 - 1461*x^23 + 4001*x^22 - 10052*x^21 + 23841*x^20 - 52926*x^19 + 109173*x^18 - 208723*x^17 + 369896*x^16 - 608215*x^15 + 913744*x^14 - 1246890*x^13 + 1584122*x^12 - 1829692*x^11 + 1878311*x^10 - 1777347*x^9 + 1518093*x^8 - 1036936*x^7 + 598732*x^6 - 344364*x^5 + 177291*x^4 - 79332*x^3 + 46992*x^2 - 15367*x + 6413, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6413, -15367, 46992, -79332, 177291, -344364, 598732, -1036936, 1518093, -1777347, 1878311, -1829692, 1584122, -1246890, 913744, -608215, 369896, -208723, 109173, -52926, 23841, -10052, 4001, -1461, 484, -147, 39, -8, 1]);
 

\( x^{28} - 8 x^{27} + 39 x^{26} - 147 x^{25} + 484 x^{24} - 1461 x^{23} + 4001 x^{22} - 10052 x^{21} + 23841 x^{20} - 52926 x^{19} + 109173 x^{18} - 208723 x^{17} + 369896 x^{16} - 608215 x^{15} + 913744 x^{14} - 1246890 x^{13} + 1584122 x^{12} - 1829692 x^{11} + 1878311 x^{10} - 1777347 x^{9} + 1518093 x^{8} - 1036936 x^{7} + 598732 x^{6} - 344364 x^{5} + 177291 x^{4} - 79332 x^{3} + 46992 x^{2} - 15367 x + 6413 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $28$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 14]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(84980457635217037392318371738994676821454381\)\(\medspace = 11^{14}\cdot 181^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $37.06$
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:  $11, 181$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{11} a^{16} + \frac{2}{11} a^{15} + \frac{2}{11} a^{14} + \frac{2}{11} a^{13} - \frac{2}{11} a^{11} + \frac{1}{11} a^{10} + \frac{4}{11} a^{9} - \frac{3}{11} a^{8} + \frac{1}{11} a^{7} - \frac{2}{11} a^{6} - \frac{5}{11} a^{5} - \frac{4}{11} a^{4} - \frac{3}{11} a^{3} + \frac{4}{11} a^{2}$, $\frac{1}{11} a^{17} - \frac{2}{11} a^{15} - \frac{2}{11} a^{14} - \frac{4}{11} a^{13} - \frac{2}{11} a^{12} + \frac{5}{11} a^{11} + \frac{2}{11} a^{10} - \frac{4}{11} a^{8} - \frac{4}{11} a^{7} - \frac{1}{11} a^{6} - \frac{5}{11} a^{5} + \frac{5}{11} a^{4} - \frac{1}{11} a^{3} + \frac{3}{11} a^{2}$, $\frac{1}{11} a^{18} + \frac{2}{11} a^{15} + \frac{2}{11} a^{13} + \frac{5}{11} a^{12} - \frac{2}{11} a^{11} + \frac{2}{11} a^{10} + \frac{4}{11} a^{9} + \frac{1}{11} a^{8} + \frac{1}{11} a^{7} + \frac{2}{11} a^{6} - \frac{5}{11} a^{5} + \frac{2}{11} a^{4} - \frac{3}{11} a^{3} - \frac{3}{11} a^{2}$, $\frac{1}{11} a^{19} - \frac{4}{11} a^{15} - \frac{2}{11} a^{14} + \frac{1}{11} a^{13} - \frac{2}{11} a^{12} - \frac{5}{11} a^{11} + \frac{2}{11} a^{10} + \frac{4}{11} a^{9} - \frac{4}{11} a^{8} - \frac{1}{11} a^{6} + \frac{1}{11} a^{5} + \frac{5}{11} a^{4} + \frac{3}{11} a^{3} + \frac{3}{11} a^{2}$, $\frac{1}{11} a^{20} - \frac{5}{11} a^{15} - \frac{2}{11} a^{14} - \frac{5}{11} a^{13} - \frac{5}{11} a^{12} + \frac{5}{11} a^{11} - \frac{3}{11} a^{10} + \frac{1}{11} a^{9} - \frac{1}{11} a^{8} + \frac{3}{11} a^{7} + \frac{4}{11} a^{6} - \frac{4}{11} a^{5} - \frac{2}{11} a^{4} + \frac{2}{11} a^{3} + \frac{5}{11} a^{2}$, $\frac{1}{11} a^{21} - \frac{3}{11} a^{15} + \frac{5}{11} a^{14} + \frac{5}{11} a^{13} + \frac{5}{11} a^{12} - \frac{2}{11} a^{11} - \frac{5}{11} a^{10} - \frac{3}{11} a^{9} - \frac{1}{11} a^{8} - \frac{2}{11} a^{7} - \frac{3}{11} a^{6} - \frac{5}{11} a^{5} + \frac{4}{11} a^{4} + \frac{1}{11} a^{3} - \frac{2}{11} a^{2}$, $\frac{1}{121} a^{22} + \frac{2}{121} a^{21} - \frac{4}{121} a^{20} + \frac{