Properties

Label 33.11.163...327.1
Degree $33$
Signature $[11, 11]$
Discriminant $-1.635\times 10^{42}$
Root discriminant $19.02$
Ramified prime $23$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{11}\times S_3$ (as 33T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^33 - 3*x^32 - 3*x^31 + 28*x^30 - 58*x^29 + 22*x^28 + 191*x^27 - 526*x^26 + 330*x^25 + 728*x^24 - 1012*x^23 + 413*x^22 - 2524*x^21 + 3437*x^20 + 6311*x^19 - 13555*x^18 - 573*x^17 + 14501*x^16 - 6805*x^15 - 2479*x^14 + 5007*x^13 - 7594*x^12 - 344*x^11 + 8602*x^10 - 626*x^9 - 5041*x^8 + 15*x^7 + 1789*x^6 + 121*x^5 - 352*x^4 - 33*x^3 + 32*x^2 + 2*x - 1)
 
gp: K = bnfinit(x^33 - 3*x^32 - 3*x^31 + 28*x^30 - 58*x^29 + 22*x^28 + 191*x^27 - 526*x^26 + 330*x^25 + 728*x^24 - 1012*x^23 + 413*x^22 - 2524*x^21 + 3437*x^20 + 6311*x^19 - 13555*x^18 - 573*x^17 + 14501*x^16 - 6805*x^15 - 2479*x^14 + 5007*x^13 - 7594*x^12 - 344*x^11 + 8602*x^10 - 626*x^9 - 5041*x^8 + 15*x^7 + 1789*x^6 + 121*x^5 - 352*x^4 - 33*x^3 + 32*x^2 + 2*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 2, 32, -33, -352, 121, 1789, 15, -5041, -626, 8602, -344, -7594, 5007, -2479, -6805, 14501, -573, -13555, 6311, 3437, -2524, 413, -1012, 728, 330, -526, 191, 22, -58, 28, -3, -3, 1]);
 

\( x^{33} - 3 x^{32} - 3 x^{31} + 28 x^{30} - 58 x^{29} + 22 x^{28} + 191 x^{27} - 526 x^{26} + 330 x^{25} + 728 x^{24} - 1012 x^{23} + 413 x^{22} - 2524 x^{21} + 3437 x^{20} + 6311 x^{19} - 13555 x^{18} - 573 x^{17} + 14501 x^{16} - 6805 x^{15} - 2479 x^{14} + 5007 x^{13} - 7594 x^{12} - 344 x^{11} + 8602 x^{10} - 626 x^{9} - 5041 x^{8} + 15 x^{7} + 1789 x^{6} + 121 x^{5} - 352 x^{4} - 33 x^{3} + 32 x^{2} + 2 x - 1 \)

