Properties

Label 26.2.214...688.1
Degree $26$
Signature $[2, 12]$
Discriminant $2.146\times 10^{50}$
Root discriminant $86.26$
Ramified primes $2, 53$
Class number $13$ (GRH)
Class group $[13]$ (GRH)
Galois group $D_{26}$ (as 26T3)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 13*x^25 + 121*x^24 - 802*x^23 + 4651*x^22 - 22825*x^21 + 98893*x^20 - 371962*x^19 + 1240872*x^18 - 3661594*x^17 + 9598744*x^16 - 22249906*x^15 + 46158003*x^14 - 86354433*x^13 + 145428273*x^12 - 217463318*x^11 + 270662252*x^10 - 252429562*x^9 + 127602004*x^8 + 63980066*x^7 - 189163921*x^6 + 167659735*x^5 - 112782103*x^4 + 89161818*x^3 - 55238589*x^2 + 18143595*x - 204972201)
 
gp: K = bnfinit(x^26 - 13*x^25 + 121*x^24 - 802*x^23 + 4651*x^22 - 22825*x^21 + 98893*x^20 - 371962*x^19 + 1240872*x^18 - 3661594*x^17 + 9598744*x^16 - 22249906*x^15 + 46158003*x^14 - 86354433*x^13 + 145428273*x^12 - 217463318*x^11 + 270662252*x^10 - 252429562*x^9 + 127602004*x^8 + 63980066*x^7 - 189163921*x^6 + 167659735*x^5 - 112782103*x^4 + 89161818*x^3 - 55238589*x^2 + 18143595*x - 204972201, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-204972201, 18143595, -55238589, 89161818, -112782103, 167659735, -189163921, 63980066, 127602004, -252429562, 270662252, -217463318, 145428273, -86354433, 46158003, -22249906, 9598744, -3661594, 1240872, -371962, 98893, -22825, 4651, -802, 121, -13, 1]);
 

\( x^{26} - 13 x^{25} + 121 x^{24} - 802 x^{23} + 4651 x^{22} - 22825 x^{21} + 98893 x^{20} - 371962 x^{19} + 1240872 x^{18} - 3661594 x^{17} + 9598744 x^{16} - 22249906 x^{15} + 46158003 x^{14} - 86354433 x^{13} + 145428273 x^{12} - 217463318 x^{11} + 270662252 x^{10} - 252429562 x^{9} + 127602004 x^{8} + 63980066 x^{7} - 189163921 x^{6} + 167659735 x^{5} - 112782103 x^{4} + 89161818 x^{3} - 55238589 x^{2} + 18143595 x - 204972201 \)

