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Properties

Label 28.2.725...875.1
Degree $28$
Signature $[2, 13]$
Discriminant $-7.259\times 10^{44}$
Root discriminant $40.01$
Ramified primes $5, 499$
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 - 5*x^27 - 3*x^26 + 57*x^25 - 13*x^24 - 420*x^23 + 259*x^22 + 2099*x^21 - 1147*x^20 - 7975*x^19 + 7844*x^18 + 15881*x^17 - 34395*x^16 - 17074*x^15 + 107194*x^14 - 103829*x^13 + 94772*x^12 - 84342*x^11 - 113950*x^10 + 177636*x^9 - 26899*x^8 + 2410*x^7 + 74025*x^6 - 120275*x^5 - 49775*x^4 + 101875*x^3 - 41875*x^2 + 31250*x - 15625)
 
gp: K = bnfinit(x^28 - 5*x^27 - 3*x^26 + 57*x^25 - 13*x^24 - 420*x^23 + 259*x^22 + 2099*x^21 - 1147*x^20 - 7975*x^19 + 7844*x^18 + 15881*x^17 - 34395*x^16 - 17074*x^15 + 107194*x^14 - 103829*x^13 + 94772*x^12 - 84342*x^11 - 113950*x^10 + 177636*x^9 - 26899*x^8 + 2410*x^7 + 74025*x^6 - 120275*x^5 - 49775*x^4 + 101875*x^3 - 41875*x^2 + 31250*x - 15625, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-15625, 31250, -41875, 101875, -49775, -120275, 74025, 2410, -26899, 177636, -113950, -84342, 94772, -103829, 107194, -17074, -34395, 15881, 7844, -7975, -1147, 2099, 259, -420, -13, 57, -3, -5, 1]);
 

\( x^{28} - 5 x^{27} - 3 x^{26} + 57 x^{25} - 13 x^{24} - 420 x^{23} + 259 x^{22} + 2099 x^{21} - 1147 x^{20} - 7975 x^{19} + 7844 x^{18} + 15881 x^{17} - 34395 x^{16} - 17074 x^{15} + 107194 x^{14} - 103829 x^{13} + 94772 x^{12} - 84342 x^{11} - 113950 x^{10} + 177636 x^{9} - 26899 x^{8} + 2410 x^{7} + 74025 x^{6} - 120275 x^{5} - 49775 x^{4} + 101875 x^{3} - 41875 x^{2} + 31250 x - 15625 \)