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

Invariants

Degree:  $33$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[11, 11]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-1635170022196481349560959748587682926364327\)\(\medspace = -\,23^{31}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $19.02$
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:  $23$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $11$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $\frac{1}{599} a^{31} - \frac{277}{599} a^{30} - \frac{244}{599} a^{29} + \frac{108}{599} a^{28} + \frac{231}{599} a^{27} + \frac{282}{599} a^{26} - \frac{77}{599} a^{25} + \frac{163}{599} a^{24} + \frac{284}{599} a^{23} + \frac{207}{599} a^{22} + \frac{198}{599} a^{21} + \frac{186}{599} a^{20} - \frac{67}{599} a^{19} - \frac{65}{599} a^{18} - \frac{209}{599} a^{17} + \frac{81}{599} a^{16} + \frac{12}{599} a^{15} - \frac{123}{599} a^{14} - \frac{251}{599} a^{13} + \frac{137}{599} a^{12} + \frac{208}{599} a^{11} + \frac{49}{599} a^{10} + \frac{56}{599} a^{9} + \frac{207}{599} a^{8} + \frac{58}{599} a^{7} + \frac{147}{599} a^{6} + \frac{235}{599} a^{5} + \frac{176}{599} a^{4} - \frac{119}{599} a^{3} + \frac{272}{599} a^{2} - \frac{218}{599} a - \frac{118}{599}$, $\frac{1}{66753596490060073962937664243030405529214290145422481} a^{32} - \frac{48162305301145250212509141197202925540031031071}{111441730367379088418927653160317872335917011928919} a^{31} + \frac{1863500054302098766710439816312490376831033690984440}{66753596490060073962937664243030405529214290145422481} a^{30} + \frac{5049309033324112788213980935494988438450471653047362}{66753596490060073962937664243030405529214290145422481} a^{29} + \frac{10190697857752620709364034742649562668085794412042791}{66753596490060073962937664243030405529214290145422481} a^{28} + \frac{15387826774467798785010946314253923128173323142759022}{66753596490060073962937664243030405529214290145422481} a^{27} + \frac{20408922851806149248023663031291959921583779388241160}{66753596490060073962937664243030405529214290145422481} a^{26} + \frac{7156445526904976778449372553661521690871380575585312}{66753596490060073962937664243030405529214290145422481} a^{25} + \frac{11026122334962817108149547153875169477697406505448223}{66753596490060073962937664243030405529214290145422481} a^{24} - \frac{31522221173769789299861908469692666274553566166354521}{66753596490060073962937664243030405529214290145422481} a^{23} + \frac{28718405049041936930791201880298662443682111872643973}{66753596490060073962937664243030405529214290145422481} a^{22} + \frac{19315034797179559549013887567098039029903883227263292}{66753596490060073962937664243030405529214290145422481} a^{21} + \frac{2829960811772751849881291285843519979538523295345354}{66753596490060073962937664243030405529214290145422481} a^{20} + \frac{5064717253054895820634907981323720899483393819654916}{66753596490060073962937664243030405529214290145422481} a^{19} + \frac{14613840097588283022372605589209227139637287065649137}{66753596490060073962937664243030405529214290145422481} a^{18} + \frac{30504101573497540941482521493244320271812216189366913}{66753596490060073962937664243030405529214290145422481} a^{17} - \frac{22696785016008744771517699558507517585459741848475781}{66753596490060073962937664243030405529214290145422481} a^{16} + \frac{13571743096902749391024199056937970452352031022017433}{66753596490060073962937664243030405529214290145422481} a^{15} - \frac{17874624152600882289018747075718513530327086696795137}{66753596490060073962937664243030405529214290145422481} a^{14} - \frac{7704098533015577358719911153568274250633374912531528}{66753596490060073962937664243030405529214290145422481} a^{13} + \frac{30014569192926784608483611859533614405463583087588937}{66753596490060073962937664243030405529214290145422481} a^{12} - \frac{744069963401288716275460854453608056905715788878916}{66753596490060073962937664243030405529214290145422481} a^{11} - \frac{6165393796370349997873121435207479129699020067436558}{66753596490060073962937664243030405529214290145422481} a^{10} + \frac{4108296104588939004268001178968999498981886372697303}{66753596490060073962937664243030405529214290145422481} a^{9} + \frac{13133976720828007926202414592470187538992165142513763}{66753596490060073962937664243030405529214290145422481} a^{8} - \frac{2899633340230204270793005839901596886565517853386723}{66753596490060073962937664243030405529214290145422481} a^{7} - \frac{17205451087316476276993560686427122188989401184512405}{66753596490060073962937664243030405529214290145422481} a^{6} + \frac{4076241814381575707746846644268022084133735080388198}{66753596490060073962937664243030405529214290145422481} a^{5} + \frac{19938112136932491763920971960727744701489860467241502}{66753596490060073962937664243030405529214290145422481} a^{4} - \frac{31069830033938897023709161542022768026041290033279643}{66753596490060073962937664243030405529214290145422481} a^{3} + \frac{27090212339070239228351817152764578311718290418923129}{66753596490060073962937664243030405529214290145422481} a^{2} - \frac{15103114135755689009556245898636359837516772106102841}{66753596490060073962937664243030405529214290145422481} a - \frac{9241615792537692556126926180262451411495908560629411}{66753596490060073962937664243030405529214290145422481}$

