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Properties

Label 28.0.194...021.1
Degree $28$
Signature $[0, 14]$
Discriminant $1.942\times 10^{41}$
Root discriminant $29.82$
Ramified primes $3, 7$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{28}$ (as 28T10)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^28 + 28*x^24 - 14*x^23 - 35*x^22 - 19*x^21 + 1470*x^20 - 4557*x^19 + 9849*x^18 - 9660*x^17 + 2863*x^16 + 12915*x^15 - 18102*x^14 + 12103*x^13 - 2891*x^12 - 21119*x^11 + 2975*x^10 - 4704*x^9 + 70*x^8 + 13698*x^7 + 6909*x^6 + 4900*x^5 + 5047*x^4 + 1015*x^3 - 7*x^2 - 7*x + 1)
 
gp: K = bnfinit(x^28 + 28*x^24 - 14*x^23 - 35*x^22 - 19*x^21 + 1470*x^20 - 4557*x^19 + 9849*x^18 - 9660*x^17 + 2863*x^16 + 12915*x^15 - 18102*x^14 + 12103*x^13 - 2891*x^12 - 21119*x^11 + 2975*x^10 - 4704*x^9 + 70*x^8 + 13698*x^7 + 6909*x^6 + 4900*x^5 + 5047*x^4 + 1015*x^3 - 7*x^2 - 7*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -7, -7, 1015, 5047, 4900, 6909, 13698, 70, -4704, 2975, -21119, -2891, 12103, -18102, 12915, 2863, -9660, 9849, -4557, 1470, -19, -35, -14, 28, 0, 0, 0, 1]);
 

\( x^{28} + 28 x^{24} - 14 x^{23} - 35 x^{22} - 19 x^{21} + 1470 x^{20} - 4557 x^{19} + 9849 x^{18} - 9660 x^{17} + 2863 x^{16} + 12915 x^{15} - 18102 x^{14} + 12103 x^{13} - 2891 x^{12} - 21119 x^{11} + 2975 x^{10} - 4704 x^{9} + 70 x^{8} + 13698 x^{7} + 6909 x^{6} + 4900 x^{5} + 5047 x^{4} + 1015 x^{3} - 7 x^{2} - 7 x + 1 \)