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

Invariants

Degree:  $28$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[2, 13]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-725917312055025700242151478457681268310546875\)\(\medspace = -\,5^{14}\cdot 499^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $40.01$
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:  $5, 499$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{9} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{6} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{7} + \frac{1}{5} a^{3}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{25} a^{14} + \frac{2}{25} a^{12} + \frac{2}{25} a^{11} + \frac{2}{25} a^{10} + \frac{1}{25} a^{9} - \frac{1}{25} a^{8} + \frac{1}{25} a^{7} + \frac{1}{5} a^{6} - \frac{3}{25} a^{5} - \frac{3}{25} a^{4} - \frac{1}{5} a^{3} - \frac{2}{5} a^{2}$, $\frac{1}{25} a^{15} + \frac{2}{25} a^{13} + \frac{2}{25} a^{12} + \frac{2}{25} a^{11} + \frac{1}{25} a^{10} - \frac{1}{25} a^{9} + \frac{1}{25} a^{8} + \frac{1}{5} a^{7} - \frac{3}{25} a^{6} - \frac{3}{25} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3}$, $\frac{1}{25} a^{16} + \frac{2}{25} a^{13} - \frac{2}{25} a^{12} + \frac{2}{25} a^{11} - \frac{1}{25} a^{9} + \frac{2}{25} a^{8} + \frac{12}{25} a^{6} + \frac{6}{25} a^{5} + \frac{6}{25} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{25} a^{17} - \frac{2}{25} a^{13} - \frac{2}{25} a^{12} + \frac{1}{25} a^{11} + \frac{2}{25} a^{8} + \frac{11}{25} a^{6} + \frac{12}{25} a^{5} + \frac{11}{25} a^{4} + \frac{2}{5} a^{3}$, $\frac{1}{25} a^{18} - \frac{2}{25} a^{13} - \frac{1}{25} a^{11} - \frac{1}{25} a^{10} - \frac{1}{25} a^{9} - \frac{2}{25} a^{8} + \frac{8}{25} a^{7} + \frac{12}{25} a^{6} - \frac{2}{5} a^{5} - \frac{11}{25} a^{4} + \frac{1}{5} a^{3}$, $\frac{1}{125} a^{19} - \frac{2}{125} a^{18} - \frac{1}{125} a^{17} + \frac{2}{125} a^{16} + \frac{1}{125} a^{15} - \frac{1}{125} a^{14} + \frac{7}{125} a^{13} + \frac{6}{125} a^{12} + \frac{8}{125} a^{11} + \frac{9}{125} a^{10} + \frac{3}{125} a^{9} + \frac{9}{125} a^{8} - \frac{8}{125} a^{7} - \frac{29}{125} a^{6} - \frac{7}{125} a^{5} - \frac{3}{25} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{125} a^{20} + \frac{1}{125} a^{15} - \frac{1}{25} a^{12} + \frac{6}{125} a^{10} + \frac{2}{25} a^{9} - \frac{1}{25} a^{8} + \frac{3}{25} a^{7} + \frac{2}{25} a^{6} - \frac{44}{125} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{125} a^{21} + \frac{1}{125} a^{16} - \frac{1}{25} a^{13} + \frac{6}{125} a^{11} + \frac{2}{25} a^{10} - \frac{1}{25} a^{9} - \frac{2}{25} a^{8} - \frac{8}{25} a^{7} + \frac{6}{125} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{2}$, $\frac{1}{625} a^{22} + \frac{2}{625} a^{21} - \frac{1}{625} a^{20} + \frac{2}{625} a^{19} - \frac{9}{625} a^{18} + \frac{4}{625} a^{17} - \frac{9}{625} a^{16} - \frac{4}{625} a^{15} - \frac{2}{625} a^{14} + \frac{39}{625} a^{13} - \frac{32}{625} a^{12} - \frac{57}{625} a^{11} + \frac{62}{625} a^{10} + \frac{56}{625} a^{9} + \frac{18}{625} a^{8} + \frac{2}{125} a^{7} + \frac{99}{625} a^{6} + \frac{1}{5} a^{5} + \frac{11}{25} a^{4} + \frac{2}{5} a^{3} - \frac{9}{25} a^{2} - \frac{2}{5} a$, $\frac{1}{625} a^{23} - \frac{1}{625} a^{20} + \frac{2}{625} a^{19} - \frac{8}{625} a^{18} - \frac{7}{625} a^{17} - \frac{1}{625} a^{16} - \frac{9}{625} a^{15} + \frac{3}{625} a^{14} + \frac{4}{125} a^{13} - \frac{53}{625} a^{12} + \frac{26}{625} a^{11} + \frac{12}{625} a^{10} + \frac{51}{625} a^{9} + \frac{34}{625} a^{8} - \frac{91}{625} a^{7} + \frac{222}{625} a^{6} + \frac{8}{125} a^{5} + \frac{6}{25} a^{4} + \frac{11}{25} a^{3} + \frac{3}{25} a^{2} + \frac{1}{5} a$, $\frac{1}{6875} a^{24} - \frac{2}{6875} a^{23} + \frac{1}{6875} a^{22} + \frac{21}{6875} a^{21} - \frac{27}{6875} a^{20} - \frac{1}{1375} a^{19} - \frac{2}{125} a^{18} - \frac{63}{6875} a^{17} - \frac{1}{625} a^{16} + \frac{17}{6875} a^{15} - \frac{93}{6875} a^{14} + \frac{481}{6875} a^{13} - \frac{109}{1375} a^{12} + \frac{263}{6875} a^{11} + \frac{79}{6875} a^{10} + \frac{353}{6875} a^{9} - \frac{521}{6875} a^{8} + \frac{1224}{6875} a^{7} + \frac{569}{1375} a^{6} - \frac{394}{1375} a^{5} - \frac{11}{25} a^{4} + \frac{6}{25} a^{3} - \frac{9}{55} a^{2} + \frac{4}{55} a + \frac{2}{11}$, $\frac{1}{34375} a^{25} - \frac{1}{34375} a^{24} - \frac{1}{34375} a^{23} + \frac{2}{3125} a^{22} - \frac{61}{34375} a^{21} - \frac{87}{34375} a^{20} - \frac{23}{6875} a^{19} - \frac{448}{34375} a^{18} - \frac{349}{34375} a^{17} + \frac{501}{34375} a^{16} + \frac{144}{34375} a^{15} + \frac{388}{34375} a^{14} + \frac{1586}{34375} a^{13} + \frac{1368}{34375} a^{12} - \frac{2463}{34375} a^{11} + \frac{1477}{34375} a^{10} + \frac{3132}{34375} a^{9} + \frac{2903}{34375} a^{8} + \frac{3244}{34375} a^{7} + \frac{1814}{6875} a^{6} - \frac{411}{1375} a^{5} - \frac{18}{125} a^{4} + \frac{186}{1375} a^{3} - \frac{126}{275} a^{2} + \frac{27}{55} a - \frac{4}{11}$, $\frac{1}{26778125} a^{26} + \frac{1}{130625} a^{25} + \frac{543}{26778125} a^{24} - \frac{6084}{26778125} a^{23} + \frac{9016}{26778125} a^{22} - \frac{100688}{26778125} a^{21} + \frac{82493}{26778125} a^{20} + \frac{100977}{26778125} a^{19} + \frac{388008}{26778125} a^{18} + \frac{303262}{26778125} a^{17} - \frac{32526}{5355625} a^{16} - \frac{342528}{26778125} a^{15} - \frac{327476}{26778125} a^{14} - \frac{850011}{26778125} a^{13} - \frac{195039}{5355625} a^{12} + \frac{2129609}{26778125} a^{11} + \frac{1035609}{26778125} a^{10} - \frac{10898}{281875} a^{9} - \frac{374853}{26778125} a^{8} - \frac{13261541}{26778125} a^{7} + \frac{2183}{5225} a^{6} - \frac{452668}{1071125} a^{5} + \frac{388508}{1071125} a^{4} - \frac{35749}{97375} a^{3} + \frac{3148}{19475} a^{2} + \frac{11653}{42845} a - \frac{2570}{8569}$, $\frac{1}{68197585161567464382778623695847273894144760890625} a^{27} + \frac{919207749021906554682959668295283150569862}{68197585161567464382778623695847273894144760890625} a^{26} + \frac{734587698440445715999305411999321280897312481}{68197585161567464382778623695847273894144760890625} a^{25} + \frac{4826882319811856074156454294079503111987396969}{68197585161567464382778623695847273894144760890625} a^{24} + \frac{9207646722489444534181629106955156618791364198}{13639517032313492876555724739169454778828952178125} a^{23} - \frac{4875381967358295922385459078525796834712883369}{13639517032313492876555724739169454778828952178125} a^{22} + \frac{132187126730934484308749726582853353476167653519}{68197585161567464382778623695847273894144760890625} a^{21} - \frac{11324727442892648458778821260631200711883032103}{68197585161567464382778623695847273894144760890625} a^{20} + \frac{213929358529516500173946625642667993254645629097}{68197585161567464382778623695847273894144760890625} a^{19} - \frac{506909480747572256568579375713705106032137010291}{68197585161567464382778623695847273894144760890625} a^{18} + \frac{1194213751807359788305778752747583407336431995932}{68197585161567464382778623695847273894144760890625} a^{17} - \frac{271632935331935729201633489422462222123468811671}{13639517032313492876555724739169454778828952178125} a^{16} - \frac{217894135214727561193887412043839427492668917019}{13639517032313492876555724739169454778828952178125} a^{15} + \frac{242909529162515945579326889464479996912842161}{151214157786180630560484753205869786904977296875} a^{14} + \frac{297774521533828449884671967739323912927228476971}{6199780469233405852979874881440661263104069171875} a^{13} - \frac{2920889944079587267285758245807321029207345104447}{68197585161567464382778623695847273894144760890625} a^{12} + \frac{4435536312246787018071535765022570441261712245678}{68197585161567464382778623695847273894144760890625} a^{11} - \frac{4585611167609831281935250915217012903022670604826}{68197585161567464382778623695847273894144760890625} a^{10} - \frac{5929001284063086406005545392381870398294116416377}{68197585161567464382778623695847273894144760890625} a^{9} - \frac{2308755217309046394643773359472379715905965540628}{68197585161567464382778623695847273894144760890625} a^{8} + \frac{837020560450204070127701393019846024785301543903}{13639517032313492876555724739169454778828952178125} a^{7} + \frac{91757338010360042892384289280444312577421781178}{247991218769336234119194995257626450524162766875} a^{6} - \frac{25163566944880033015224760979024859569205484878}{2727903406462698575311144947833890955765790435625} a^{5} + \frac{731243363267716860050651594519464739000747358633}{2727903406462698575311144947833890955765790435625} a^{4} + \frac{89361547003539180550559924511010001672313391821}{545580681292539715062228989566778191153158087125} a^{3} - \frac{18714866590684908613001126609036829179188699936}{109116136258507943012445797913355638230631617425} a^{2} + \frac{3313184066430435527920984601531430103617420404}{21823227251701588602489159582671127646126323485} a - \frac{2094308493428239412423457222950719576375792571}{4364645450340317720497831916534225529225264697}$

