Properties

Label 26.2.233...497.1
Degree $26$
Signature $[2, 12]$
Discriminant $2.335\times 10^{43}$
Root discriminant $46.56$
Ramified primes $17, 191$
Class number not computed
Class group not computed
Galois group $D_{26}$ (as 26T3)

Related objects

Downloads

Learn more about

Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^26 - x^25 - 50*x^24 + 48*x^23 + 1188*x^22 - 1066*x^21 - 17960*x^20 + 14231*x^19 + 194023*x^18 - 126696*x^17 - 1579415*x^16 + 794998*x^15 + 9883811*x^14 - 3670079*x^13 - 47524550*x^12 + 13070104*x^11 + 173418938*x^10 - 37145018*x^9 - 469905596*x^8 + 84248201*x^7 + 912345834*x^6 - 152992905*x^5 - 1188256605*x^4 + 203114817*x^3 + 913828487*x^2 - 132782216*x - 329129521)
 
gp: K = bnfinit(x^26 - x^25 - 50*x^24 + 48*x^23 + 1188*x^22 - 1066*x^21 - 17960*x^20 + 14231*x^19 + 194023*x^18 - 126696*x^17 - 1579415*x^16 + 794998*x^15 + 9883811*x^14 - 3670079*x^13 - 47524550*x^12 + 13070104*x^11 + 173418938*x^10 - 37145018*x^9 - 469905596*x^8 + 84248201*x^7 + 912345834*x^6 - 152992905*x^5 - 1188256605*x^4 + 203114817*x^3 + 913828487*x^2 - 132782216*x - 329129521, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-329129521, -132782216, 913828487, 203114817, -1188256605, -152992905, 912345834, 84248201, -469905596, -37145018, 173418938, 13070104, -47524550, -3670079, 9883811, 794998, -1579415, -126696, 194023, 14231, -17960, -1066, 1188, 48, -50, -1, 1]);
 

\( x^{26} - x^{25} - 50 x^{24} + 48 x^{23} + 1188 x^{22} - 1066 x^{21} - 17960 x^{20} + 14231 x^{19} + 194023 x^{18} - 126696 x^{17} - 1579415 x^{16} + 794998 x^{15} + 9883811 x^{14} - 3670079 x^{13} - 47524550 x^{12} + 13070104 x^{11} + 173418938 x^{10} - 37145018 x^{9} - 469905596 x^{8} + 84248201 x^{7} + 912345834 x^{6} - 152992905 x^{5} - 1188256605 x^{4} + 203114817 x^{3} + 913828487 x^{2} - 132782216 x - 329129521 \)