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:  $21$
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:  \( 255776199.8931762 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{11}\cdot(2\pi)^{11}\cdot 255776199.8931762 \cdot 1}{2\sqrt{1635170022196481349560959748587682926364327}}\approx 0.123411940469866$ (assuming GRH)

Galois group

$C_{11}\times S_3$ (as 33T2):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 66
The 33 conjugacy class representatives for $C_{11}\times S_3$
Character table for $C_{11}\times S_3$ is not computed

Intermediate fields

3.1.23.1, \(\Q(\zeta_{23})^+\)

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

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $33$ $33$ $22{,}\,{\href{/LocalNumberField/5.11.0.1}{11} }$ $22{,}\,{\href{/LocalNumberField/7.11.0.1}{11} }$ $22{,}\,{\href{/LocalNumberField/11.11.0.1}{11} }$ $33$ $22{,}\,{\href{/LocalNumberField/17.11.0.1}{11} }$ $22{,}\,{\href{/LocalNumberField/19.11.0.1}{11} }$ R $33$ $33$ $22{,}\,{\href{/LocalNumberField/37.11.0.1}{11} }$ $33$ $22{,}\,{\href{/LocalNumberField/43.11.0.1}{11} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{11}$ $22{,}\,{\href{/LocalNumberField/53.11.0.1}{11} }$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{3}$

