Properties

Label 29.1.190...729.1
Degree $29$
Signature $[1, 14]$
Discriminant $1.904\times 10^{47}$
Root discriminant $42.69$
Ramified prime $2383$
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 - 8*x^28 + 42*x^27 - 91*x^26 + 71*x^25 - 172*x^24 + 980*x^23 - 2477*x^22 + 4032*x^21 - 5998*x^20 + 8995*x^19 - 12396*x^18 + 15464*x^17 - 17477*x^16 + 17661*x^15 - 16777*x^14 + 14899*x^13 - 10749*x^12 + 6197*x^11 - 2594*x^10 - 1233*x^9 + 2318*x^8 - 113*x^7 - 855*x^6 + 1459*x^5 - 2625*x^4 + 2079*x^3 - 978*x^2 + 419*x + 1)
 
gp: K = bnfinit(x^29 - 8*x^28 + 42*x^27 - 91*x^26 + 71*x^25 - 172*x^24 + 980*x^23 - 2477*x^22 + 4032*x^21 - 5998*x^20 + 8995*x^19 - 12396*x^18 + 15464*x^17 - 17477*x^16 + 17661*x^15 - 16777*x^14 + 14899*x^13 - 10749*x^12 + 6197*x^11 - 2594*x^10 - 1233*x^9 + 2318*x^8 - 113*x^7 - 855*x^6 + 1459*x^5 - 2625*x^4 + 2079*x^3 - 978*x^2 + 419*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 419, -978, 2079, -2625, 1459, -855, -113, 2318, -1233, -2594, 6197, -10749, 14899, -16777, 17661, -17477, 15464, -12396, 8995, -5998, 4032, -2477, 980, -172, 71, -91, 42, -8, 1]);
 

\( x^{29} - 8 x^{28} + 42 x^{27} - 91 x^{26} + 71 x^{25} - 172 x^{24} + 980 x^{23} - 2477 x^{22} + 4032 x^{21} - 5998 x^{20} + 8995 x^{19} - 12396 x^{18} + 15464 x^{17} - 17477 x^{16} + 17661 x^{15} - 16777 x^{14} + 14899 x^{13} - 10749 x^{12} + 6197 x^{11} - 2594 x^{10} - 1233 x^{9} + 2318 x^{8} - 113 x^{7} - 855 x^{6} + 1459 x^{5} - 2625 x^{4} + 2079 x^{3} - 978 x^{2} + 419 x + 1 \)

