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Properties

Label 29.1.756...121.1
Degree $29$
Signature $[1, 14]$
Discriminant $7.568\times 10^{46}$
Root discriminant $41.35$
Ramified primes $23, 97$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{29}$ (as 29T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^29 - 9*x^28 + 56*x^27 - 308*x^26 + 1574*x^25 - 7243*x^24 + 28966*x^23 - 99448*x^22 + 293090*x^21 - 744900*x^20 + 1641343*x^19 - 3148829*x^18 + 5274672*x^17 - 7725970*x^16 + 9897023*x^15 - 11084677*x^14 + 10859104*x^13 - 9337717*x^12 + 7122047*x^11 - 4916951*x^10 + 3150459*x^9 - 1890622*x^8 + 1028653*x^7 - 470622*x^6 + 165361*x^5 - 41848*x^4 + 8324*x^3 - 1810*x^2 + 323*x - 1)
 
gp: K = bnfinit(x^29 - 9*x^28 + 56*x^27 - 308*x^26 + 1574*x^25 - 7243*x^24 + 28966*x^23 - 99448*x^22 + 293090*x^21 - 744900*x^20 + 1641343*x^19 - 3148829*x^18 + 5274672*x^17 - 7725970*x^16 + 9897023*x^15 - 11084677*x^14 + 10859104*x^13 - 9337717*x^12 + 7122047*x^11 - 4916951*x^10 + 3150459*x^9 - 1890622*x^8 + 1028653*x^7 - 470622*x^6 + 165361*x^5 - 41848*x^4 + 8324*x^3 - 1810*x^2 + 323*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 323, -1810, 8324, -41848, 165361, -470622, 1028653, -1890622, 3150459, -4916951, 7122047, -9337717, 10859104, -11084677, 9897023, -7725970, 5274672, -3148829, 1641343, -744900, 293090, -99448, 28966, -7243, 1574, -308, 56, -9, 1]);
 

\( x^{29} - 9 x^{28} + 56 x^{27} - 308 x^{26} + 1574 x^{25} - 7243 x^{24} + 28966 x^{23} - 99448 x^{22} + 293090 x^{21} - 744900 x^{20} + 1641343 x^{19} - 3148829 x^{18} + 5274672 x^{17} - 7725970 x^{16} + 9897023 x^{15} - 11084677 x^{14} + 10859104 x^{13} - 9337717 x^{12} + 7122047 x^{11} - 4916951 x^{10} + 3150459 x^{9} - 1890622 x^{8} + 1028653 x^{7} - 470622 x^{6} + 165361 x^{5} - 41848 x^{4} + 8324 x^{3} - 1810 x^{2} + 323 x - 1 \)