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
23Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.23.2t1.a.a$1$ $ 23 $ $x^{2} - x + 6$ $C_2$ (as 2T1) $1$ $-1$
* 1.23.11t1.a.a$1$ $ 23 $ $x^{11} - x^{10} - 10 x^{9} + 9 x^{8} + 36 x^{7} - 28 x^{6} - 56 x^{5} + 35 x^{4} + 35 x^{3} - 15 x^{2} - 6 x + 1$ $C_{11}$ (as 11T1) $0$ $1$
1.23.22t1.a.a$1$ $ 23 $ $x^{22} - x^{21} + x^{20} - x^{19} + x^{18} - x^{17} + x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$ $C_{22}$ (as 22T1) $0$ $-1$
1.23.22t1.a.b$1$ $ 23 $ $x^{22} - x^{21} + x^{20} - x^{19} + x^{18} - x^{17} + x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$ $C_{22}$ (as 22T1) $0$ $-1$
* 1.23.11t1.a.b$1$ $ 23 $ $x^{11} - x^{10} - 10 x^{9} + 9 x^{8} + 36 x^{7} - 28 x^{6} - 56 x^{5} + 35 x^{4} + 35 x^{3} - 15 x^{2} - 6 x + 1$ $C_{11}$ (as 11T1) $0$ $1$
* 1.23.11t1.a.c$1$ $ 23 $ $x^{11} - x^{10} - 10 x^{9} + 9 x^{8} + 36 x^{7} - 28 x^{6} - 56 x^{5} + 35 x^{4} + 35 x^{3} - 15 x^{2} - 6 x + 1$ $C_{11}$ (as 11T1) $0$ $1$
* 1.23.11t1.a.d$1$ $ 23 $ $x^{11} - x^{10} - 10 x^{9} + 9 x^{8} + 36 x^{7} - 28 x^{6} - 56 x^{5} + 35 x^{4} + 35 x^{3} - 15 x^{2} - 6 x + 1$ $C_{11}$ (as 11T1) $0$ $1$
* 1.23.11t1.a.e$1$ $ 23 $ $x^{11} - x^{10} - 10 x^{9} + 9 x^{8} + 36 x^{7} - 28 x^{6} - 56 x^{5} + 35 x^{4} + 35 x^{3} - 15 x^{2} - 6 x + 1$ $C_{11}$ (as 11T1) $0$ $1$
* 1.23.11t1.a.f$1$ $ 23 $ $x^{11} - x^{10} - 10 x^{9} + 9 x^{8} + 36 x^{7} - 28 x^{6} - 56 x^{5} + 35 x^{4} + 35 x^{3} - 15 x^{2} - 6 x + 1$ $C_{11}$ (as 11T1) $0$ $1$
1.23.22t1.a.c$1$ $ 23 $ $x^{22} - x^{21} + x^{20} - x^{19} + x^{18} - x^{17} + x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$ $C_{22}$ (as 22T1) $0$ $-1$
1.23.22t1.a.d$1$ $ 23 $ $x^{22} - x^{21} + x^{20} - x^{19} + x^{18} - x^{17} + x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$ $C_{22}$ (as 22T1) $0$ $-1$
1.23.22t1.a.e$1$ $ 23 $ $x^{22} - x^{21} + x^{20} - x^{19} + x^{18} - x^{17} + x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$ $C_{22}$ (as 22T1) $0$ $-1$
1.23.22t1.a.f$1$ $ 23 $ $x^{22} - x^{21} + x^{20} - x^{19} + x^{18} - x^{17} + x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$ $C_{22}$ (as 22T1) $0$ $-1$
1.23.22t1.a.g$1$ $ 23 $ $x^{22} - x^{21} + x^{20} - x^{19} + x^{18} - x^{17} + x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$ $C_{22}$ (as 22T1) $0$ $-1$
* 1.23.11t1.a.g$1$ $ 23 $ $x^{11} - x^{10} - 10 x^{9} + 9 x^{8} + 36 x^{7} - 28 x^{6} - 56 x^{5} + 35 x^{4} + 35 x^{3} - 15 x^{2} - 6 x + 1$ $C_{11}$ (as 11T1) $0$ $1$
1.23.22t1.a.h$1$ $ 23 $ $x^{22} - x^{21} + x^{20} - x^{19} + x^{18} - x^{17} + x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$ $C_{22}$ (as 22T1) $0$ $-1$
1.23.22t1.a.i$1$ $ 23 $ $x^{22} - x^{21} + x^{20} - x^{19} + x^{18} - x^{17} + x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$ $C_{22}$ (as 22T1) $0$ $-1$