1}{121} a^{19} - \frac{3}{121} a^{18} + \frac{4}{121} a^{17} - \frac{3}{121} a^{16} + \frac{23}{121} a^{15} - \frac{42}{121} a^{14} + \frac{14}{121} a^{13} + \frac{3}{121} a^{12} + \frac{36}{121} a^{11} + \frac{14}{121} a^{10} - \frac{8}{121} a^{9} + \frac{10}{121} a^{8} + \frac{6}{121} a^{7} + \frac{50}{121} a^{6} + \frac{6}{121} a^{5} + \frac{58}{121} a^{4} - \frac{5}{11} a^{3} - \frac{2}{11} a^{2}$, $\frac{1}{121} a^{23} + \frac{3}{121} a^{21} - \frac{2}{121} a^{20} - \frac{5}{121} a^{19} - \frac{1}{121} a^{18} - \frac{4}{121} a^{16} - \frac{5}{11} a^{15} - \frac{34}{121} a^{14} - \frac{47}{121} a^{13} - \frac{58}{121} a^{12} + \frac{8}{121} a^{11} + \frac{30}{121} a^{10} + \frac{48}{121} a^{9} + \frac{30}{121} a^{8} + \frac{16}{121} a^{7} - \frac{17}{121} a^{6} - \frac{42}{121} a^{5} + \frac{60}{121} a^{4} - \frac{4}{11} a^{3} + \frac{2}{11} a^{2}$, $\frac{1}{121} a^{24} + \frac{3}{121} a^{21} - \frac{4}{121} a^{20} - \frac{4}{121} a^{19} - \frac{2}{121} a^{18} - \frac{5}{121} a^{17} - \frac{2}{121} a^{16} - \frac{37}{121} a^{15} - \frac{20}{121} a^{14} + \frac{32}{121} a^{13} + \frac{32}{121} a^{12} - \frac{45}{121} a^{11} + \frac{28}{121} a^{10} + \frac{21}{121} a^{9} + \frac{41}{121} a^{8} + \frac{20}{121} a^{7} - \frac{27}{121} a^{6} + \frac{53}{121} a^{5} - \frac{53}{121} a^{4} - \frac{5}{11} a^{3} - \frac{1}{11} a^{2}$, $\frac{1}{1573} a^{25} - \frac{6}{1573} a^{23} + \frac{6}{1573} a^{22} + \frac{61}{1573} a^{21} - \frac{70}{1573} a^{20} - \frac{35}{1573} a^{19} - \frac{30}{1573} a^{18} + \frac{43}{1573} a^{17} - \frac{6}{143} a^{16} + \frac{60}{1573} a^{15} + \frac{2}{13} a^{14} - \frac{711}{1573} a^{13} + \frac{620}{1573} a^{12} + \frac{43}{143} a^{11} - \frac{700}{1573} a^{10} - \frac{733}{1573} a^{9} + \frac{101}{1573} a^{8} - \frac{171}{1573} a^{7} + \frac{492}{1573} a^{6} + \frac{74}{1573} a^{5} - \frac{439}{1573} a^{4} - \frac{3}{13} a^{3} + \frac{40}{143} a^{2} - \frac{5}{13} a + \frac{2}{13}$, $\frac{1}{72032389} a^{26} - \frac{7643}{72032389} a^{25} + \frac{189586}{72032389} a^{24} + \frac{985}{503723} a^{23} + \frac{11195}{72032389} a^{22} - \frac{2706050}{72032389} a^{21} - \frac{2374503}{72032389} a^{20} + \frac{85144}{5540953} a^{19} - \frac{222363}{5540953} a^{18} - \frac{2385}{284713} a^{17} - \frac{310940}{72032389} a^{16} - \frac{21195444}{72032389} a^{15} + \frac{14028182}{72032389} a^{14} + \frac{397821}{5540953} a^{13} + \frac{9147971}{72032389} a^{12} + \frac{9858193}{72032389} a^{11} - \frac{29140413}{72032389} a^{10} - \frac{17455628}{72032389} a^{9} + \frac{32715938}{72032389} a^{8} - \frac{14910042}{72032389} a^{7} - \frac{31894024}{72032389} a^{6} - \frac{30330158}{72032389} a^{5} + \frac{26940996}{72032389} a^{4} - \frac{2411116}{6548399} a^{3} - \frac{3263288}{6548399} a^{2} - \frac{2434}{595309} a - \frac{280057}{595309}$, $\frac{1}{6948452831464791556490402061655278578013457256259205576961} a^{27} - \frac{24611249133424600620044742893774564635796556960118}{6948452831464791556490402061655278578013457256259205576961} a^{26} + \frac{1024372656857620692369180076050200764985662113193875171}{6948452831464791556490402061655278578013457256259205576961} a^{25} + \frac{18454629505371194223036694782994163759182652366812906433}{6948452831464791556490402061655278578013457256259205576961} a^{24} - \frac{15444417966829341165728786960211038986574175652524677769}{6948452831