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

Invariants

Degree:  $26$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[2, 12]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(214593535987174144573161771120178694372847609970688\)\(\medspace = 2^{24}\cdot 53^{25}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $86.26$
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:  $2, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a + \frac{1}{8}$, $\frac{1}{24} a^{9} + \frac{1}{8} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{24} a - \frac{1}{4}$, $\frac{1}{48} a^{10} - \frac{1}{48} a^{9} - \frac{1}{16} a^{8} - \frac{1}{8} a^{7} + \frac{1}{16} a^{6} + \frac{1}{16} a^{5} + \frac{1}{16} a^{4} + \frac{1}{4} a^{3} + \frac{23}{48} a^{2} - \frac{23}{48} a + \frac{3}{16}$, $\frac{1}{48} a^{11} - \frac{1}{16} a^{8} - \frac{1}{16} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{16} a^{4} + \frac{11}{48} a^{3} - \frac{1}{2} a^{2} + \frac{1}{8} a - \frac{7}{16}$, $\frac{1}{48} a^{12} - \frac{1}{48} a^{9} - \frac{1}{16} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{16} a^{5} - \frac{1}{48} a^{4} - \frac{1}{4} a^{3} + \frac{3}{8} a^{2} - \frac{23}{48} a - \frac{1}{4}$, $\frac{1}{48} a^{13} - \frac{1}{16} a^{8} - \frac{1}{8} a^{6} + \frac{1}{24} a^{5} + \frac{3}{16} a^{4} - \frac{1}{8} a^{3} + \frac{1}{4} a^{2} - \frac{5}{16} a - \frac{3}{16}$, $\frac{1}{288} a^{14} - \frac{1}{288} a^{13} - \frac{1}{144} a^{12} + \frac{1}{288} a^{11} + \frac{1}{288} a^{10} - \frac{1}{32} a^{8} + \frac{11}{96} a^{7} + \frac{5}{288} a^{6} + \frac{13}{72} a^{5} - \frac{1}{288} a^{4} - \frac{43}{288} a^{3} - \frac{5}{36} a^{2} - \frac{41}{96} a + \frac{5}{32}$, $\frac{1}{288} a^{15} - \frac{1}{96} a^{13} - \frac{1}{288} a^{12} + \frac{1}{144} a^{11} + \frac{1}{288} a^{10} + \frac{1}{96} a^{9} - \frac{1}{24} a^{8} - \frac{17}{144} a^{7} - \frac{5}{96} a^{6} + \frac{5}{96} a^{5} + \frac{2}{9} a^{4} - \frac{83}{288} a^{3} + \frac{125}{288} a^{2} - \frac{5}{16} a - \frac{7}{32}$, $\frac{1}{1152} a^{16} - \frac{1}{288} a^{13} - \frac{5}{576} a^{12} + \frac{1}{288} a^{11} - \frac{1}{48} a^{9} + \frac{11}{1152} a^{8} + \frac{7}{96} a^{7} - \frac{1}{48} a^{6} - \frac{13}{144} a^{5} + \frac{77}{576} a^{4} - \frac{91}{288} a^{3} - \frac{5}{24} a^{2} - \frac{47}{96} a - \frac{11}{128}$, $\frac{1}{1152} a^{17} + \frac{5}{576} a^{13} - \frac{1}{288} a^{12} + \frac{1}{288} a^{11} + \frac{1}{288} a^{10} - \frac{13}{1152} a^{9} + \frac{1}{24} a^{8} - \frac{1}{32} a^{7} + \frac{11}{96} a^{6} - \frac{47}{576} a^{5} + \frac{1}{18} a^{4} - \frac{67}{288} a^{3} + \frac{29}{288} a^{2} + \frac{25}{128} a - \frac{15}{32}$, $\frac{1}{4608} a^{18} - \frac{1}{4608} a^{17} - \frac{1}{4608} a^{16} + \frac{1}{1152} a^{15} - \frac{1}{768} a^{14} - \frac{5}{768} a^{13} + \frac{23}{2304} a^{12} - \frac{1}{128} a^{11} + \frac{43}{4608} a^{10} - \frac{23}{4608} a^{9} - \frac{215}{4608} a^{8} - \frac{91}{1152} a^{7} - \frac{7}{256} a^{6} + \frac{45}{256} a^{5} + \frac{289}{2304} a^{4} - \frac{7}{192} a^{3} - \frac{887}{4608} a^{2} - \frac{571}{1536} a + \frac{35}{512}$, $\frac{1}{13824} a^{19} + \frac{1}{13824} a^{18} - \frac{1}{4608} a^{17} - \frac{1}{2304} a^{16} - \frac{7}{6912} a^{15} + \frac{1}{2304} a^{14} + \frac{19}{2304} a^{13} - \frac{5}{1728} a^{12} - \frac{5}{1536} a^{11} + \frac{95}{13824} a^{10} - \frac{7}{4608} a^{9} + \frac{5}{2304} a^{8} - \frac{563}{6912} a^{7} - \frac{179}{2304} a^{6} + \frac{97}{2304} a^{5} - \frac{377}{3456} a^{4} - \frac{3079}{13824} a^{3} + \frac{283}{4608} a^{2} + \frac{171}{512} a - \frac{59}{256}$, $\frac{1}{41969664} a^{20} - \frac{5}{20984832} a^{19} - \frac{265}{3815424} a^{18} - \frac{413}{1748736} a^{17} + \frac{8617}{41969664} a^{16} + \frac{14323}{10492416} a^{15} - \frac{2377}{2331648} a^{14} + \frac{30181}{5246208} a^{13} + \frac{365573}{41969664} a^{12} - \frac{182135}{20984832} a^{11} - \frac{278977}{41969664} a^{10} - \frac{3833}{874368} a^{9} + \frac{1576643}{41969664} a^{8} + \frac{713495}{10492416} a^{7} + \frac{824467}{6994944} a^{6} - \frac{608137}{5246208} a^{5} + \frac{3136985}{13989888} a^{4} - \frac{4470725}{20984832} a^{3} - \frac{6517861}{13989888} a^{2} - \frac{127141}{582912} a + \frac{52015}{518144}$, $\frac{1}{41969664} a^{21} + \frac{7}{13989888} a^{19} + \frac{203}{20984832} a^{18} + \frac{9685}{41969664} a^{17} + \frac{725}{1907712} a^{16} + \frac{2293}{1907712} a^{15} + \frac{12599}{10492416} a^{14} - \frac{152723}{41969664} a^{13} + \frac{82081}{10492416} a^{12} + \frac{313543}{41969664} a^{11} + \frac{96397}{20984832} a^{10} + \frac{39829}{3815424} a^{9} + \frac{885305}{20984832} a^{8} + \frac{169777}{20984832} a^{7} - \frac{539387}{10492416} a^{6} + \frac{5152387}{41969664} a^{5} - \frac{454843}{10492416} a^{4} + \frac{12402089}{41969664} a^{3} + \frac{1295389}{6994944} a^{2} - \frac{1575773}{4663296} a - \frac{48585}{259072}$, $\frac{1}{671514624} a^{22} + \frac{5}{671514624} a^{21} + \frac{1}{167878656} a^{20} + \frac{227}{223838208} a^{19} - \frac{2597}{30523392} a^{18} + \frac{132691}{671514624} a^{17} - \frac{38827}{223838208} a^{16} + \frac{229027}{335757312} a^{15} - \frac{65965}{671514624} a^{14} + \frac{1489589}{671514624} a^{13} + \frac{2724775}{335757312} a^{12} + \frac{74299}{20348928} a^{11} - \frac{784889}{335757312} a^{10} + \frac{12216737}{671514624} a^{9} + \frac{4442173}{74612736} a^{8} + \frac{25987817}{335757312} a^{7} + \frac{15963631}{223838208} a^{6} - \frac{38514109}{671514624} a^{5} - \frac{18040777}{111919104} a^{4} - \frac{108400873}{223838208} a^{3} - \frac{317923}{1695744} a^{2} + \frac{2494641}{8290304} a + \frac{246481}{753664}$, $\frac{1}{2014543872} a^{23} - \frac{1}{2014543872} a^{22} - \frac{5}{1007271936} a^{21} + \frac{1}{2014543872} a^{20} - \frac{503}{18653184} a^{19} + \frac{3019}{29196288} a^{18} + \frac{352669}{2014543872} a^{17} - \frac{76489}{503635968} a^{16} + \frac{2491}{7962624} a^{15} - \frac{426461}{2014543872} a^{14} - \frac{571937}{62954496} a^{13} - \frac{579113}{61046784} a^{12} + \frac{3733487}{503635968} a^{11} + \frac{6045631}{671514624} a^{10} + \frac{40317455}{2014543872} a^{9} - \frac{1865701}{45785088} a^{8} + \frac{112887361}{2014543872} a^{7} - \frac{83316473}{671514624} a^{6} + \frac{34957405}{251817984} a^{5} + \frac{17432345}{2014543872} a^{4} + \frac{3603383}{1007271936} a^{3} + \frac{33306403}{671514624} a^{2} + \frac{103905773}{223838208} a - \frac{1665139}{4145152}$, $\frac{1}{2513036488746776505216270336} a^{24} - \frac{1}{209419707395564708768022528} a^{23} - \frac{62626808931384965}{157064780546673531576016896} a^{22} + \frac{55668274605675527}{12692103478519073258668032} a^{21} - \frac{1113129661552425443}{2513036488746776505216270336} a^{20} - \frac{62441641066967858537}{418839414791129417536045056} a^{19} + \frac{83357969044588527146471}{1256518244373388252608135168} a^{18} + \frac{115648052617587086754631}{418839414791129417536045056} a^{17} - \frac{62607849724336315896263}{209419707395564708768022528} a^{16} - \frac{21539201541445405093015}{13657807004058567963131904} a^{15} + \frac{195045415844677448687501}{209419707395564708768022528} a^{14} + \frac{133740133962185114341195}{66132539177546750137270272} a^{13} + \frac{1134006746844635199744587}{132265078355093500274540544} a^{12} - \frac{12915410416508051164337945}{1256518244373388252608135168} a^{11} - \frac{3107000773230910761153407}{1256518244373388252608135168} a^{10} + \frac{623206484626389943264967}{139613138263709805845348352} a^{9} + \frac{43364093958543273847675}{701182056011935408821504} a^{8} - \frac{257582918293013300962103}{13657807004058567963131904} a^{7} + \frac{20527088615153582736012905}{628259122186694126304067584} a^{6} - \frac{2217834017608804906209203}{114228931306671659328012288} a^{5} + \frac{28167110632783111472949773}{119668404226036976438870016} a^{4} - \frac{582513516904196252423925355}{1256518244373388252608135168} a^{3} + \frac{59009885218690415280970313}{418839414791129417536045056} a^{2} + \frac{48821684100524824994197915}{139613138263709805845348352} a - \frac{486703238327787710272817}{1477387706494283659739136}$, $\frac{1}{1490901618569333856920141053427712} a^{25} + \frac{296621}{1490901618569333856920141053427712} a^{24} + \frac{6114414250008877750987}{53246486377476209175719323336704} a^{23} - \frac{22934758957887661200557}{745450809284666928460070526713856} a^{22} + \frac{300220227696604086494245}{212985945509904836702877293346816} a^{21} + \frac{36513071021702514765791}{1490901618569333856920141053427712} a^{20} - \frac{8286432671377977875596728839}{372725404642333464230035263356928} a^{19} - \frac{5328210364256281089069569179}{93181351160583366057508815839232} a^{18} + \frac{101388322397123755699240284775}{248483603094888976153356842237952} a^{17} + \frac{8176472518686054044636071105}{186362702321166732115017631678464} a^{16} + \frac{260466090504523085099503219171}{372725404642333464230035263356928} a^{15} + \frac{1076771298402241851242549895985}{745450809284666928460070526713856} a^{14} + \frac{27446587412467343346355190279}{26156168746830418542458614972416} a^{13} + \frac{4530512589086438878024917794597}{496967206189777952306713684475904} a^{12} - \frac{45097738812103514455063372643}{11294709231585862552425311010816} a^{11} - \frac{35015619596676896790936945541}{26623243188738104587859661668352} a^{10} - \frac{8121454266611009219111569171955}{745450809284666928460070526713856} a^{9} - \frac{194079368230481790330301755941}{33884127694757587657275933032448} a^{8} + \frac{6182020082148516352176763408711}{53246486377476209175719323336704} a^{7} + \frac{2590954682904466120185785315021}{67768255389515175314551866064896} a^{6} - \frac{17367876423665074917906081426887}{135536510779030350629103732129792} a^{5} + \frac{57794791864912852985541836999995}{1490901618569333856920141053427712} a^{4} - \frac{6632753170667524601109766186195}{19617126560122813906843961229312} a^{3} - \frac{278947765563449382983059992563}{1613529890226551793203615858688} a^{2} - \frac{65482006253912739544848048603227}{165655735396592650768904561491968} a + \frac{355502312539302504869988563963}{876485372468744183962457997312}$