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:  \(190430537333205962921320156798047381287357677729\)\(\medspace = 2383^{14}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $42.69$
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:  $2383$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{6}$, $\frac{1}{9} a^{15} + \frac{1}{9} a^{14} + \frac{1}{9} a^{13} + \frac{1}{9} a^{12} + \frac{1}{9} a^{11} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{9} a^{8} - \frac{4}{9} a^{7} - \frac{4}{9} a^{6} - \frac{4}{9} a^{5} - \frac{4}{9} a^{4} - \frac{4}{9} a^{3} - \frac{4}{9} a^{2} - \frac{4}{9} a - \frac{4}{9}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{8} - \frac{2}{9}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{9} - \frac{2}{9} a$, $\frac{1}{9} a^{18} + \frac{1}{9} a^{10} - \frac{2}{9} a^{2}$, $\frac{1}{9} a^{19} + \frac{1}{9} a^{11} - \frac{2}{9} a^{3}$, $\frac{1}{45} a^{20} - \frac{1}{45} a^{19} + \frac{2}{45} a^{18} + \frac{2}{45} a^{17} - \frac{1}{45} a^{16} - \frac{1}{45} a^{15} - \frac{7}{45} a^{14} - \frac{1}{45} a^{13} - \frac{1}{9} a^{11} + \frac{7}{45} a^{10} + \frac{4}{45} a^{9} - \frac{2}{45} a^{8} - \frac{14}{45} a^{7} + \frac{2}{9} a^{6} + \frac{22}{45} a^{5} + \frac{4}{9} a^{4} - \frac{1}{5} a^{3} - \frac{2}{15} a^{2} + \frac{1}{3} a - \frac{7}{15}$, $\frac{1}{45} a^{21} + \frac{1}{45} a^{19} - \frac{1}{45} a^{18} + \frac{1}{45} a^{17} - \frac{2}{45} a^{16} + \frac{2}{45} a^{15} + \frac{2}{45} a^{14} - \frac{2}{15} a^{13} + \frac{1}{9} a^{12} - \frac{1}{15} a^{11} + \frac{1}{45} a^{10} - \frac{1}{15} a^{9} - \frac{2}{15} a^{8} + \frac{1}{45} a^{7} - \frac{8}{45} a^{6} + \frac{17}{45} a^{5} + \frac{16}{45} a^{4} + \frac{1}{9} a^{3} - \frac{2}{15} a^{2} + \frac{14}{45} a - \frac{16}{45}$, $\frac{1}{135} a^{22} - \frac{1}{135} a^{21} + \frac{4}{135} a^{19} + \frac{1}{27} a^{18} - \frac{1}{27} a^{16} - \frac{4}{135} a^{15} + \frac{1}{15} a^{14} - \frac{8}{135} a^{13} - \frac{13}{135} a^{12} - \frac{2}{45} a^{11} - \frac{11}{135} a^{10} - \frac{22}{135} a^{9} - \frac{2}{45} a^{8} - \frac{13}{27} a^{7} - \frac{5}{27} a^{6} - \frac{11}{45} a^{5} - \frac{56}{135} a^{4} - \frac{67}{135} a^{3} - \frac{1}{15} a^{2} - \frac{4}{27} a - \frac{58}{135}$, $\frac{1}{135} a^{23} - \frac{1}{135} a^{21} + \frac{1}{135} a^{20} - \frac{1}{45} a^{19} - \frac{1}{135} a^{18} + \frac{4}{135} a^{17} - \frac{2}{45} a^{16} - \frac{7}{135} a^{15} + \frac{7}{135} a^{14} + \frac{4}{45} a^{13} + \frac{11}{135} a^{12} + \frac{13}{135} a^{11} + \frac{7}{45} a^{10} + \frac{1}{27} a^{9} + \frac{2}{27} a^{8} + \frac{4}{45} a^{7} - \frac{28}{135} a^{6} - \frac{1}{27} a^{5} - \frac{11}{45} a^{4} - \frac{4}{135} a^{3} - \frac{41}{135} a^{2} - \frac{1}{45} a - \frac{5}{27}$, $\frac{1}{135} a^{24} + \frac{1}{9} a^{8} - \frac{16}{135}$, $\frac{1}{2295} a^{25} + \frac{2}{2295} a^{24} - \frac{7}{2295} a^{23} + \frac{2}{2295} a^{22} + \frac{8}{2295} a^{21} + \frac{2}{2295} a^{20} - \frac{22}{2295} a^{19} - \frac{43}{2295} a^{18} - \frac{52}{2295} a^{17} + \frac{122}{2295} a^{16} + \frac{53}{2295} a^{15} - \frac{298}{2295} a^{14} - \frac{67}{2295} a^{13} - \frac{28}{2295} a^{12} - \frac{7}{2295} a^{11} - \frac{73}{2295} a^{10} - \frac{112}{2295} a^{9} - \frac{103}{2295} a^{8} - \frac{1072}{2295} a^{7} - \frac{28}{2295} a^{6} + \frac{653}{2295} a^{5} + \frac{512}{2295} a^{4} - \frac{322}{2295} a^{3} - \frac{478}{2295} a^{2} - \frac{458}{2295} a - \frac{5}{153}$, $\frac{1}{20655} a^{26} - \frac{1}{6885} a^{25} - \frac{2}{1215} a^{24} + \frac{4}{4131} a^{23} - \frac{2}{20655} a^{22} + \frac{1}{255} a^{21} + \frac{2}{20655} a^{20} - \frac{341}{20655} a^{19} + \frac{7}{153} a^{18} + \frac{518}{20655} a^{17} + \frac{164}{4131} a^{16} - \frac{182}{6885} a^{15} - \frac{2164}{20655} a^{14} - \frac{622}{4131} a^{13} + \frac{1087}{6885} a^{12} - \frac{470}{4131} a^{11} - \frac{286}{4131} a^{10} + \frac{205}{1377} a^{9} + \frac{2639}{20655} a^{8} + \frac{10126}{20655} a^{7} + \frac{916}{6885} a^{6} + \frac{8093}{20655} a^{5} + \frac{5176}{20655} a^{4} + \frac{587}{6885} a^{3} + \frac{3496}{20655} a^{2} - \frac{2171}{20655} a + \frac{7039}{20655}$, $\frac{1}{119117385} a^{27} - \frac{2564}{119117385} a^{26} - \frac{22546}{119117385} a^{25} + \frac{163486}{119117385} a^{24} + \frac{53738}{39705795} a^{23} + \frac{400906}{119117385} a^{22} + \frac{1144649}{119117385} a^{21} - \frac{100477}{13235265} a^{20} + \frac{62698}{119117385} a^{19} + \frac{3106148}{119117385} a^{18} + \frac{1372672}{39705795} a^{17} + \frac{1246973}{23823477} a^{16} + \frac{444833}{119117385} a^{15} - \frac{2936167}{39705795} a^{14} - \frac{4276}{7006905} a^{13} - \frac{3153319}{119117385} a^{12} - \frac{1041382}{39705795} a^{11} - \frac{15334577}{119117385} a^{10} + \frac{1573253}{119117385} a^{9} - \frac{9031}{60435} a^{8} - \frac{4949653}{23823477} a^{7} - \frac{59524924}{119117385} a^{6} + \frac{6300764}{13235265} a^{5} + \frac{16270786}{119117385} a^{4} + \frac{2873897}{7006905} a^{3} - \frac{16947886}{119117385} a^{2} + \frac{2148520}{23823477} a + \frac{58725184}{119117385}$, $\frac{1}{249485287787876730958538085} a^{28} - \frac{12669999357061084}{27720587531986303439837565} a^{27} + \frac{1955010382106624234269}{83161762595958910319512695} a^{26} + \frac{10133047450293764679955}{49897057557575346191707617} a^{25} + \frac{226304621386182419966537}{83161762595958910319512695} a^{24} + \frac{920606886874985467171901}{249485287787876730958538085} a^{23} - \frac{17319393523191119475470}{5544117506397260687967513} a^{22} - \frac{120368892651015679005854}{249485287787876730958538085} a^{21} + \frac{1453400523682921380313481}{249485287787876730958538085} a^{20} - \frac{3730207808083354140846967}{83161762595958910319512695} a^{19} + \frac{26129358191281929732398}{1393772557474171681332615} a^{18} + \frac{2796364479861213752296766}{249485287787876730958538085} a^{17} - \frac{515616947887515660035258}{16632352519191782063902539} a^{16} + \frac{10815053086580310778040617}{249485287787876730958538085} a^{15} - \frac{38054509308166383014981119}{249485287787876730958538085} a^{14} - \frac{5019358150436410071781}{110148029928422397774189} a^{13} - \frac{20308792111813689394580144}{249485287787876730958538085} a^{12} + \frac{21814107273652042405982}{1182394728852496355253735} a^{11} + \frac{13497808965053574825476011}{83161762595958910319512695} a^{10} + \frac{19875513589401603476471989}{249485287787876730958538085} a^{9} - \frac{3976663776765842695889512}{249485287787876730958538085} a^{8} + \frac{38842611928869969847844728}{83161762595958910319512695} a^{7} + \frac{2440890337599319550160673}{6742845615888560296176705} a^{6} - \frac{114138405353724200534068576}{249485287787876730958538085} a^{5} - \frac{90564790424432587186039492}{249485287787876730958538085} a^{4} - \frac{193192605214265157060094}{14675605163992748879914005} a^{3} + \frac{17674841176971540557318908}{49897057557575346191707617} a^{2} - \frac{21115999905643669366708135}{49897057557575346191707617} a + \frac{16645904323499286496410652}{249485287787876730958538085}$