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:  \(23347285006439981249999555372409414171948497\)\(\medspace = 17^{13}\cdot 191^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $46.56$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{7} a^{16} + \frac{3}{7} a^{15} - \frac{2}{7} a^{14} + \frac{1}{7} a^{13} - \frac{1}{7} a^{12} + \frac{2}{7} a^{11} + \frac{1}{7} a^{10} + \frac{3}{7} a^{9} - \frac{3}{7} a^{8} - \frac{3}{7} a^{7} - \frac{2}{7} a^{6} + \frac{1}{7} a^{5} - \frac{1}{7} a^{4} + \frac{1}{7} a^{3} - \frac{3}{7} a^{2} + \frac{2}{7} a$, $\frac{1}{7} a^{17} + \frac{3}{7} a^{15} + \frac{3}{7} a^{13} - \frac{2}{7} a^{12} + \frac{2}{7} a^{11} + \frac{2}{7} a^{9} - \frac{1}{7} a^{8} + \frac{3}{7} a^{5} - \frac{3}{7} a^{4} + \frac{1}{7} a^{3} - \frac{3}{7} a^{2} + \frac{1}{7} a$, $\frac{1}{7} a^{18} - \frac{2}{7} a^{15} + \frac{2}{7} a^{14} + \frac{2}{7} a^{13} - \frac{2}{7} a^{12} + \frac{1}{7} a^{11} - \frac{1}{7} a^{10} - \frac{3}{7} a^{9} + \frac{2}{7} a^{8} + \frac{2}{7} a^{7} + \frac{2}{7} a^{6} + \frac{1}{7} a^{5} - \frac{3}{7} a^{4} + \frac{1}{7} a^{3} + \frac{3}{7} a^{2} + \frac{1}{7} a$, $\frac{1}{7} a^{19} + \frac{1}{7} a^{15} - \frac{2}{7} a^{14} - \frac{1}{7} a^{12} + \frac{3}{7} a^{11} - \frac{1}{7} a^{10} + \frac{1}{7} a^{9} + \frac{3}{7} a^{8} + \frac{3}{7} a^{7} - \frac{3}{7} a^{6} - \frac{1}{7} a^{5} - \frac{1}{7} a^{4} - \frac{2}{7} a^{3} + \frac{2}{7} a^{2} - \frac{3}{7} a$, $\frac{1}{7} a^{20} + \frac{2}{7} a^{15} + \frac{2}{7} a^{14} - \frac{2}{7} a^{13} - \frac{3}{7} a^{12} - \frac{3}{7} a^{11} - \frac{1}{7} a^{8} + \frac{1}{7} a^{6} - \frac{2}{7} a^{5} - \frac{1}{7} a^{4} + \frac{1}{7} a^{3} - \frac{2}{7} a$, $\frac{1}{7} a^{21} + \frac{3}{7} a^{15} + \frac{2}{7} a^{14} + \frac{2}{7} a^{13} - \frac{1}{7} a^{12} + \frac{3}{7} a^{11} - \frac{2}{7} a^{10} - \frac{1}{7} a^{8} + \frac{2}{7} a^{6} - \frac{3}{7} a^{5} + \frac{3}{7} a^{4} - \frac{2}{7} a^{3} - \frac{3}{7} a^{2} + \frac{3}{7} a$, $\frac{1}{7} a^{22} + \frac{1}{7} a^{14} + \frac{3}{7} a^{13} - \frac{1}{7} a^{12} - \frac{1}{7} a^{11} - \frac{3}{7} a^{10} - \frac{3}{7} a^{9} + \frac{2}{7} a^{8} - \frac{3}{7} a^{7} + \frac{3}{7} a^{6} + \frac{1}{7} a^{4} + \frac{1}{7} a^{3} - \frac{2}{7} a^{2} + \frac{1}{7} a$, $\frac{1}{163009} a^{23} - \frac{10869}{163009} a^{22} - \frac{4679}{163009} a^{21} + \frac{953}{163009} a^{20} + \frac{5333}{163009} a^{19} - \frac{4801}{163009} a^{18} - \frac{2903}{163009} a^{17} + \frac{58}{5621} a^{16} + \frac{73286}{163009} a^{15} - \frac{49132}{163009} a^{14} - \frac{9078}{23287} a^{13} - \frac{55954}{163009} a^{12} + \frac{21843}{163009} a^{11} - \frac{29387}{163009} a^{10} - \frac{16091}{163009} a^{9} - \frac{632}{2233} a^{8} - \frac{66048}{163009} a^{7} + \frac{72025}{163009} a^{6} - \frac{69709}{163009} a^{5} - \frac{3408}{14819} a^{4} + \frac{5512}{23287} a^{3} - \frac{59288}{163009} a^{2} - \frac{56006}{163009} a - \frac{415}{23287}$, $\frac{1}{35372953} a^{24} + \frac{38}{35372953} a^{23} + \frac{1538197}{35372953} a^{22} + \frac{1502572}{35372953} a^{21} + \frac{2342402}{35372953} a^{20} - \frac{51143}{5053279} a^{19} - \frac{69698}{1141063} a^{18} + \frac{2477403}{35372953} a^{17} + \frac{136230}{5053279} a^{16} - \frac{15787017}{35372953} a^{15} - \frac{3931468}{35372953} a^{14} - \frac{14918301}{35372953} a^{13} + \frac{10073958}{35372953} a^{12} + \frac{4131290}{35372953} a^{11} + \frac{16748808}{35372953} a^{10} - \frac{16244306}{35372953} a^{9} - \frac{14803288}{35372953} a^{8} - \frac{16557084}{35372953} a^{7} - \frac{14611067}{35372953} a^{6} + \frac{98093}{1141063} a^{5} - \frac{16829674}{35372953} a^{4} + \frac{167570}{3215723} a^{3} + \frac{6606563}{35372953} a^{2} - \frac{18653}{163009} a + \frac{118515}{721897}$, $\frac{1}{66595059839277063965733825282108200302372275197325767451661824147975051117178056918917} a^{25} + \frac{81228747778157298766336125167645830967297969991460564105395675811544560415839}{9513579977039580566533403611729742900338896456760823921665974878282150159596865274131} a^{24} - \frac{122368490765312381810536387242653720646725423817474979500398642029580849520479847}{66595059839277063965733825282108200302372275197325767451661824147975051117178056918917} a^{23} + \frac{4258567952574708601032643676373139787976877574837961664066089622546960003671410016943}{66595059839277063965733825282108200302372275197325767451661824147975051117178056918917} a^{22} + \frac{133991884208597129485408153953326578500301921980547365500951528324210444646760088718}{66595059839277063965733825282108200302372275197325767451661824147975051117178056918917} a^{21} - \frac{539379108677981958709230079183218364216350496430988941796573793982081648165040870316}{66595059839277063965733825282108200302372275197325767451661824147975051117178056918917} a^{20} - \frac{2019088890995769931183176267616816766573601739331016992789203853547112317488501711037}{66595059839277063965733825282108200302372275197325767451661824147975051117178056918917} a^{19} - \frac{4750151328136286053308318296780649425448370159327222172012952865287029508054081456183}{66595059839277063965733825282108200302372275197325767451661824147975051117178056918917} a^{18} - \frac{11176550909253766280286208510625481109100210805535944581102825687463361487056060948}{6054096349025187633248529571100745482033843199756887950151074922543186465198005174447} a^{17} + \frac{2310997361101964822042733071906076648178012479154443449343679031582095778492928650509}{66595059839277063965733825282108200302372275197325767451661824147975051117178056918917} a^{16} + \frac{2801801609959891066067397895921751203920895493474669788890236128285402021378864496248}{6054096349025187633248529571100745482033843199756887950151074922543186465198005174447} a^{15} - \frac{3884265680689510516752661729066412997568935232838794364561538710574845902806833294355}{9513579977039580566533403611729742900338896456760823921665974878282150159596865274131} a^{14} - \frac{3739648237417283950085228839081235786625139506839740887839386714545974302287930384889}{66595059839277063965733825282108200302372275197325767451661824147975051117178056918917} a^{13} - \frac{10928599450799558569359338724606365786259043952603786440614500522105959847639866110934}{66595059839277063965733825282108200302372275197325767451661824147975051117178056918917} a^{12} + \frac{3675551723257075228708312390154677540122533288607951954794529452169451408453899003412}{9513579977039580566533403611729742900338896456760823921665974878282150159596865274131} a^{11} + \frac{198217919235856785958340195918440430319586584204434358250413296155672648445228342000}{937958589285592450221603172987439440878482749258109400727631326027817621368705027027} a^{10} - \frac{277497295833235444848526188307349360423444996598290980237063673696175187004588953045}{1359082853862797223790486230247106128619842350965831988809424982611735737085266467733} a^{9} + \frac{16559058916476016616097480042401342911601279922429613135320739707883590668795856916143}{66595059839277063965733825282108200302372275197325767451661824147975051117178056918917} a^{8} - \frac{18725822961939545087693072269645118874170167971208377886118769072982261771409893024577}{66595059839277063965733825282108200302372275197325767451661824147975051117178056918917} a^{7} + \frac{3020929572741896167888155759062830875246528815209141447611106869301751044962441289830}{6054096349025187633248529571100745482033843199756887950151074922543186465198005174447} a^{6} - \frac{10691686472543941638069758095223232220773662737152535820278355222030908417160636791570}{66595059839277063965733825282108200302372275197325767451661824147975051117178056918917} a^{5} + \frac{16379407463217327245158368551897817367429952695051682307182221093710725058205970454505}{66595059839277063965733825282108200302372275197325767451661824147975051117178056918917} a^{4} + \frac{27667445491370508326528132688920981533472601018661052450448117542603234190429503388462}{66595059839277063965733825282108200302372275197325767451661824147975051117178056918917} a^{3} + \frac{18262282478176953919572524801020400738861855065453546378104510172180940705516440309514}{66595059839277063965733825282108200302372275197325767451661824147975051117178056918917} a^{2} + \frac{3173482666396580323725343425319677144203225402024543759060407373134554094002711963093}{9513579977039580566533403611729742900338896456760823921665974878282150159596865274131} a - \frac{575696147268037436371477533602341883186955364661130850108220325310487558382430734411}{1359082853862797223790486230247106128619842350965831988809424982611735737085266467733}$