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

Class group and class number

$C_{13}$, which has order $13$ (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:  \( 1441989777359363.2 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{12}\cdot 1441989777359363.2 \cdot 13}{2\sqrt{214593535987174144573161771120178694372847609970688}}\approx 9.68914893995216$ (assuming GRH)

Galois group

$D_{26}$ (as 26T3):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 52
The 16 conjugacy class representatives for $D_{26}$
Character table for $D_{26}$

Intermediate fields

\(\Q(\sqrt{53}) \), 13.1.2012196471835550329409536.1

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

Sibling fields

Degree 26 sibling: 26.0.858374143948696578292647084480714777491390439882752.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/LocalNumberField/3.2.0.1}{2} }^{13}$ $26$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/17.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/29.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/37.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/59.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
53Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.212.2t1.a.a$1$ $ 2^{2} \cdot 53 $ \(\Q(\sqrt{-53}) \) $C_2$ (as 2T1) $1$ $-1$
1.4.2t1.a.a$1$ $ 2^{2}$ \(\Q(\sqrt{-1}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.53.2t1.a.a$1$ $ 53 $ \(\Q(\sqrt{53}) \) $C_2$ (as 2T1) $1$ $1$
* 2.11236.26t3.b.e$2$ $ 2^{2} \cdot 53^{2}$ 26.2.214593535987174144573161771120178694372847609970688.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.11236.13t2.a.c$2$ $ 2^{2} \cdot 53^{2}$ 13.1.2012196471835550329409536.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.11236.13t2.a.b$2$ $ 2^{2} \cdot 53^{2}$ 13.1.2012196471835550329409536.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.11236.26t3.b.d$2$ $ 2^{2} \cdot 53^{2}$ 26.2.214593535987174144573161771120178694372847609970688.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.11236.13t2.a.a$2$ $ 2^{2} \cdot 53^{2}$ 13.1.2012196471835550329409536.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.11236.26t3.b.b$2$ $ 2^{2} \cdot 53^{2}$ 26.2.214593535987174144573161771120178694372847609970688.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.11236.13t2.a.f$2$ $ 2^{2} \cdot 53^{2}$ 13.1.2012196471835550329409536.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.11236.13t2.a.d$2$ $ 2^{2} \cdot 53^{2}$ 13.1.2012196471835550329409536.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.11236.13t2.a.e$2$ $ 2^{2} \cdot 53^{2}$ 13.1.2012196471835550329409536.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.11236.26t3.b.c$2$ $ 2^{2} \cdot 53^{2}$ 26.2.214593535987174144573161771120178694372847609970688.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.11236.26t3.b.f$2$ $ 2^{2} \cdot 53^{2}$ 26.2.214593535987174144573161771120178694372847609970688.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.11236.26t3.b.a$2$ $ 2^{2} \cdot 53^{2}$ 26.2.214593535987174144573161771120178694372847609970688.1 $D_{26}$ (as 26T3) $1$ $0$

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 *.