Properties

Label 26.0.445...927.1
Degree $26$
Signature $[0, 13]$
Discriminant $-4.459\times 10^{45}$
Root discriminant $56.98$
Ramified primes $17, 191$
Class number not computed
Class group not computed
Galois group $D_{26}$ (as 26T3)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 2*x^25 + 17*x^24 - 98*x^23 + 152*x^22 - 696*x^21 + 1597*x^20 + 1896*x^19 + 20067*x^18 + 156392*x^17 + 357813*x^16 - 241572*x^15 - 6441785*x^14 - 21049242*x^13 - 6337440*x^12 + 144534568*x^11 + 455484358*x^10 + 307770160*x^9 - 848362260*x^8 - 452511774*x^7 + 6860115833*x^6 + 21793527372*x^5 + 15854969370*x^4 - 22167126630*x^3 - 34371344111*x^2 - 12604846614*x + 29070259637)
 
gp: K = bnfinit(x^26 - 2*x^25 + 17*x^24 - 98*x^23 + 152*x^22 - 696*x^21 + 1597*x^20 + 1896*x^19 + 20067*x^18 + 156392*x^17 + 357813*x^16 - 241572*x^15 - 6441785*x^14 - 21049242*x^13 - 6337440*x^12 + 144534568*x^11 + 455484358*x^10 + 307770160*x^9 - 848362260*x^8 - 452511774*x^7 + 6860115833*x^6 + 21793527372*x^5 + 15854969370*x^4 - 22167126630*x^3 - 34371344111*x^2 - 12604846614*x + 29070259637, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![29070259637, -12604846614, -34371344111, -22167126630, 15854969370, 21793527372, 6860115833, -452511774, -848362260, 307770160, 455484358, 144534568, -6337440, -21049242, -6441785, -241572, 357813, 156392, 20067, 1896, 1597, -696, 152, -98, 17, -2, 1]);
 

\( x^{26} - 2 x^{25} + 17 x^{24} - 98 x^{23} + 152 x^{22} - 696 x^{21} + 1597 x^{20} + 1896 x^{19} + 20067 x^{18} + 156392 x^{17} + 357813 x^{16} - 241572 x^{15} - 6441785 x^{14} - 21049242 x^{13} - 6337440 x^{12} + 144534568 x^{11} + 455484358 x^{10} + 307770160 x^{9} - 848362260 x^{8} - 452511774 x^{7} + 6860115833 x^{6} + 21793527372 x^{5} + 15854969370 x^{4} - 22167126630 x^{3} - 34371344111 x^{2} - 12604846614 x + 29070259637 \)