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:  $13$
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:  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

\(\Q(\sqrt{17}) \), 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: 26.0.4459331436230036418749915076130198106842162927.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.13.0.1}{13} }^{2}$ $26$ $26$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/13.13.0.1}{13} }^{2}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $26$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/43.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.13.0.1}{13} }^{2}$

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$$\Q_{191}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{191}$$x + 2$$1$$1$$0$Trivial$[\ ]$
191.2.1.2$x^{2} + 382$$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 Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.3247.2t1.a.a$1$ $ 17 \cdot 191 $ $x^{2} - x + 812$ $C_2$ (as 2T1) $1$ $-1$
1.191.2t1.a.a$1$ $ 191 $ $x^{2} - x + 48$ $C_2$ (as 2T1) $1$ $-1$
* 1.17.2t1.a.a$1$ $ 17 $ $x^{2} - x - 4$ $C_2$ (as 2T1) $1$ $1$
* 2.55199.26t3.b.f$2$ $ 17^{2} \cdot 191 $ $x^{26} - x^{25} - 50 x^{24} + 48 x^{23} + 1188 x^{22} - 1066 x^{21} - 17960 x^{20} + 14231 x^{19} + 194023 x^{18} - 126696 x^{17} - 1579415 x^{16} + 794998 x^{15} + 9883811 x^{14} - 3670079 x^{13} - 47524550 x^{12} + 13070104 x^{11} + 173418938 x^{10} - 37145018 x^{9} - 469905596 x^{8} + 84248201 x^{7} + 912345834 x^{6} - 152992905 x^{5} - 1188256605 x^{4} + 203114817 x^{3} + 913828487 x^{2} - 132782216 x - 329129521$ $D_{26}$ (as 26T3) $1$ $0$
* 2.191.13t2.a.b$2$ $ 191 $ $x^{13} - 2 x^{12} + 4 x^{10} - 5 x^{9} + x^{8} + 5 x^{7} - 11 x^{6} + 19 x^{5} - 22 x^{4} + 16 x^{3} - 10 x^{2} + 6 x - 1$ $D_{13}$ (as 13T2) $1$ $0$
* 2.191.13t2.a.a$2$ $ 191 $ $x^{13} - 2 x^{12} + 4 x^{10} - 5 x^{9} + x^{8} + 5 x^{7} - 11 x^{6} + 19 x^{5} - 22 x^{4} + 16 x^{3} - 10 x^{2} + 6 x - 1$ $D_{13}$ (as 13T2) $1$ $0$
* 2.55199.26t3.b.a$2$ $ 17^{2} \cdot 191 $ $x^{26} - x^{25} - 50 x^{24} + 48 x^{23} + 1188 x^{22} - 1066 x^{21} - 17960 x^{20} + 14231 x^{19} + 194023 x^{18} - 126696 x^{17} - 1579415 x^{16} + 794998 x^{15} + 9883811 x^{14} - 3670079 x^{13} - 47524550 x^{12} + 13070104 x^{11} + 173418938 x^{10} - 37145018 x^{9} - 469905596 x^{8} + 84248201 x^{7} + 912345834 x^{6} - 152992905 x^{5} - 1188256605 x^{4} + 203114817 x^{3} + 913828487 x^{2} - 132782216 x - 329129521$ $D_{26}$ (as 26T3) $1$ $0$
