Properties

Label 28.2.295...963.1
Degree $28$
Signature $[2, 13]$
Discriminant $-2.958\times 10^{50}$
Root discriminant $63.47$
Ramified primes $7, 29$
Class number $7$ (GRH)
Class group $[7]$ (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 - 14*x^27 + 80*x^26 - 221*x^25 + 397*x^24 - 1904*x^23 + 10213*x^22 - 26576*x^21 + 32909*x^20 - 65079*x^19 + 350623*x^18 - 1046531*x^17 + 1678487*x^16 - 1771438*x^15 + 2670493*x^14 - 7195176*x^13 + 14367013*x^12 - 15585945*x^11 + 7205726*x^10 - 3208396*x^9 + 37523327*x^8 - 120026000*x^7 + 201372903*x^6 - 215912334*x^5 + 156455683*x^4 - 76691304*x^3 + 25200801*x^2 - 5337738*x + 544563)
 
gp: K = bnfinit(x^28 - 14*x^27 + 80*x^26 - 221*x^25 + 397*x^24 - 1904*x^23 + 10213*x^22 - 26576*x^21 + 32909*x^20 - 65079*x^19 + 350623*x^18 - 1046531*x^17 + 1678487*x^16 - 1771438*x^15 + 2670493*x^14 - 7195176*x^13 + 14367013*x^12 - 15585945*x^11 + 7205726*x^10 - 3208396*x^9 + 37523327*x^8 - 120026000*x^7 + 201372903*x^6 - 215912334*x^5 + 156455683*x^4 - 76691304*x^3 + 25200801*x^2 - 5337738*x + 544563, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![544563, -5337738, 25200801, -76691304, 156455683, -215912334, 201372903, -120026000, 37523327, -3208396, 7205726, -15585945, 14367013, -7195176, 2670493, -1771438, 1678487, -1046531, 350623, -65079, 32909, -26576, 10213, -1904, 397, -221, 80, -14, 1]);
 

\( x^{28} - 14 x^{27} + 80 x^{26} - 221 x^{25} + 397 x^{24} - 1904 x^{23} + 10213 x^{22} - 26576 x^{21} + 32909 x^{20} - 65079 x^{19} + 350623 x^{18} - 1046531 x^{17} + 1678487 x^{16} - 1771438 x^{15} + 2670493 x^{14} - 7195176 x^{13} + 14367013 x^{12} - 15585945 x^{11} + 7205726 x^{10} - 3208396 x^{9} + 37523327 x^{8} - 120026000 x^{7} + 201372903 x^{6} - 215912334 x^{5} + 156455683 x^{4} - 76691304 x^{3} + 25200801 x^{2} - 5337738 x + 544563 \)