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

Invariants

Degree:  $29$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[1, 14]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(75682241113219898520171301845005641468074963121\)\(\medspace = 23^{14}\cdot 97^{14}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $41.35$
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:  $23, 97$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{23} a^{11} + \frac{8}{23} a^{10} + \frac{9}{23} a^{9} - \frac{4}{23} a^{8} + \frac{1}{23} a^{7} - \frac{1}{23} a^{6} - \frac{5}{23} a^{5} + \frac{10}{23} a^{4} - \frac{11}{23} a^{3} - \frac{8}{23} a^{2} - \frac{4}{23} a - \frac{1}{23}$, $\frac{1}{23} a^{12} - \frac{9}{23} a^{10} - \frac{7}{23} a^{9} + \frac{10}{23} a^{8} - \frac{9}{23} a^{7} + \frac{3}{23} a^{6} + \frac{4}{23} a^{5} + \frac{1}{23} a^{4} + \frac{11}{23} a^{3} - \frac{9}{23} a^{2} + \frac{8}{23} a + \frac{8}{23}$, $\frac{1}{23} a^{13} - \frac{4}{23} a^{10} - \frac{1}{23} a^{9} + \frac{1}{23} a^{8} - \frac{11}{23} a^{7} - \frac{5}{23} a^{6} + \frac{2}{23} a^{5} + \frac{9}{23} a^{4} + \frac{7}{23} a^{3} + \frac{5}{23} a^{2} - \frac{5}{23} a - \frac{9}{23}$, $\frac{1}{23} a^{14} + \frac{8}{23} a^{10} - \frac{9}{23} a^{9} - \frac{4}{23} a^{8} - \frac{1}{23} a^{7} - \frac{2}{23} a^{6} - \frac{11}{23} a^{5} + \frac{1}{23} a^{4} + \frac{7}{23} a^{3} + \frac{9}{23} a^{2} - \frac{2}{23} a - \frac{4}{23}$, $\frac{1}{23} a^{15} - \frac{4}{23} a^{10} - \frac{7}{23} a^{9} + \frac{8}{23} a^{8} - \frac{10}{23} a^{7} - \frac{3}{23} a^{6} - \frac{5}{23} a^{5} - \frac{4}{23} a^{4} + \frac{5}{23} a^{3} - \frac{7}{23} a^{2} + \frac{5}{23} a + \frac{8}{23}$, $\frac{1}{23} a^{16} + \frac{2}{23} a^{10} - \frac{2}{23} a^{9} - \frac{3}{23} a^{8} + \frac{1}{23} a^{7} - \frac{9}{23} a^{6} - \frac{1}{23} a^{5} - \frac{1}{23} a^{4} - \frac{5}{23} a^{3} - \frac{4}{23} a^{2} - \frac{8}{23} a - \frac{4}{23}$, $\frac{1}{23} a^{17} + \frac{5}{23} a^{10} + \frac{2}{23} a^{9} + \frac{9}{23} a^{8} - \frac{11}{23} a^{7} + \frac{1}{23} a^{6} + \frac{9}{23} a^{5} - \frac{2}{23} a^{4} - \frac{5}{23} a^{3} + \frac{8}{23} a^{2} + \frac{4}{23} a + \frac{2}{23}$, $\frac{1}{23} a^{18} + \frac{8}{23} a^{10} + \frac{10}{23} a^{9} + \frac{9}{23} a^{8} - \frac{4}{23} a^{7} - \frac{9}{23} a^{6} - \frac{9}{23} a^{4} - \frac{6}{23} a^{3} - \frac{2}{23} a^{2} - \frac{1}{23} a + \frac{5}{23}$, $\frac{1}{23} a^{19} - \frac{8}{23} a^{10} + \frac{6}{23} a^{9} + \frac{5}{23} a^{8} + \frac{6}{23} a^{7} + \frac{8}{23} a^{6} + \frac{8}{23} a^{5} + \frac{6}{23} a^{4} - \frac{6}{23} a^{3} - \frac{6}{23} a^{2} - \frac{9}{23} a + \frac{8}{23}$, $\frac{1}{299} a^{20} + \frac{3}{299} a^{19} - \frac{2}{299} a^{18} - \frac{1}{299} a^{17} - \frac{3}{299} a^{16} + \frac{1}{299} a^{15} - \frac{1}{299} a^{14} - \frac{3}{299} a^{13} + \frac{3}{299} a^{12} - \frac{5}{299} a^{11} - \frac{25}{299} a^{10} - \frac{120}{299} a^{9} - \frac{85}{299} a^{8} - \frac{50}{299} a^{7} + \frac{27}{299} a^{6} + \frac{90}{299} a^{5} + \frac{82}{299} a^{4} + \frac{54}{299} a^{3} - \frac{9}{299} a^{2} - \frac{147}{299} a + \frac{15}{299}$, $\frac{1}{299} a^{21} + \frac{2}{299} a^{19} + \frac{5}{299} a^{18} - \frac{3}{299} a^{16} - \frac{4}{299} a^{15} - \frac{1}{299} a^{13} - \frac{1}{299} a^{12} + \frac{3}{299} a^{11} - \frac{136}{299} a^{10} + \frac{119}{299} a^{9} + \frac{75}{299} a^{8} - \frac{18}{299} a^{7} + \frac{22}{299} a^{6} - \frac{110}{299} a^{5} - \frac{75}{299} a^{4} + \frac{24}{299} a^{3} - \frac{133}{299} a^{2} - \frac{38}{299} a + \frac{20}{299}$, $\frac{1}{6877} a^{22} - \frac{7}{6877} a^{21} - \frac{10}{6877} a^{20} - \frac{71}{6877} a^{19} - \frac{50}{6877} a^{18} + \frac{126}{6877} a^{17} - \frac{38}{6877} a^{16} + \frac{29}{6877} a^{15} + \frac{11}{6877} a^{14} + \frac{3}{6877} a^{13} - \frac{6}{529} a^{12} + \frac{2}{299} a^{11} - \frac{1996}{6877} a^{10} - \frac{1437}{6877} a^{9} - \frac{602}{6877} a^{8} + \frac{2672}{6877} a^{7} + \frac{2142}{6877} a^{6} + \frac{980}{6877} a^{5} - \frac{2099}{6877} a^{4} - \frac{142}{529} a^{3} + \frac{29}{529} a^{2} + \frac{399}{6877} a - \frac{2621}{6877}$, $\frac{1}{75647} a^{23} - \frac{2}{75647} a^{22} + \frac{1}{75647} a^{21} + \frac{109}{75647} a^{20} - \frac{40}{5819} a^{19} - \frac{5}{6877} a^{18} + \frac{1558}{75647} a^{17} - \frac{4}{3289} a^{16} - \frac{1293}{75647} a^{15} - \frac{1069}{75647} a^{14} + \frac{1294}{75647} a^{13} - \frac{298}{75647} a^{12} - \frac{386}{75647} a^{11} - \frac{4885}{75647} a^{10} - \frac{783}{5819} a^{9} - \frac{25109}{75647} a^{8} + \frac{22}{299} a^{7} + \frac{673}{75647} a^{6} - \frac{19831}{75647} a^{5} + \frac{4564}{75647} a^{4} + \frac{7063}{75647} a^{3} - \frac{16668}{75647} a^{2} + \frac{593}{75647} a - \frac{36243}{75647}$, $\frac{1}{75647} a^{24} - \frac{3}{75647} a^{22} + \frac{111}{75647} a^{21} - \frac{49}{75647} a^{20} - \frac{336}{75647} a^{19} + \frac{942}{75647} a^{18} - \frac{518}{75647} a^{17} + \frac{81}{5819} a^{16} - \frac{113}{75647} a^{15} - \frac{1097}{75647} a^{14} + \frac{1531}{75647} a^{13} - \frac{223}{75647} a^{12} - \frac{344}{75647} a^{11} + \frac{3327}{75647} a^{10} + \frac{22843}{75647} a^{9} - \frac{2306}{5819} a^{8} + \frac{947}{5819} a^{7} + \frac{14658}{75647} a^{6} - \frac{822}{3289} a^{5} + \frac{17203}{75647} a^{4} + \frac{30854}{75647} a^{3} + \frac{1159}{75647} a^{2} + \frac{2701}{6877} a + \frac{6956}{75647}$, $\frac{1}{75647} a^{25} - \frac{5}{75647} a^{22} - \frac{35}{75647} a^{21} + \frac{79}{75647} a^{20} - \frac{651}{75647} a^{19} - \frac{243}{75647} a^{18} - \frac{543}{75647} a^{17} - \frac{763}{75647} a^{16} + \frac{436}{75647} a^{15} + \frac{1415}{75647} a^{14} + \frac{42}{5819} a^{13} - \frac{1513}{75647} a^{12} - \frac{108}{75647} a^{11} + \frac{1060}{75647} a^{10} + \frac{790}{5819} a^{9} - \frac{33481}{75647} a^{8} + \frac{179}{5819} a^{7} + \frac{15926}{75647} a^{6} + \frac{3481}{75647} a^{5} + \frac{15858}{75647} a^{4} - \frac{5328}{75647} a^{3} + \frac{25269}{75647} a^{2} - \frac{15421}{75647} a - \frac{21807}{75647}$, $\frac{1}{75647} a^{26} - \frac{1}{75647} a^{22} + \frac{29}{75647} a^{21} - \frac{40}{75647} a^{20} - \frac{654}{75647} a^{19} + \frac{524}{75647} a^{18} - \frac{1091}{75647} a^{17} - \frac{684}{75647} a^{16} - \frac{991}{75647} a^{15} - \frac{1532}{75647} a^{14} + \frac{29}{75647} a^{13} - \frac{476}{75647} a^{12} - \frac{617}{75647} a^{11} + \frac{15413}{75647} a^{10} - \frac{300}{5819} a^{9} + \frac{13732}{75647} a^{8} + \frac{29511}{75647} a^{7} + \frac{8496}{75647} a^{6} - \frac{12600}{75647} a^{5} + \frac{277}{75647} a^{4} - \frac{400}{75647} a^{3} - \frac{1972}{75647} a^{2} + \frac{6810}{75647} a + \frac{670}{3289}$, $\frac{1}{34116797} a^{27} + \frac{142}{34116797} a^{26} + \frac{171}{34116797} a^{25} - \frac{163}{34116797} a^{24} - \frac{13}{2624369} a^{23} - \frac{150}{3101527} a^{22} - \frac{2029}{34116797} a^{21} - \frac{44752}{34116797} a^{20} - \frac{89642}{34116797} a^{19} - \frac{359514}{34116797} a^{18} - \frac{463598}{34116797} a^{17} + \frac{102591}{34116797} a^{16} - \frac{232488}{34116797} a^{15} + \frac{603843}{34116797} a^{14} - \frac{452129}{34116797} a^{13} + \frac{179615}{34116797} a^{12} + \frac{41542}{2624369} a^{11} - \frac{16170411}{34116797} a^{10} + \frac{9348450}{34116797} a^{9} - \frac{13850392}{34116797} a^{8} - \frac{14567122}{34116797} a^{7} - \frac{10494323}{34116797} a^{6} + \frac{244523}{34116797} a^{5} + \frac{3721095}{34116797} a^{4} - \frac{16599589}{34116797} a^{3} + \frac{148624}{3101527} a^{2} + \frac{7398970}{34116797} a + \frac{3110377}{34116797}$, $\frac{1}{123541800638971} a^{28} - \frac{469561}{123541800638971} a^{27} - \frac{678001997}{123541800638971} a^{26} + \frac{582923028}{123541800638971} a^{25} + \frac{769804344}{123541800638971} a^{24} - \frac{197090255}{123541800638971} a^{23} + \frac{1189880115}{123541800638971} a^{22} - \frac{14553236240}{11231072785361} a^{21} + \frac{43249794894}{123541800638971} a^{20} + \frac{1289915540449}{123541800638971} a^{19} + \frac{220025269941}{11231072785361} a^{18} + \frac{1452946398357}{123541800638971} a^{17} - \frac{294583839535}{123541800638971} a^{16} - \frac{1323438835756}{123541800638971} a^{15} + \frac{461640572491}{123541800638971} a^{14} + \frac{148168981912}{123541800638971} a^{13} - \frac{786099422491}{123541800638971} a^{12} - \frac{1580938999482}{123541800638971} a^{11} + \frac{57466471633221}{123541800638971} a^{10} + \frac{12461019357635}{123541800638971} a^{9} + \frac{34085266141364}{123541800638971} a^{8} + \frac{1216802954142}{9503215433767} a^{7} - \frac{30651663472073}{123541800638971} a^{6} + \frac{1528505098804}{4260062090999} a^{5} + \frac{46804345207426}{123541800638971} a^{4} - \frac{28093874868327}{123541800638971} a^{3} + \frac{580486533304}{1388110119539} a^{2} + \frac{961944788385}{9503215433767} a + \frac{36508927200805}{123541800638971}$