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:  \(194166288728873478373956441365593381326021\)\(\medspace = 3^{21}\cdot 7^{37}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $29.82$
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:  $3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{15} + \frac{1}{3} a^{14} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{14} - \frac{1}{3} a^{13} - \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{15} a^{18} + \frac{2}{15} a^{16} - \frac{1}{3} a^{15} + \frac{7}{15} a^{14} + \frac{1}{3} a^{13} + \frac{2}{5} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{2}{5} a^{6} + \frac{1}{3} a^{5} + \frac{2}{15} a^{4} - \frac{1}{3} a^{3} + \frac{7}{15} a^{2} - \frac{4}{15}$, $\frac{1}{45} a^{19} - \frac{1}{15} a^{17} - \frac{1}{9} a^{16} + \frac{22}{45} a^{15} - \frac{4}{9} a^{14} - \frac{4}{45} a^{13} - \frac{4}{9} a^{11} - \frac{1}{3} a^{10} - \frac{1}{9} a^{8} - \frac{19}{45} a^{7} + \frac{2}{9} a^{6} - \frac{13}{45} a^{5} + \frac{4}{9} a^{4} + \frac{2}{45} a^{3} - \frac{4}{9} a^{2} + \frac{1}{45} a + \frac{4}{9}$, $\frac{1}{45} a^{20} - \frac{1}{9} a^{17} - \frac{2}{45} a^{16} - \frac{4}{9} a^{15} - \frac{13}{45} a^{14} + \frac{1}{3} a^{13} - \frac{17}{45} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{2}{9} a^{9} - \frac{19}{45} a^{8} + \frac{2}{9} a^{7} + \frac{1}{9} a^{6} + \frac{4}{9} a^{5} - \frac{7}{45} a^{4} + \frac{2}{9} a^{3} - \frac{8}{45} a^{2} + \frac{1}{9} a + \frac{1}{15}$, $\frac{1}{315} a^{21} + \frac{1}{45} a^{18} + \frac{4}{45} a^{17} + \frac{7}{45} a^{16} - \frac{4}{45} a^{15} - \frac{52}{105} a^{14} - \frac{11}{45} a^{13} - \frac{1}{5} a^{12} + \frac{1}{3} a^{11} - \frac{4}{9} a^{10} - \frac{22}{45} a^{9} - \frac{4}{9} a^{8} + \frac{10}{63} a^{7} - \frac{4}{45} a^{6} + \frac{14}{45} a^{5} + \frac{22}{45} a^{4} + \frac{1}{45} a^{3} + \frac{17}{45} a^{2} + \frac{1}{5} a - \frac{11}{105}$, $\frac{1}{315} a^{22} + \frac{1}{45} a^{18} - \frac{1}{9} a^{17} - \frac{1}{9} a^{16} + \frac{22}{63} a^{15} + \frac{1}{15} a^{14} - \frac{1}{9} a^{13} + \frac{4}{15} a^{12} - \frac{1}{3} a^{11} - \frac{7}{45} a^{10} + \frac{2}{9} a^{9} - \frac{25}{63} a^{8} + \frac{1}{45} a^{6} + \frac{4}{9} a^{5} + \frac{1}{9} a^{4} + \frac{1}{3} a^{3} - \frac{7}{45} a^{2} + \frac{13}{63} a + \frac{7}{45}$, $\frac{1}{2205} a^{23} + \frac{1}{2205} a^{22} + \frac{2}{2205} a^{21} + \frac{1}{45} a^{18} + \frac{1}{315} a^{17} + \frac{19}{2205} a^{16} - \frac{709}{2205} a^{15} - \frac{67}{147} a^{14} - \frac{8}{45} a^{13} - \frac{7}{15} a^{12} + \frac{118}{315} a^{11} + \frac{113}{315} a^{10} - \frac{226}{735} a^{9} + \frac{115}{441} a^{8} + \frac{37}{147} a^{7} + \frac{7}{15} a^{6} - \frac{23}{105} a^{5} - \frac{4}{45} a^{4} - \frac{82}{315} a^{3} - \frac{211}{441} a^{2} - \frac{397}{2205} a - \frac{724}{2205}$, $\frac{1}{2205} a^{24} + \frac{1}{2205} a^{22} - \frac{2}{2205} a^{21} - \frac{2}{105} a^{18} + \frac{53}{735} a^{17} + \frac{4}{35} a^{16} - \frac{71}{245} a^{15} + \frac{41}{735} a^{14} - \frac{1}{5} a^{13} - \frac{31}{63} a^{12} - \frac{5}{21} a^{11} - \frac{734}{2205} a^{10} + \frac{74}{315} a^{9} + \frac{64}{147} a^{8} - \frac{13}{441} a^{7} + \frac{29}{315} a^{6} + \frac{3}{35} a^{5} + \frac{16}{315} a^{4} - \frac{193}{735} a^{3} + \frac{8}{105} a^{2} + \frac{1094}{2205} a + \frac{479}{2205}$, $\frac{1}{2205} a^{25} - \frac{1}{735} a^{22} - \frac{2}{2205} a^{21} + \frac{1}{315} a^{19} - \frac{37}{2205} a^{18} + \frac{2}{45} a^{17} + \frac{13}{105} a^{16} - \frac{59}{441} a^{15} + \frac{5}{441} a^{14} + \frac{83}{315} a^{13} + \frac{17}{105} a^{12} - \frac{67}{441} a^{11} + \frac{22}{105} a^{10} + \frac{43}{105} a^{9} - \frac{59}{147} a^{8} + \frac{922}{2205} a^{7} + \frac{139}{315} a^{6} - \frac{2}{105} a^{5} + \frac{346}{735} a^{4} - \frac{2}{7} a^{3} - \frac{17}{63} a^{2} - \frac{109}{441} a - \frac{72}{245}$, $\frac{1}{46293975} a^{26} + \frac{1063}{6613425} a^{25} - \frac{334}{5143775} a^{24} - \frac{58}{270725} a^{23} + \frac{2164}{3561075} a^{22} + \frac{2152}{6613425} a^{21} + \frac{8963}{1322685} a^{20} - \frac{64717}{46293975} a^{19} - \frac{188861}{6613425} a^{18} + \frac{97513}{712215} a^{17} - \frac{2502677}{46293975} a^{16} + \frac{19216598}{46293975} a^{15} - \frac{1454}{188955} a^{14} - \frac{493231}{2204475} a^{13} + \frac{22378672}{46293975} a^{12} - \frac{365048}{2204475} a^{11} + \frac{7403584}{15431325} a^{10} - \frac{2054524}{5143775} a^{9} + \frac{141486}{1028755} a^{8} + \frac{99646}{734825} a^{7} - \frac{968152}{2204475} a^{6} + \frac{4557776}{9258795} a^{5} - \frac{238}{4275} a^{4} - \frac{2536879}{15431325} a^{3} + \frac{4833194}{46293975} a^{2} + \frac{1869851}{15431325} a - \frac{2853551}{6613425}$, $\frac{1}{56886505789770608186789192755949152872825} a^{27} - \frac{35771743980121858223672198187002}{3346265046457094599222893691526420757225} a^{26} + \frac{4890967528896389031317022851379939359}{56886505789770608186789192755949152872825} a^{25} - \frac{10625099556366108249359044203804188398}{56886505789770608186789192755949152872825} a^{24} - \frac{2898963745687758145722200931706792168}{56886505789770608186789192755949152872825} a^{23} - \frac{8385805029321695344524879033491837722}{18962168596590202728929730918649717624275} a^{22} - \frac{5124770479904835024267019529648667907}{11377301157954121637357838551189830574565} a^{21} - \frac{143836718842553444371475236126409852552}{56886505789770608186789192755949152872825} a^{20} + \frac{168580481679482036512749836525218516786}{18962168596590202728929730918649717624275} a^{19} - \frac{334761206129850742795660742558684736832}{11377301157954121637357838551189830574565} a^{18} - \frac{7869364020393380425596957173485158316252}{56886505789770608186789192755949152872825} a^{17} - \frac{5425147797802850197501811098570350477697}{56886505789770608186789192755949152872825} a^{16} + \frac{2917378281513395641795291922337403653542}{11377301157954121637357838551189830574565} a^{15} - \frac{18065210444992952018802679796622872038}{221348271555527658314354835626261295225} a^{14} - \frac{2040828868053594711079579570229354116771}{18962168596590202728929730918649717624275} a^{13} - \frac{17179018384327708843272430107136975957028}{56886505789770608186789192755949152872825} a^{12} + \frac{137501011742625634361966853596660760722}{1387475750970014833824126652584125679825} a^{11} - \frac{8350654401805098700956246159228759086282}{18962168596590202728929730918649717624275} a^{10} - \frac{451528972547034656445223624322980676}{7115260261384691455508341808123721435} a^{9} - \frac{11839730086098826147007280825158147948687}{56886505789770608186789192755949152872825} a^{8} + \frac{158577169665303741681732277334348833648}{1387475750970014833824126652584125679825} a^{7} - \frac{36956580658832449995648257351001278381}{133850601858283783968915747661056830289} a^{6} - \frac{445411079924388290016561841179722674727}{56886505789770608186789192755949152872825} a^{5} + \frac{11402197122221631326657307754067073030943}{56886505789770608186789192755949152872825} a^{4} - \frac{2830905091693953200741502271210034700722}{18962168596590202728929730918649717624275} a^{3} - \frac{7375988574927877501630875490748423933012}{56886505789770608186789192755949152872825} a^{2} + \frac{9456026709750301306867170715593909019418}{56886505789770608186789192755949152872825} a + \frac{300956330243363765190411252983147228718}{1264144573106013515261982061243314508285}$