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:  \(-295815184798509371659078776791937755272938949172963\)\(\medspace = -\,7^{13}\cdot 29^{27}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $63.47$
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:  $7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{21} a^{10} + \frac{2}{21} a^{9} + \frac{2}{21} a^{8} + \frac{1}{21} a^{7} + \frac{5}{21} a^{6} - \frac{8}{21} a^{5} + \frac{8}{21} a^{4} - \frac{2}{21} a^{3} + \frac{4}{21} a^{2} + \frac{8}{21} a - \frac{1}{7}$, $\frac{1}{21} a^{11} - \frac{2}{21} a^{9} - \frac{1}{7} a^{8} + \frac{1}{7} a^{7} + \frac{1}{7} a^{6} + \frac{1}{7} a^{5} + \frac{1}{7} a^{4} + \frac{8}{21} a^{3} + \frac{2}{21} a + \frac{2}{7}$, $\frac{1}{21} a^{12} + \frac{1}{21} a^{9} - \frac{3}{7} a^{7} + \frac{2}{7} a^{6} - \frac{2}{7} a^{5} - \frac{4}{21} a^{4} + \frac{1}{7} a^{3} + \frac{1}{7} a^{2} + \frac{8}{21} a - \frac{2}{7}$, $\frac{1}{21} a^{13} - \frac{2}{21} a^{9} + \frac{1}{7} a^{8} - \frac{3}{7} a^{7} + \frac{1}{7} a^{6} - \frac{10}{21} a^{5} + \frac{3}{7} a^{4} - \frac{3}{7} a^{3} - \frac{1}{7} a^{2} - \frac{1}{3} a + \frac{1}{7}$, $\frac{1}{63} a^{14} - \frac{1}{63} a^{13} - \frac{1}{63} a^{12} + \frac{1}{63} a^{11} + \frac{2}{21} a^{9} + \frac{10}{63} a^{8} + \frac{26}{63} a^{7} - \frac{3}{7} a^{6} - \frac{1}{7} a^{5} - \frac{16}{63} a^{4} + \frac{1}{9} a^{3} + \frac{1}{63} a^{2} + \frac{20}{63} a - \frac{2}{7}$, $\frac{1}{63} a^{15} + \frac{1}{63} a^{13} + \frac{1}{63} a^{11} - \frac{2}{63} a^{9} - \frac{1}{7} a^{8} + \frac{8}{63} a^{7} + \frac{3}{7} a^{6} - \frac{4}{9} a^{5} - \frac{1}{7} a^{4} - \frac{4}{9} a^{3} + \frac{1}{7} a^{2} - \frac{25}{63} a + \frac{1}{7}$, $\frac{1}{2205} a^{16} - \frac{8}{2205} a^{15} - \frac{4}{2205} a^{14} - \frac{2}{105} a^{13} + \frac{8}{735} a^{12} + \frac{38}{2205} a^{11} - \frac{38}{2205} a^{10} - \frac{206}{2205} a^{9} - \frac{97}{735} a^{8} - \frac{827}{2205} a^{7} + \frac{76}{441} a^{6} - \frac{928}{2205} a^{5} + \frac{94}{315} a^{4} + \frac{62}{245} a^{3} - \frac{6}{245} a^{2} - \frac{206}{2205} a - \frac{103}{245}$, $\frac{1}{2205} a^{17} + \frac{2}{2205} a^{15} - \frac{4}{2205} a^{14} + \frac{1}{735} a^{13} - \frac{10}{441} a^{12} - \frac{2}{315} a^{11} + \frac{1}{147} a^{10} + \frac{2}{35} a^{9} + \frac{34}{441} a^{8} + \frac{1079}{2205} a^{7} + \frac{319}{735} a^{6} + \frac{934}{2205} a^{5} - \frac{338}{2205} a^{4} - \frac{5}{21} a^{3} + \frac{587}{2205} a^{2} + \frac{5}{49} a + \frac{121}{245}$, $\frac{1}{6615} a^{18} + \frac{1}{6615} a^{16} + \frac{4}{6615} a^{15} + \frac{2}{315} a^{14} - \frac{148}{6615} a^{13} + \frac{137}{6615} a^{12} + \frac{4}{2205} a^{11} - \frac{151}{6615} a^{10} - \frac{359}{6615} a^{9} - \frac{139}{1323} a^{8} + \frac{233}{2205} a^{7} - \frac{2491}{6615} a^{6} - \frac{71}{1323} a^{5} - \frac{148}{315} a^{4} - \frac{2666}{6615} a^{3} + \frac{2939}{6615} a^{2} - \frac{1}{21} a - \frac{94}{245}$, $\frac{1}{6615} a^{19} + \frac{1}{6615} a^{17} + \frac{1}{6615} a^{16} - \frac{13}{2205} a^{15} - \frac{31}{6615} a^{14} + \frac{53}{6615} a^{13} + \frac{10}{441} a^{12} + \frac{10}{1323} a^{11} + \frac{2}{189} a^{10} + \frac{22}{135} a^{9} + \frac{349}{2205} a^{8} + \frac{502}{1323} a^{7} + \frac{331}{1323} a^{6} + \frac{137}{2205} a^{5} + \frac{248}{1323} a^{4} - \frac{650}{1323} a^{3} - \frac{331}{2205} a^{2} - \frac{170}{441} a - \frac{37}{245}$, $\frac{1}{19845} a^{20} - \frac{1}{19845} a^{19} - \frac{1}{6615} a^{17} + \frac{4}{19845} a^{16} - \frac{152}{19845} a^{15} - \frac{2}{315} a^{14} - \frac{184}{19845} a^{13} + \frac{16}{6615} a^{12} + \frac{79}{3969} a^{11} + \frac{34}{19845} a^{10} - \frac{436}{3969} a^{9} + \frac{838}{19845} a^{8} + \frac{418}{6615} a^{7} + \frac{2246}{19845} a^{6} + \frac{2402}{19845} a^{5} + \frac{7507}{19845} a^{4} - \frac{2239}{6615} a^{3} + \frac{964}{2835} a^{2} + \frac{94}{2205} a + \frac{12}{49}$, $\frac{1}{19845} a^{21} - \frac{1}{19845} a^{19} + \frac{1}{19845} a^{17} - \frac{1}{19845} a^{16} + \frac{157}{19845} a^{15} - \frac{26}{3969} a^{14} - \frac{13}{19845} a^{13} - \frac{20}{3969} a^{12} - \frac{121}{6615} a^{11} + \frac{62}{2835} a^{10} + \frac{677}{19845} a^{9} + \frac{3148}{19845} a^{8} + \frac{146}{405} a^{7} + \frac{551}{3969} a^{6} + \frac{719}{6615} a^{5} + \frac{5263}{19845} a^{4} - \frac{1777}{3969} a^{3} + \frac{530}{3969} a^{2} + \frac{27}{245} a + \frac{103}{245}$, $\frac{1}{99225} a^{22} - \frac{1}{99225} a^{21} + \frac{1}{99225} a^{20} + \frac{1}{19845} a^{19} - \frac{2}{99225} a^{18} + \frac{16}{99225} a^{17} - \frac{4}{19845} a^{16} - \frac{26}{11025} a^{15} + \frac{184}{33075} a^{14} - \frac{1466}{99225} a^{13} + \frac{1186}{99225} a^{12} + \frac{449}{33075} a^{11} + \frac{2021}{99225} a^{10} - \frac{8}{2025} a^{9} - \frac{458}{11025} a^{8} + \frac{8822}{99225} a^{7} + \frac{3076}{33075} a^{6} + \frac{1493}{3969} a^{5} + \frac{8774}{99225} a^{4} - \frac{257}{11025} a^{3} + \frac{925}{3969} a^{2} - \frac{386}{1575} a + \frac{62}{1225}$, $\frac{1}{99225} a^{23} + \frac{1}{99225} a^{20} - \frac{1}{14175} a^{19} - \frac{1}{99225} a^{18} - \frac{4}{99225} a^{17} + \frac{11}{99225} a^{16} + \frac{94}{14175} a^{15} - \frac{134}{99225} a^{14} - \frac{7}{405} a^{13} - \frac{752}{99225} a^{12} + \frac{1408}{99225} a^{11} + \frac{22}{14175} a^{10} - \frac{6289}{99225} a^{9} - \frac{113}{2205} a^{8} - \frac{4247}{19845} a^{7} - \frac{26162}{99225} a^{6} - \frac{1832}{33075} a^{5} + \frac{1397}{33075} a^{4} - \frac{44918}{99225} a^{3} + \frac{29117}{99225} a^{2} - \frac{4874}{11025} a - \frac{388}{1225}$, $\frac{1}{106269975} a^{24} - \frac{4}{35423325} a^{23} - \frac{166}{106269975} a^{22} + \frac{2332}{106269975} a^{21} - \frac{76}{7084665} a^{20} + \frac{6563}{106269975} a^{19} - \frac{157}{21253995} a^{18} - \frac{1693}{11807775} a^{17} + \frac{9}{145775} a^{16} - \frac{408196}{106269975} a^{15} + \frac{811061}{106269975} a^{14} - \frac{437011}{106269975} a^{13} + \frac{768511}{106269975} a^{12} - \frac{11227}{15181425} a^{11} - \frac{162541}{35423325} a^{10} + \frac{336512}{4250799} a^{9} + \frac{4243084}{35423325} a^{8} - \frac{82466}{562275} a^{7} + \frac{372584}{787185} a^{6} - \frac{2186651}{6251175} a^{5} - \frac{2314211}{11807775} a^{4} + \frac{30960841}{106269975} a^{3} - \frac{44986}{112455} a^{2} + \frac{209887}{1311975} a - \frac{18974}{145775}$, $\frac{1}{743889825} a^{25} - \frac{2}{743889825} a^{24} + \frac{1856}{743889825} a^{23} - \frac{2}{1012095} a^{22} + \frac{13612}{743889825} a^{21} - \frac{208}{15181425} a^{20} + \frac{779}{82654425} a^{19} + \frac{1090}{29755593} a^{18} - \frac{253}{4862025} a^{17} - \frac{94114}{743889825} a^{16} - \frac{262408}{148777965} a^{15} + \frac{1813087}{743889825} a^{14} - \frac{553649}{247963275} a^{13} - \frac{292829}{16530885} a^{12} + \frac{10172264}{743889825} a^{11} - \frac{5196364}{743889825} a^{10} - \frac{122541382}{743889825} a^{9} - \frac{4638496}{49592655} a^{8} + \frac{26971604}{82654425} a^{7} + \frac{274834613}{743889825} a^{6} - \frac{146107546}{743889825} a^{5} - \frac{5682547}{148777965} a^{4} - \frac{47759113}{148777965} a^{3} + \frac{28240189}{82654425} a^{2} - \frac{3264937}{9183825} a + \frac{51302}{204085}$, $\frac{1}{444288725390527299426348225} a^{26} - \frac{13}{444288725390527299426348225} a^{25} + \frac{14929591166253548}{3291027595485387403158135} a^{24} - \frac{4837187537866149422}{88857745078105459885269645} a^{23} - \frac{1424962676308954556}{2454633841936614913957725} a^{22} + \frac{428549006788656483554}{49365413932280811047372025} a^{21} - \frac{4519612930980753080758}{444288725390527299426348225} a^{20} - \frac{1190874478291723993592}{17771549015621091977053929} a^{19} - \frac{12637344902030300735327}{444288725390527299426348225} a^{18} - \frac{11436094070821868902652}{88857745078105459885269645} a^{17} - \frac{9613232122789449724951}{49365413932280811047372025} a^{16} + \frac{1928884623113938311527969}{444288725390527299426348225} a^{15} + \frac{894390855275860368662986}{444288725390527299426348225} a^{14} + \frac{362957771903177040569473}{16455137977426937015790675} a^{13} - \frac{1082615351607159116520134}{88857745078105459885269645} a^{12} + \frac{1003682099084240376214066}{63469817912932471346621175} a^{11} - \frac{823165327470834689084524}{88857745078105459885269645} a^{10} - \frac{3844990545137738083269848}{26134630905325135260373425} a^{9} + \frac{4327098881031525015437164}{148096241796842433142116075} a^{8} - \frac{3033684371714490356905663}{444288725390527299426348225} a^{7} - \frac{147753220879114397096222252}{444288725390527299426348225} a^{6} - \frac{26202055523342651437050316}{148096241796842433142116075} a^{5} + \frac{53868902129361448885785893}{444288725390527299426348225} a^{4} + \frac{2757307838320837965014458}{63469817912932471346621175} a^{3} + \frac{3177156674015960535679489}{7052201990325830149624575} a^{2} - \frac{1272086132929636374643721}{5485045992475645671930225} a - \frac{193332964091403385094826}{609449554719516185770025}$, $\frac{1}{12225492856571139698314824107325} a^{27} + \frac{2749}{2445098571314227939662964821465} a^{26} - \frac{1642364868824682848261}{4075164285523713232771608035775} a^{25} + \frac{3533105636869562842271}{1358388095174571077590536011925} a^{24} + \frac{13020465448257778136217598}{12225492856571139698314824107325} a^{23} - \frac{19247602425155918307387877}{12225492856571139698314824107325} a^{22} + \frac{57621620900428029924574946}{4075164285523713232771608035775} a^{21} - \frac{278675052925011521893351}{13508831885713966517474943765} a^{20} - \frac{18103439270698948299637304}{452796031724857025863512003975} a^{19} - \frac{78813956366367572557945942}{1358388095174571077590536011925} a^{18} + \frac{700105620071893632025134757}{4075164285523713232771608035775} a^{17} + \frac{826794563041247787095819}{13131571274512502361240412575} a^{16} - \frac{31677702668318389338773876204}{4075164285523713232771608035775} a^{15} + \frac{73339012450063484717084552339}{12225492856571139698314824107325} a^{14} + \frac{10799170935428610825526082839}{12225492856571139698314824107325} a^{13} + \frac{163456612000985413318118332}{260116869288747653155634555475} a^{12} - \frac{25643936923199007905512015109}{4075164285523713232771608035775} a^{11} - \frac{496256221091575093596383851}{1746498979510162814044974872475} a^{10} + \frac{968247976242895252470974369}{7862053283968578584125288815} a^{9} + \frac{1704625582763321274733874959163}{12225492856571139698314824107325} a^{8} + \frac{659254194517136852085636820369}{1746498979510162814044974872475} a^{7} - \frac{551606572118155316990969295611}{4075164285523713232771608035775} a^{6} + \frac{295326609152010477372270587573}{719146638621831746959695535725} a^{5} + \frac{436447747898837174994034202843}{2445098571314227939662964821465} a^{4} - \frac{214124629504963747745523656096}{1746498979510162814044974872475} a^{3} + \frac{199129207888007937190047152963}{452796031724857025863512003975} a^{2} + \frac{841953186457373010238115365}{2012426807666031226060053351} a - \frac{3340135953342914918791303414}{16770223397216926883833777925}$