464791556490402061655278578013457256259205576961} a^{23} + \frac{457304266420827298907436331717989615483686049927179647}{131102883612543236914913246446326010905914287853947275037} a^{22} - \frac{286366443570279109534867885041114204144929149407153021364}{6948452831464791556490402061655278578013457256259205576961} a^{21} - \frac{191368196449251482725504801433329017971458950820374640964}{6948452831464791556490402061655278578013457256259205576961} a^{20} - \frac{59468641445398164073044355389236773223652391873630894951}{6948452831464791556490402061655278578013457256259205576961} a^{19} + \frac{148482743597142294775312683899186015393950718752768791769}{6948452831464791556490402061655278578013457256259205576961} a^{18} + \frac{117823364893778367290214898666607699418114970162460833383}{6948452831464791556490402061655278578013457256259205576961} a^{17} + \frac{159896230233363941926573240127991064360639535972510697045}{6948452831464791556490402061655278578013457256259205576961} a^{16} + \frac{8064729618736763914701218028103163542515918275570386875}{27464240440572298642254553603380547739183625518811089237} a^{15} + \frac{2938041375327781142045642911927849159523513989666293886699}{6948452831464791556490402061655278578013457256259205576961} a^{14} + \frac{3339427010933309203764910296763810062718098851601585037348}{6948452831464791556490402061655278578013457256259205576961} a^{13} + \frac{3278783967714673017687955622380479031793948717160615296620}{6948452831464791556490402061655278578013457256259205576961} a^{12} + \frac{3462862950614304844082617446257190915083799648506676369083}{6948452831464791556490402061655278578013457256259205576961} a^{11} + \frac{172723147161653664646531485442644164465981006773542210994}{631677530133162868771854732877752598001223386932655052451} a^{10} + \frac{337921078195356134633916402815526302899298938101606984260}{6948452831464791556490402061655278578013457256259205576961} a^{9} + \frac{1072860002898812187615549224272460045885486282087472578884}{6948452831464791556490402061655278578013457256259205576961} a^{8} - \frac{112575074519457302645416789038612713001328144646626259104}{302106644846295285064800089637186025131019880706921981607} a^{7} + \frac{1166410881763028004974030221998639285982002858941396448107}{6948452831464791556490402061655278578013457256259205576961} a^{6} + \frac{986581369356939051508359288954916252794702395316661753623}{6948452831464791556490402061655278578013457256259205576961} a^{5} - \frac{2543988599662322792197724164748930677844254263700152574130}{6948452831464791556490402061655278578013457256259205576961} a^{4} + \frac{1974359437739808383309909626467753639689757149375774514}{6132791554690901638561696435706335902924498902258786917} a^{3} - \frac{4503775796893363667725814790647611917614140183107988722}{27464240440572298642254553603380547739183625518811089237} a^{2} - \frac{12776073526367268736610260209487864456110782021381745821}{57425230012105715342895884807068418000111216993877732041} a - \frac{252715611982334774608544012812388797251535874580369708}{1083494905888787081941431788812611660379456924412787397}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 81264745077.87146 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{14}\cdot 81264745077.87146 \cdot 1}{2\sqrt{84980457635217037392318371738994676821454381}}\approx 0.658765438022473$ (assuming GRH)