1.23.22t1.a.j$1$ $ 23 $ $x^{22} - x^{21} + x^{20} - x^{19} + x^{18} - x^{17} + x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$ $C_{22}$ (as 22T1) $0$ $-1$
* 1.23.11t1.a.h$1$ $ 23 $ $x^{11} - x^{10} - 10 x^{9} + 9 x^{8} + 36 x^{7} - 28 x^{6} - 56 x^{5} + 35 x^{4} + 35 x^{3} - 15 x^{2} - 6 x + 1$ $C_{11}$ (as 11T1) $0$ $1$
* 1.23.11t1.a.i$1$ $ 23 $ $x^{11} - x^{10} - 10 x^{9} + 9 x^{8} + 36 x^{7} - 28 x^{6} - 56 x^{5} + 35 x^{4} + 35 x^{3} - 15 x^{2} - 6 x + 1$ $C_{11}$ (as 11T1) $0$ $1$
* 1.23.11t1.a.j$1$ $ 23 $ $x^{11} - x^{10} - 10 x^{9} + 9 x^{8} + 36 x^{7} - 28 x^{6} - 56 x^{5} + 35 x^{4} + 35 x^{3} - 15 x^{2} - 6 x + 1$ $C_{11}$ (as 11T1) $0$ $1$
* 2.23.3t2.b.a$2$ $ 23 $ $x^{3} - x^{2} + 1$ $S_3$ (as 3T2) $1$ $0$
* 2.529.33t2.a.a$2$ $ 23^{2}$ $x^{33} - 3 x^{32} - 3 x^{31} + 28 x^{30} - 58 x^{29} + 22 x^{28} + 191 x^{27} - 526 x^{26} + 330 x^{25} + 728 x^{24} - 1012 x^{23} + 413 x^{22} - 2524 x^{21} + 3437 x^{20} + 6311 x^{19} - 13555 x^{18} - 573 x^{17} + 14501 x^{16} - 6805 x^{15} - 2479 x^{14} + 5007 x^{13} - 7594 x^{12} - 344 x^{11} + 8602 x^{10} - 626 x^{9} - 5041 x^{8} + 15 x^{7} + 1789 x^{6} + 121 x^{5} - 352 x^{4} - 33 x^{3} + 32 x^{2} + 2 x - 1$ $C_{11}\times S_3$ (as 33T2) $0$ $0$
* 2.529.33t2.a.b$2$ $ 23^{2}$ $x^{33} - 3 x^{32} - 3 x^{31} + 28 x^{30} - 58 x^{29} + 22 x^{28} + 191 x^{27} - 526 x^{26} + 330 x^{25} + 728 x^{24} - 1012 x^{23} + 413 x^{22} - 2524 x^{21} + 3437 x^{20} + 6311 x^{19} - 13555 x^{18} - 573 x^{17} + 14501 x^{16} - 6805 x^{15} - 2479 x^{14} + 5007 x^{13} - 7594 x^{12} - 344 x^{11} + 8602 x^{10} - 626 x^{9} - 5041 x^{8} + 15 x^{7} + 1789 x^{6} + 121 x^{5} - 352 x^{4} - 33 x^{3} + 32 x^{2} + 2 x - 1$ $C_{11}\times S_3$ (as 33T2) $0$ $0$
* 2.529.33t2.a.c$2$ $ 23^{2}$ $x^{33} - 3 x^{32} - 3 x^{31} + 28 x^{30} - 58 x^{29} + 22 x^{28} + 191 x^{27} - 526 x^{26} + 330 x^{25} + 728 x^{24} - 1012 x^{23} + 413 x^{22} - 2524 x^{21} + 3437 x^{20} + 6311 x^{19} - 13555 x^{18} - 573 x^{17} + 14501 x^{16} - 6805 x^{15} - 2479 x^{14} + 5007 x^{13} - 7594 x^{12} - 344 x^{11} + 8602 x^{10} - 626 x^{9} - 5041 x^{8} + 15 x^{7} + 1789 x^{6} + 121 x^{5} - 352 x^{4} - 33 x^{3} + 32 x^{2} + 2 x - 1$ $C_{11}\times S_3$ (as 33T2) $0$ $0$
* 2.529.33t2.a.d$2$ $ 23^{2}$ $x^{33} - 3 x^{32} - 3 x^{31} + 28 x^{30} - 58 x^{29} + 22 x^{28} + 191 x^{27} - 526 x^{26} + 330 x^{25} + 728 x^{24} - 1012 x^{23} + 413 x^{22} - 2524 x^{21} + 3437 x^{20} + 6311 x^{19} - 13555 x^{18} - 573 x^{17} + 14501 x^{16} - 6805 x^{15} - 2479 x^{14} + 5007 x^{13} - 7594 x^{12} - 344 x^{11} + 8602 x^{10} - 626 x^{9} - 5041 x^{8} + 15 x^{7} + 1789 x^{6} + 121 x^{5} - 352 x^{4} - 33 x^{3} + 32 x^{2} + 2 x - 1$ $C_{11}\times S_3$ (as 33T2) $0$ $0$