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:  \( 737823739010.7355 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{14}\cdot 737823739010.7355 \cdot 1}{2\sqrt{75682241113219898520171301845005641468074963121}}\approx 0.400842525094451$ (assuming GRH)

Galois group

$D_{29}$ (as 29T2):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 58
The 16 conjugacy class representatives for $D_{29}$
Character table for $D_{29}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $29$ $29$ $29$ $29$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ $29$ $29$ R ${\href{/LocalNumberField/29.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ $29$ $29$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $29$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
$97$$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.2231.2t1.a.a$1$ $ 23 \cdot 97 $ \(\Q(\sqrt{-2231}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.2231.29t2.a.h$2$ $ 23 \cdot 97 $ 29.1.75682241113219898520171301845005641468074963121.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2231.29t2.a.a$2$ $ 23 \cdot 97 $ 29.1.75682241113219898520171301845005641468074963121.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2231.29t2.a.f$2$ $ 23 \cdot 97 $ 29.1.75682241113219898520171301845005641468074963121.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2231.29t2.a.g$2$ $ 23 \cdot 97 $ 29.1.75682241113219898520171301845005641468074963121.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2231.29t2.a.b$2$ $ 23 \cdot 97 $ 29.1.75682241113219898520171301845005641468074963121.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2231.29t2.a.m$2$ $ 23 \cdot 97 $ 29.1.75682241113219898520171301845005641468074963121.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2231.29t2.a.n$2$ $ 23 \cdot 97 $ 29.1.75682241113219898520171301845005641468074963121.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2231.29t2.a.c$2$ $ 23 \cdot 97 $ 29.1.75682241113219898520171301845005641468074963121.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2231.29t2.a.l$2$ $ 23 \cdot 97 $ 29.1.75682241113219898520171301845005641468074963121.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2231.29t2.a.i$2$ $ 23 \cdot 97 $ 29.1.75682241113219898520171301845005641468074963121.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2231.29t2.a.k$2$ $ 23 \cdot 97 $ 29.1.75682241113219898520171301845005641468074963121.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2231.29t2.a.j$2$ $ 23 \cdot 97 $ 29.1.75682241113219898520171301845005641468074963121.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2231.29t2.a.e$2$ $ 23 \cdot 97 $ 29.1.75682241113219898520171301845005641468074963121.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2231.29t2.a.d$2$ $ 23 \cdot 97 $ 29.1.75682241113219898520171301845005641468074963121.1 $D_{29}$ (as 29T2) $1$ $0$

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 *.