Galois group

$D_{28}$ (as 28T10):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 56
The 17 conjugacy class representatives for $D_{28}$
Character table for $D_{28}$

Intermediate fields

\(\Q(\sqrt{-11}) \), 4.0.21901.1, 7.1.7892485271.1, 14.0.685204561282471377851.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 28 sibling: Deg 28

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $28$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/5.14.0.1}{14} }^{2}$ $28$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{14}$ $28$ $28$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{4}$ $28$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }^{2}$

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
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
$181$$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
181.2.1.1$x^{2} - 181$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.1$x^{2} - 181$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.1$x^{2} - 181$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.1$x^{2} - 181$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.1$x^{2} - 181$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.1$x^{2} - 181$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.1$x^{2} - 181$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.1$x^{2} - 181$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.1$x^{2} - 181$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.1$x^{2} - 181$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.1$x^{2} - 181$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.1$x^{2} - 181$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.1$x^{2} - 181$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.181.2t1.a.a$1$ $ 181 $ \(\Q(\sqrt{181}) \) $C_2$ (as 2T1) $1$ $1$
1.1991.2t1.a.a$1$ $ 11 \cdot 181 $ \(\Q(\sqrt{-1991}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.11.2t1.a.a$1$ $ 11 $ \(\Q(\sqrt{-11}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.1991.4t3.c.a$2$ $ 11 \cdot 181 $ 4.2.360371.1 $D_{4}$ (as 4T3) $1$ $0$
* 2.1991.14t3.a.a$2$ $ 11 \cdot 181 $ 14.2.11274729599284301762821.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.1991.14t3.a.c$2$ $ 11 \cdot 181 $ 14.2.11274729599284301762821.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.1991.7t2.a.b$2$ $ 11 \cdot 181 $ 7.1.7892485271.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.1991.7t2.a.a$2$ $ 11 \cdot 181 $ 7.1.7892485271.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.1991.14t3.a.b$2$ $ 11 \cdot 181 $ 14.2.11274729599284301762821.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.1991.7t2.a.c$2$ $ 11 \cdot 181 $ 7.1.7892485271.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.1991.28t10.a.e$2$ $ 11 \cdot 181 $ 28.0.84980457635217037392318371738994676821454381.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.1991.28t10.a.b$2$ $ 11 \cdot 181 $ 28.0.84980457635217037392318371738994676821454381.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.1991.28t10.a.a$2$ $ 11 \cdot 181 $ 28.0.84980457635217037392318371738994676821454381.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.1991.28t10.a.c$2$ $ 11 \cdot 181 $ 28.0.84980457635217037392318371738994676821454381.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.1991.28t10.a.f$2$ $ 11 \cdot 181 $ 28.0.84980457635217037392318371738994676821454381.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.1991.28t10.a.d$2$ $ 11 \cdot 181 $ 28.0.84980457635217037392318371738994676821454381.1 $D_{28}$ (as 28T10) $1$ $0$

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 *.