* 2.529.33t2.a.e$2$ $ 23^{2}$ $x^{33} - 3 x^{32} - 3 x^{31} + 28 x^{30} - 58 x^{29} + 22 x^{28} + 191 x^{27} - 526 x^{26} + 330 x^{25} + 728 x^{24} - 1012 x^{23} + 413 x^{22} - 2524 x^{21} + 3437 x^{20} + 6311 x^{19} - 13555 x^{18} - 573 x^{17} + 14501 x^{16} - 6805 x^{15} - 2479 x^{14} + 5007 x^{13} - 7594 x^{12} - 344 x^{11} + 8602 x^{10} - 626 x^{9} - 5041 x^{8} + 15 x^{7} + 1789 x^{6} + 121 x^{5} - 352 x^{4} - 33 x^{3} + 32 x^{2} + 2 x - 1$ $C_{11}\times S_3$ (as 33T2) $0$ $0$
* 2.529.33t2.a.f$2$ $ 23^{2}$ $x^{33} - 3 x^{32} - 3 x^{31} + 28 x^{30} - 58 x^{29} + 22 x^{28} + 191 x^{27} - 526 x^{26} + 330 x^{25} + 728 x^{24} - 1012 x^{23} + 413 x^{22} - 2524 x^{21} + 3437 x^{20} + 6311 x^{19} - 13555 x^{18} - 573 x^{17} + 14501 x^{16} - 6805 x^{15} - 2479 x^{14} + 5007 x^{13} - 7594 x^{12} - 344 x^{11} + 8602 x^{10} - 626 x^{9} - 5041 x^{8} + 15 x^{7} + 1789 x^{6} + 121 x^{5} - 352 x^{4} - 33 x^{3} + 32 x^{2} + 2 x - 1$ $C_{11}\times S_3$ (as 33T2) $0$ $0$
* 2.529.33t2.a.g$2$ $ 23^{2}$ $x^{33} - 3 x^{32} - 3 x^{31} + 28 x^{30} - 58 x^{29} + 22 x^{28} + 191 x^{27} - 526 x^{26} + 330 x^{25} + 728 x^{24} - 1012 x^{23} + 413 x^{22} - 2524 x^{21} + 3437 x^{20} + 6311 x^{19} - 13555 x^{18} - 573 x^{17} + 14501 x^{16} - 6805 x^{15} - 2479 x^{14} + 5007 x^{13} - 7594 x^{12} - 344 x^{11} + 8602 x^{10} - 626 x^{9} - 5041 x^{8} + 15 x^{7} + 1789 x^{6} + 121 x^{5} - 352 x^{4} - 33 x^{3} + 32 x^{2} + 2 x - 1$ $C_{11}\times S_3$ (as 33T2) $0$ $0$
* 2.529.33t2.a.h$2$ $ 23^{2}$ $x^{33} - 3 x^{32} - 3 x^{31} + 28 x^{30} - 58 x^{29} + 22 x^{28} + 191 x^{27} - 526 x^{26} + 330 x^{25} + 728 x^{24} - 1012 x^{23} + 413 x^{22} - 2524 x^{21} + 3437 x^{20} + 6311 x^{19} - 13555 x^{18} - 573 x^{17} + 14501 x^{16} - 6805 x^{15} - 2479 x^{14} + 5007 x^{13} - 7594 x^{12} - 344 x^{11} + 8602 x^{10} - 626 x^{9} - 5041 x^{8} + 15 x^{7} + 1789 x^{6} + 121 x^{5} - 352 x^{4} - 33 x^{3} + 32 x^{2} + 2 x - 1$ $C_{11}\times S_3$ (as 33T2) $0$ $0$
* 2.529.33t2.a.i$2$ $ 23^{2}$ $x^{33} - 3 x^{32} - 3 x^{31} + 28 x^{30} - 58 x^{29} + 22 x^{28} + 191 x^{27} - 526 x^{26} + 330 x^{25} + 728 x^{24} - 1012 x^{23} + 413 x^{22} - 2524 x^{21} + 3437 x^{20} + 6311 x^{19} - 13555 x^{18} - 573 x^{17} + 14501 x^{16} - 6805 x^{15} - 2479 x^{14} + 5007 x^{13} - 7594 x^{12} - 344 x^{11} + 8602 x^{10} - 626 x^{9} - 5041 x^{8} + 15 x^{7} + 1789 x^{6} + 121 x^{5} - 352 x^{4} - 33 x^{3} + 32 x^{2} + 2 x - 1$ $C_{11}\times S_3$ (as 33T2) $0$ $0$
* 2.529.33t2.a.j$2$ $ 23^{2}$ $x^{33} - 3 x^{32} - 3 x^{31} + 28 x^{30} - 58 x^{29} + 22 x^{28} + 191 x^{27} - 526 x^{26} + 330 x^{25} + 728 x^{24} - 1012 x^{23} + 413 x^{22} - 2524 x^{21} + 3437 x^{20} + 6311 x^{19} - 13555 x^{18} - 573 x^{17} + 14501 x^{16} - 6805 x^{15} - 2479 x^{14} + 5007 x^{13} - 7594 x^{12} - 344 x^{11} + 8602 x^{10} - 626 x^{9} - 5041 x^{8} + 15 x^{7} + 1789 x^{6} + 121 x^{5} - 352 x^{4} - 33 x^{3} + 32 x^{2} + 2 x - 1$ $C_{11}\times S_3$ (as 33T2) $0$ $0$

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 *.