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:  \( 6346995452652.517 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{14}\cdot 6346995452652.517 \cdot 1}{2\sqrt{190430537333205962921320156798047381287357677729}}\approx 2.17379230843342$ (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 $29$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ $29$ $29$ $29$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ $29$ $29$ $29$ $29$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $29$ $29$ $29$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ $29$

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
2383Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.2383.2t1.a.a$1$ $ 2383 $ \(\Q(\sqrt{-2383}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.2383.29t2.a.b$2$ $ 2383 $ 29.1.190430537333205962921320156798047381287357677729.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2383.29t2.a.l$2$ $ 2383 $ 29.1.190430537333205962921320156798047381287357677729.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2383.29t2.a.k$2$ $ 2383 $ 29.1.190430537333205962921320156798047381287357677729.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2383.29t2.a.e$2$ $ 2383 $ 29.1.190430537333205962921320156798047381287357677729.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2383.29t2.a.f$2$ $ 2383 $ 29.1.190430537333205962921320156798047381287357677729.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2383.29t2.a.c$2$ $ 2383 $ 29.1.190430537333205962921320156798047381287357677729.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2383.29t2.a.i$2$ $ 2383 $ 29.1.190430537333205962921320156798047381287357677729.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2383.29t2.a.a$2$ $ 2383 $ 29.1.190430537333205962921320156798047381287357677729.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2383.29t2.a.d$2$ $ 2383 $ 29.1.190430537333205962921320156798047381287357677729.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2383.29t2.a.h$2$ $ 2383 $ 29.1.190430537333205962921320156798047381287357677729.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2383.29t2.a.j$2$ $ 2383 $ 29.1.190430537333205962921320156798047381287357677729.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2383.29t2.a.n$2$ $ 2383 $ 29.1.190430537333205962921320156798047381287357677729.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2383.29t2.a.m$2$ $ 2383 $ 29.1.190430537333205962921320156798047381287357677729.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2383.29t2.a.g$2$ $ 2383 $ 29.1.190430537333205962921320156798047381287357677729.1 $D_{29}$ (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 *.