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

Invariants

Degree:  $26$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 13]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-4459331436230036418749915076130198106842162927\)\(\medspace = -\,17^{13}\cdot 191^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $56.98$
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:  $17, 191$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} + \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{15} - \frac{1}{2} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{16} - \frac{1}{2} a^{12} - \frac{1}{4} a^{11} + \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{28} a^{17} - \frac{1}{28} a^{16} + \frac{1}{14} a^{15} + \frac{1}{28} a^{14} + \frac{3}{14} a^{12} - \frac{11}{28} a^{11} - \frac{1}{4} a^{10} - \frac{1}{28} a^{9} + \frac{5}{28} a^{8} - \frac{2}{7} a^{7} + \frac{3}{14} a^{6} + \frac{5}{28} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{14} a^{2} - \frac{9}{28} a - \frac{1}{28}$, $\frac{1}{28} a^{18} + \frac{1}{28} a^{16} + \frac{3}{28} a^{15} + \frac{1}{28} a^{14} - \frac{1}{28} a^{13} + \frac{1}{14} a^{12} + \frac{5}{14} a^{11} - \frac{1}{28} a^{10} - \frac{3}{28} a^{9} + \frac{1}{7} a^{8} + \frac{5}{28} a^{7} - \frac{3}{28} a^{6} + \frac{3}{7} a^{5} + \frac{3}{7} a^{3} - \frac{11}{28} a^{2} + \frac{11}{28} a - \frac{2}{7}$, $\frac{1}{28} a^{19} - \frac{3}{28} a^{16} - \frac{1}{28} a^{15} - \frac{1}{14} a^{14} + \frac{1}{14} a^{13} - \frac{5}{14} a^{12} - \frac{11}{28} a^{11} - \frac{3}{28} a^{10} - \frac{9}{28} a^{9} - \frac{1}{4} a^{8} - \frac{9}{28} a^{7} - \frac{1}{28} a^{6} + \frac{1}{14} a^{5} - \frac{9}{28} a^{4} + \frac{5}{14} a^{3} - \frac{1}{28} a^{2} + \frac{2}{7} a - \frac{13}{28}$, $\frac{1}{28} a^{20} + \frac{3}{28} a^{16} - \frac{3}{28} a^{15} - \frac{1}{14} a^{14} - \frac{3}{28} a^{13} + \frac{1}{4} a^{12} - \frac{2}{7} a^{11} + \frac{5}{28} a^{10} - \frac{5}{14} a^{9} - \frac{2}{7} a^{8} + \frac{5}{14} a^{7} + \frac{13}{28} a^{6} - \frac{2}{7} a^{5} + \frac{3}{28} a^{4} - \frac{2}{7} a^{3} - \frac{5}{28} a^{2} - \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{28} a^{21} - \frac{1}{28} a^{15} + \frac{1}{28} a^{14} - \frac{3}{7} a^{12} + \frac{3}{28} a^{11} + \frac{11}{28} a^{10} + \frac{9}{28} a^{9} - \frac{3}{7} a^{8} - \frac{3}{7} a^{7} - \frac{3}{7} a^{6} - \frac{5}{28} a^{5} - \frac{1}{28} a^{4} + \frac{1}{14} a^{3} - \frac{13}{28} a^{2} - \frac{1}{7} a + \frac{5}{14}$, $\frac{1}{532} a^{22} - \frac{1}{76} a^{21} + \frac{9}{532} a^{20} + \frac{2}{133} a^{19} - \frac{1}{76} a^{18} + \frac{3}{532} a^{17} + \frac{12}{133} a^{16} + \frac{1}{76} a^{15} - \frac{17}{532} a^{14} + \frac{13}{266} a^{13} - \frac{47}{266} a^{12} + \frac{29}{76} a^{11} - \frac{17}{133} a^{10} - \frac{127}{266} a^{9} + \frac{85}{532} a^{8} - \frac{1}{133} a^{7} + \frac{31}{532} a^{6} - \frac{13}{38} a^{5} + \frac{101}{266} a^{4} - \frac{31}{133} a^{3} - \frac{15}{76} a^{2} - \frac{41}{266} a - \frac{25}{266}$, $\frac{1}{3724} a^{23} + \frac{3}{3724} a^{22} - \frac{61}{3724} a^{21} - \frac{27}{1862} a^{20} + \frac{27}{1862} a^{19} + \frac{1}{133} a^{18} - \frac{55}{3724} a^{17} + \frac{449}{3724} a^{16} + \frac{207}{1862} a^{15} - \frac{239}{3724} a^{14} + \frac{223}{3724} a^{13} - \frac{5}{38} a^{12} + \frac{639}{1862} a^{11} + \frac{1327}{3724} a^{10} - \frac{1847}{3724} a^{9} + \frac{979}{3724} a^{8} - \frac{617}{3724} a^{7} + \frac{59}{532} a^{6} + \frac{157}{532} a^{5} - \frac{591}{1862} a^{4} + \frac{555}{3724} a^{3} - \frac{1531}{3724} a^{2} - \frac{15}{3724} a + \frac{160}{931}$, $\frac{1}{1269884} a^{24} + \frac{3}{90706} a^{23} - \frac{3}{4123} a^{22} + \frac{130}{16709} a^{21} - \frac{11155}{634942} a^{20} + \frac{5067}{1269884} a^{19} + \frac{3483}{634942} a^{18} - \frac{45}{33418} a^{17} + \frac{11175}{317471} a^{16} + \frac{28496}{317471} a^{15} + \frac{950}{16709} a^{14} + \frac{30827}{317471} a^{13} + \frac{11475}{634942} a^{12} + \frac{70355}{317471} a^{11} - \frac{7951}{33418} a^{10} - \frac{74169}{1269884} a^{9} + \frac{138993}{634942} a^{8} + \frac{83180}{317471} a^{7} - \frac{59965}{181412} a^{6} + \frac{31125}{66836} a^{5} + \frac{543325}{1269884} a^{4} + \frac{133061}{634942} a^{3} - \frac{417}{5852} a^{2} + \frac{196125}{1269884} a + \frac{351265}{1269884}$, $\frac{1}{28968642326862789116880046079963666853809705012707038429502596479631703072901655051310975391309821130985304256444055721882996} a^{25} - \frac{2491722710280256002544074309615497265782782046416095453974224791627539949385322113171444112555307810692765503251529899}{28968642326862789116880046079963666853809705012707038429502596479631703072901655051310975391309821130985304256444055721882996} a^{24} - \frac{504091359578123019661633215622760803922234551623782368069779301168576813582066599790653187759930291274108464678785026267}{28968642326862789116880046079963666853809705012707038429502596479631703072901655051310975391309821130985304256444055721882996} a^{23} - \frac{1339586854104363768273122445308713914271986661127836338089913515064127427584181446886463175572561887383967511022042810243}{28968642326862789116880046079963666853809705012707038429502596479631703072901655051310975391309821130985304256444055721882996} a^{22} - \frac{472363477979597840053149114155441308873989212565548767172843413222764223558737230565545789823525674979963432978437642947577}{28968642326862789116880046079963666853809705012707038429502596479631703072901655051310975391309821130985304256444055721882996} a^{21} - \frac{55926180572127815192188222042748492868134