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:  \( -\frac{2483015689689046132493535544}{135590657338272663144140439575} a^{27} + \frac{2550543468967063270793151703}{135590657338272663144140439575} a^{26} - \frac{399125827023381464749367464}{135590657338272663144140439575} a^{25} + \frac{5339565635006596601382103}{4785552611939035169793191985} a^{24} - \frac{69618072586628078637744406409}{135590657338272663144140439575} a^{23} + \frac{106253445256546974600845969303}{135590657338272663144140439575} a^{22} + \frac{119850001201105857689343127274}{406771972014817989432421318725} a^{21} - \frac{1894113248788320169802516891}{7975921019898391949655319975} a^{20} - \frac{3689312465566995731063620029961}{135590657338272663144140439575} a^{19} + \frac{15070188960599642672111541589406}{135590657338272663144140439575} a^{18} - \frac{36667504056741658515065598846192}{135590657338272663144140439575} a^{17} + \frac{51149522515922769936197617155879}{135590657338272663144140439575} a^{16} - \frac{36506418125527823753562191525819}{135590657338272663144140439575} a^{15} - \frac{4321473322594020558373154494}{31034712139682458948075175} a^{14} + \frac{14781353780222136425805463800152}{27118131467654532628828087915} a^{13} - \frac{236578907182029997695256974094232}{406771972014817989432421318725} a^{12} + \frac{1100707315458103496336303720491}{3307089203372503979125376575} a^{11} + \frac{38185084449299036945595814437488}{135590657338272663144140439575} a^{10} - \frac{73537725712684747062956184953}{174057326493289683111861925} a^{9} + \frac{25372018618811115752087887460923}{135590657338272663144140439575} a^{8} - \frac{356887161289165830050959144831}{3307089203372503979125376575} a^{7} - \frac{31218436408085924827447862317339}{135590657338272663144140439575} a^{6} + \frac{16821201697778245653747351112203}{135590657338272663144140439575} a^{5} + \frac{607109558819869352566853417219}{135590657338272663144140439575} a^{4} - \frac{764906593331357705290854969772}{81354394402963597886484263745} a^{3} + \frac{452436643392682952353581058149}{7136350386224877007586338925} a^{2} + \frac{1098898729747356602719239949854}{135590657338272663144140439575} a - \frac{3881275378066986026342599727}{406771972014817989432421318725} \) (order $6$)
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:  \( 13724323267.71925 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{14}\cdot 13724323267.71925 \cdot 1}{6\sqrt{194166288728873478373956441365593381326021}}\approx 0.775837792097827$ (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