* 2.55199.26t3.b.e$2$ $ 17^{2} \cdot 191 $ $x^{26} - x^{25} - 50 x^{24} + 48 x^{23} + 1188 x^{22} - 1066 x^{21} - 17960 x^{20} + 14231 x^{19} + 194023 x^{18} - 126696 x^{17} - 1579415 x^{16} + 794998 x^{15} + 9883811 x^{14} - 3670079 x^{13} - 47524550 x^{12} + 13070104 x^{11} + 173418938 x^{10} - 37145018 x^{9} - 469905596 x^{8} + 84248201 x^{7} + 912345834 x^{6} - 152992905 x^{5} - 1188256605 x^{4} + 203114817 x^{3} + 913828487 x^{2} - 132782216 x - 329129521$ $D_{26}$ (as 26T3) $1$ $0$
* 2.55199.26t3.b.c$2$ $ 17^{2} \cdot 191 $ $x^{26} - x^{25} - 50 x^{24} + 48 x^{23} + 1188 x^{22} - 1066 x^{21} - 17960 x^{20} + 14231 x^{19} + 194023 x^{18} - 126696 x^{17} - 1579415 x^{16} + 794998 x^{15} + 9883811 x^{14} - 3670079 x^{13} - 47524550 x^{12} + 13070104 x^{11} + 173418938 x^{10} - 37145018 x^{9} - 469905596 x^{8} + 84248201 x^{7} + 912345834 x^{6} - 152992905 x^{5} - 1188256605 x^{4} + 203114817 x^{3} + 913828487 x^{2} - 132782216 x - 329129521$ $D_{26}$ (as 26T3) $1$ $0$
* 2.55199.26t3.b.b$2$ $ 17^{2} \cdot 191 $ $x^{26} - x^{25} - 50 x^{24} + 48 x^{23} + 1188 x^{22} - 1066 x^{21} - 17960 x^{20} + 14231 x^{19} + 194023 x^{18} - 126696 x^{17} - 1579415 x^{16} + 794998 x^{15} + 9883811 x^{14} - 3670079 x^{13} - 47524550 x^{12} + 13070104 x^{11} + 173418938 x^{10} - 37145018 x^{9} - 469905596 x^{8} + 84248201 x^{7} + 912345834 x^{6} - 152992905 x^{5} - 1188256605 x^{4} + 203114817 x^{3} + 913828487 x^{2} - 132782216 x - 329129521$ $D_{26}$ (as 26T3) $1$ $0$
* 2.191.13t2.a.f$2$ $ 191 $ $x^{13} - 2 x^{12} + 4 x^{10} - 5 x^{9} + x^{8} + 5 x^{7} - 11 x^{6} + 19 x^{5} - 22 x^{4} + 16 x^{3} - 10 x^{2} + 6 x - 1$ $D_{13}$ (as 13T2) $1$ $0$
* 2.191.13t2.a.d$2$ $ 191 $ $x^{13} - 2 x^{12} + 4 x^{10} - 5 x^{9} + x^{8} + 5 x^{7} - 11 x^{6} + 19 x^{5} - 22 x^{4} + 16 x^{3} - 10 x^{2} + 6 x - 1$ $D_{13}$ (as 13T2) $1$ $0$
* 2.191.13t2.a.e$2$ $ 191 $ $x^{13} - 2 x^{12} + 4 x^{10} - 5 x^{9} + x^{8} + 5 x^{7} - 11 x^{6} + 19 x^{5} - 22 x^{4} + 16 x^{3} - 10 x^{2} + 6 x - 1$ $D_{13}$ (as 13T2) $1$ $0$
* 2.55199.26t3.b.d$2$ $ 17^{2} \cdot 191 $ $x^{26} - x^{25} - 50 x^{24} + 48 x^{23} + 1188 x^{22} - 1066 x^{21} - 17960 x^{20} + 14231 x^{19} + 194023 x^{18} - 126696 x^{17} - 1579415 x^{16} + 794998 x^{15} + 9883811 x^{14} - 3670079 x^{13} - 47524550 x^{12} + 13070104 x^{11} + 173418938 x^{10} - 37145018 x^{9} - 469905596 x^{8} + 84248201 x^{7} + 912345834 x^{6} - 152992905 x^{5} - 1188256605 x^{4} + 203114817 x^{3} + 913828487 x^{2} - 132782216 x - 329129521$ $D_{26}$ (as 26T3) $1$ $0$
* 2.191.13t2.a.c$2$ $ 191 $ $x^{13} - 2 x^{12} + 4 x^{10} - 5 x^{9} + x^{8} + 5 x^{7} - 11 x^{6} + 19 x^{5} - 22 x^{4} + 16 x^{3} - 10 x^{2} + 6 x - 1$ $D_{13}$ (as 13T2) $1$ $0$

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 *.