392904377294996349504512421345682460434831129608426243473152424330619113343759680}{7242160581715697279220011519990916713452426253176759607375649119907925768225413762827743847827455282746326064111013930470749} a^{20} - \frac{258276757545953246671565253493785317775918873786970361071737147639198774140501622100569719566848552283495222897141189823809}{28968642326862789116880046079963666853809705012707038429502596479631703072901655051310975391309821130985304256444055721882996} a^{19} - \frac{70027965308874236462178076268606576167164290310491428876682159464047779371192588275029199053833176544168215150664779002457}{28968642326862789116880046079963666853809705012707038429502596479631703072901655051310975391309821130985304256444055721882996} a^{18} - \frac{16676970278257755484679443260128785227908810704373407413656483386218322725542924068441787581025382984598831602978908264103}{14484321163431394558440023039981833426904852506353519214751298239815851536450827525655487695654910565492652128222027860941498} a^{17} + \frac{3173800515927744801784509943477986630094399437999946815436124116666155304119910348897097781940499582324994471509244716270685}{28968642326862789116880046079963666853809705012707038429502596479631703072901655051310975391309821130985304256444055721882996} a^{16} + \frac{3310857887144385740763347072644089218153096300722299193704414215112806115015792857485496881905857932633137035779681689651535}{28968642326862789116880046079963666853809705012707038429502596479631703072901655051310975391309821130985304256444055721882996} a^{15} - \frac{62771249253203294245036346338116901371742295654576512630644903359023899642994668171715902767395449060204573139022849050861}{28968642326862789116880046079963666853809705012707038429502596479631703072901655051310975391309821130985304256444055721882996} a^{14} - \frac{1600342048570762472601040236385281460345075797339087724232579797787410377318838600808794596246455988610038426645960165172885}{14484321163431394558440023039981833426904852506353519214751298239815851536450827525655487695654910565492652128222027860941498} a^{13} - \frac{2547766235897505227961297778895798267207289969632393171616515994016175043962654123408100291275031186326685641911069605240697}{7242160581715697279220011519990916713452426253176759607375649119907925768225413762827743847827455282746326064111013930470749} a^{12} + \frac{3491490493771456321964636644029278449608330929817579279462967752614372285253765042181378271650581154753100611436042965201087}{7242160581715697279220011519990916713452426253176759607375649119907925768225413762827743847827455282746326064111013930470749} a^{11} - \frac{5760269436803548509401919424339659956180816179574337974418847476838660282360952225882712751943309804858787945193886113585}{188108067057550578681039260259504330219543539043552197594172704413192877096763993839681658385128708642761715950935426765474} a^{10} + \frac{1572693191590566079851658744787066733247520445923493517800405818768410607051622154150240491569646197984453992213528095104573}{7242160581715697279220011519990916713452426253176759607375649119907925768225413762827743847827455282746326064111013930470749} a^{9} - \frac{46296445505742453034209906560777102990230663399234662203843676159103797727952715132348589171873916409512154654089498667719}{7242160581715697279220011519990916713452426253176759607375649119907925768225413762827743847827455282746326064111013930470749} a^{8} + \frac{11980593618298748125457894896422073492027782829935073895956745209809173463755438243497563662052753958938691982056245388764133}{28968642326862789116880046079963666853809705012707038429502596479631703072901655051310975391309821130985304256444055721882996} a^{7} + \frac{168061839104010147872057702904922544849640254113860121534661589333369778730041358403145294899612631790702700293009129478889}{467236166562303050272258807741349465384027500204952232733912846445672630208091210505015732117900340822343617039420253578758} a^{6} - \frac{307445184866913005860677394032407380113707231227262890817390497871085856573912744207034877969837284811136789519695606423511}{14484321163431394558440023039981833426904852506353519214751298239815851536450827525655487695654910565492652128222027860941498} a^{5} - \frac{804968627853538022004753196478716066702246210632069049312310399509196426809543138080375793925786908584580481778867347540464}{7242160581715697279220011519990916713452426253176759607375649119907925768225413762827743847827455282746326064111013930470749} a^{4} - \frac{6882292213251634518638750594920930797223574649608410628563901640087279784793768022308657678795252408168622582915909376596143}{14484321163431394558440023039981833426904852506353519214751298239815851536450827525655487695654910565492652128222027860941498} a^{3} - \frac{1567779845707909031743706618061313588107642974783837855650798706104711205644232790596080844508433510885986763429943214538369}{4138377475266112730982863725709095264829957858958148347071799497090243296128807864472996484472831590140757750920579388840428} a^{2} + \frac{3938495404510776367681958170896468605404552168872215832787585802869687440260414816867067353531645899462346311511469617468451}{28968642326862789116880046079963666853809705012707038429502596479631703072901655051310975391309821130985304256444055721882996} a + \frac{3558217921619928379629572882550417832323699715213873978068247019374562181636375577055729595986000930267218654721499318771295}{7242160581715697279220011519990916713452426253176759607375649119907925768225413762827743847827455282746326064111013930470749}$