\(\Q(\sqrt{-3}) \), 4.0.189.1, 7.1.40353607.1, 14.0.3561340538630151963.1

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 $28$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{14}$ R $28$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $28$ $28$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{14}$ $28$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$

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
$3$3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$7$7.14.18.23$x^{14} + 294 x^{13} + 21 x^{12} + 322 x^{11} + 224 x^{10} + 315 x^{9} + 63 x^{8} + 111 x^{7} + 273 x^{6} + 140 x^{5} + 189 x^{4} + 63 x^{3} + 182 x^{2} + 119 x + 258$$7$$2$$18$$D_{14}$$[3/2]_{2}^{2}$
7.14.19.4$x^{14} + 14 x^{12} + 14 x^{11} + 21 x^{10} - 7 x^{9} + 14 x^{8} - 7 x^{7} + 21 x^{6} - 14$$14$$1$$19$$D_{14}$$[3/2]_{2}^{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.3.2t1.a.a$1$ $ 3 $ \(\Q(\sqrt{-3}) \) $C_2$ (as 2T1) $1$ $-1$
1.7.2t1.a.a$1$ $ 7 $ \(\Q(\sqrt{-7}) \) $C_2$ (as 2T1) $1$ $-1$
1.21.2t1.a.a$1$ $ 3 \cdot 7 $ \(\Q(\sqrt{21}) \) $C_2$ (as 2T1) $1$ $1$
* 2.63.4t3.a.a$2$ $ 3^{2} \cdot 7 $ 4.0.189.1 $D_{4}$ (as 4T3) $1$ $0$
* 2.343.7t2.a.c$2$ $ 7^{3}$ 7.1.40353607.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.3087.14t3.a.b$2$ $ 3^{2} \cdot 7^{3}$ 14.0.3561340538630151963.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.343.7t2.a.a$2$ $ 7^{3}$ 7.1.40353607.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.3087.14t3.a.a$2$ $ 3^{2} \cdot 7^{3}$ 14.0.3561340538630151963.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.343.7t2.a.b$2$ $ 7^{3}$ 7.1.40353607.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.3087.14t3.a.c$2$ $ 3^{2} \cdot 7^{3}$ 14.0.3561340538630151963.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.3087.28t10.a.a$2$ $ 3^{2} \cdot 7^{3}$ 28.0.194166288728873478373956441365593381326021.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.3087.28t10.a.f$2$ $ 3^{2} \cdot 7^{3}$ 28.0.194166288728873478373956441365593381326021.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.3087.28t10.a.c$2$ $ 3^{2} \cdot 7^{3}$ 28.0.194166288728873478373956441365593381326021.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.3087.28t10.a.e$2$ $ 3^{2} \cdot 7^{3}$ 28.0.194166288728873478373956441365593381326021.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.3087.28t10.a.d$2$ $ 3^{2} \cdot 7^{3}$ 28.0.194166288728873478373956441365593381326021.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.3087.28t10.a.b$2$ $ 3^{2} \cdot 7^{3}$ 28.0.194166288728873478373956441365593381326021.1 $D_{28}$ (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 *.