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

Class group and class number

$C_{7}$, which has order $7$ (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:  \( 140953741391215.78 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{13}\cdot 140953741391215.78 \cdot 7}{2\sqrt{295815184798509371659078776791937755272938949172963}}\approx 2.72918327152723$ (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

\(\Q(\sqrt{29}) \), 4.2.170723.1, 7.1.204024399103.1, 14.2.1207152707450866588933661.1

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 $28$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ R $28$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{14}$ $28$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{7}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.29.2t1.a.a$1$ $ 29 $ \(\Q(\sqrt{29}) \) $C_2$ (as 2T1) $1$ $1$
1.7.2t1.a.a$1$ $ 7 $ \(\Q(\sqrt{-7}) \) $C_2$ (as 2T1) $1$ $-1$
1.203.2t1.a.a$1$ $ 7 \cdot 29 $ \(\Q(\sqrt{-203}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.5887.4t3.c.a$2$ $ 7 \cdot 29^{2}$ 4.2.170723.1 $D_{4}$ (as 4T3) $1$ $0$
* 2.5887.14t3.b.b$2$ $ 7 \cdot 29^{2}$ 14.2.1207152707450866588933661.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.5887.14t3.b.c$2$ $ 7 \cdot 29^{2}$ 14.2.1207152707450866588933661.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.5887.7t2.a.a$2$ $ 7 \cdot 29^{2}$ 7.1.204024399103.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.5887.14t3.b.a$2$ $ 7 \cdot 29^{2}$ 14.2.1207152707450866588933661.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.5887.7t2.a.b$2$ $ 7 \cdot 29^{2}$ 7.1.204024399103.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.5887.7t2.a.c$2$ $ 7 \cdot 29^{2}$ 7.1.204024399103.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.5887.28t10.a.f$2$ $ 7 \cdot 29^{2}$ 28.2.295815184798509371659078776791937755272938949172963.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.5887.28t10.a.b$2$ $ 7 \cdot 29^{2}$ 28.2.295815184798509371659078776791937755272938949172963.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.5887.28t10.a.a$2$ $ 7 \cdot 29^{2}$ 28.2.295815184798509371659078776791937755272938949172963.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.5887.28t10.a.d$2$ $ 7 \cdot 29^{2}$ 28.2.295815184798509371659078776791937755272938949172963.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.5887.28t10.a.c$2$ $ 7 \cdot 29^{2}$ 28.2.295815184798509371659078776791937755272938949172963.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.5887.28t10.a.e$2$ $ 7 \cdot 29^{2}$ 28.2.295815184798509371659078776791937755272938949172963.1 $D_{28}$ (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 *.