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

Class group and class number

not computed

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $12$
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:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) $ not computed

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

\(\Q(\sqrt{-3247}) \), 13.1.48551226272641.1

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

Sibling fields

Degree 26 sibling: 26.2.23347285006439981249999555372409414171948497.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.13.0.1}{13} }^{2}$ $26$ $26$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.13.0.1}{13} }^{2}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{13}$ $26$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/59.13.0.1}{13} }^{2}$

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
17Data not computed
$191$191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.3247.2t1.a.a$1$ $ 17 \cdot 191 $ \(\Q(\sqrt{-3247}) \) $C_2$ (as 2T1) $1$ $-1$
1.191.2t1.a.a$1$ $ 191 $ \(\Q(\sqrt{-191}) \) $C_2$ (as 2T1) $1$ $-1$
1.17.2t1.a.a$1$ $ 17 $ \(\Q(\sqrt{17}) \) $C_2$ (as 2T1) $1$ $1$
* 2.55199.26t3.a.d$2$ $ 17^{2} \cdot 191 $ 26.0.4459331436230036418749915076130198106842162927.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.191.13t2.a.c$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.191.13t2.a.b$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.55199.26t3.a.a$2$ $ 17^{2} \cdot 191 $ 26.0.4459331436230036418749915076130198106842162927.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.55199.26t3.a.e$2$ $ 17^{2} \cdot 191 $ 26.0.4459331436230036418749915076130198106842162927.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.191.13t2.a.a$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.191.13t2.a.d$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.191.13t2.a.f$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.55199.26t3.a.c$2$ $ 17^{2} \cdot 191 $ 26.0.4459331436230036418749915076130198106842162927.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.55199.26t3.a.f$2$ $ 17^{2} \cdot 191 $ 26.0.4459331436230036418749915076130198106842162927.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.191.13t2.a.e$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.55199.26t3.a.b$2$ $ 17^{2} \cdot 191 $ 26.0.4459331436230036418749915076130198106842162927.1 $D_{26}$ (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 *.