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:  $14$
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:  \( 2346058429844.763 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{13}\cdot 2346058429844.763 \cdot 1}{2\sqrt{725917312055025700242151478457681268310546875}}\approx 4.14250967749672$ (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

\(\Q(\sqrt{5}) \), 4.2.12475.1, 7.1.15531437375.1, 14.2.1206127734667734453125.1

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 $28$ $28$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{7}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $28$ $28$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $28$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{4}$ $28$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{14}$ $28$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
499Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.499.2t1.a.a$1$ $ 499 $ \(\Q(\sqrt{-499}) \) $C_2$ (as 2T1) $1$ $-1$
1.2495.2t1.a.a$1$ $ 5 \cdot 499 $ \(\Q(\sqrt{-2495}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
* 2.2495.4t3.c.a$2$ $ 5 \cdot 499 $ 4.0.1245005.1 $D_{4}$ (as 4T3) $1$ $0$
* 2.2495.7t2.a.a$2$ $ 5 \cdot 499 $ 7.1.15531437375.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.2495.14t3.a.a$2$ $ 5 \cdot 499 $ 14.0.120371547919839898421875.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.2495.14t3.a.b$2$ $ 5 \cdot 499 $ 14.0.120371547919839898421875.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.2495.7t2.a.c$2$ $ 5 \cdot 499 $ 7.1.15531437375.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.2495.7t2.a.b$2$ $ 5 \cdot 499 $ 7.1.15531437375.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.2495.14t3.a.c$2$ $ 5 \cdot 499 $ 14.0.120371547919839898421875.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.2495.28t10.a.e$2$ $ 5 \cdot 499 $ 28.2.725917312055025700242151478457681268310546875.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.2495.28t10.a.d$2$ $ 5 \cdot 499 $ 28.2.725917312055025700242151478457681268310546875.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.2495.28t10.a.b$2$ $ 5 \cdot 499 $ 28.2.725917312055025700242151478457681268310546875.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.2495.28t10.a.c$2$ $ 5 \cdot 499 $ 28.2.725917312055025700242151478457681268310546875.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.2495.28t10.a.f$2$ $ 5 \cdot 499 $ 28.2.725917312055025700242151478457681268310546875.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.2495.28t10.a.a$2$ $ 5 \cdot 499 $ 28.2.725917312055025700242151478457681268310546875.1 $D_{28}$ (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 *.