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Properties

Label 29.1.574...129.1
Degree $29$
Signature $[1, 14]$
Discriminant $5.747\times 10^{49}$
Root discriminant $51.98$
Ramified prime $3583$
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 - x^28 + 18*x^27 + 12*x^26 + 179*x^25 + 123*x^24 + 1662*x^23 - 196*x^22 + 11119*x^21 - 9104*x^20 + 47466*x^19 - 44547*x^18 + 83114*x^17 - 23886*x^16 - 121329*x^15 + 248495*x^14 - 275546*x^13 - 188738*x^12 + 573921*x^11 - 476142*x^10 + 379865*x^9 - 32019*x^8 + 20658*x^7 - 6163*x^6 - 27930*x^5 + 2012*x^4 - 1404*x^3 + 2898*x^2 - 594*x + 81)
 
gp: K = bnfinit(x^29 - x^28 + 18*x^27 + 12*x^26 + 179*x^25 + 123*x^24 + 1662*x^23 - 196*x^22 + 11119*x^21 - 9104*x^20 + 47466*x^19 - 44547*x^18 + 83114*x^17 - 23886*x^16 - 121329*x^15 + 248495*x^14 - 275546*x^13 - 188738*x^12 + 573921*x^11 - 476142*x^10 + 379865*x^9 - 32019*x^8 + 20658*x^7 - 6163*x^6 - 27930*x^5 + 2012*x^4 - 1404*x^3 + 2898*x^2 - 594*x + 81, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, -594, 2898, -1404, 2012, -27930, -6163, 20658, -32019, 379865, -476142, 573921, -188738, -275546, 248495, -121329, -23886, 83114, -44547, 47466, -9104, 11119, -196, 1662, 123, 179, 12, 18, -1, 1]);
 

\( x^{29} - x^{28} + 18 x^{27} + 12 x^{26} + 179 x^{25} + 123 x^{24} + 1662 x^{23} - 196 x^{22} + 11119 x^{21} - 9104 x^{20} + 47466 x^{19} - 44547 x^{18} + 83114 x^{17} - 23886 x^{16} - 121329 x^{15} + 248495 x^{14} - 275546 x^{13} - 188738 x^{12} + 573921 x^{11} - 476142 x^{10} + 379865 x^{9} - 32019 x^{8} + 20658 x^{7} - 6163 x^{6} - 27930 x^{5} + 2012 x^{4} - 1404 x^{3} + 2898 x^{2} - 594 x + 81 \)

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:  \(57471868152223727924656865491755923007187161136129\)\(\medspace = 3583^{14}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $51.98$
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:  $3583$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{6}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{7}$, $\frac{1}{15} a^{16} + \frac{1}{15} a^{15} + \frac{1}{15} a^{14} + \frac{2}{15} a^{13} + \frac{2}{15} a^{12} + \frac{1}{15} a^{11} + \frac{2}{15} a^{10} + \frac{1}{15} a^{9} - \frac{4}{15} a^{8} - \frac{1}{15} a^{7} + \frac{1}{3} a^{6} - \frac{2}{15} a^{5} + \frac{7}{15} a^{4} - \frac{1}{15} a^{3} + \frac{4}{15} a^{2} - \frac{1}{15} a - \frac{1}{5}$, $\frac{1}{15} a^{17} + \frac{1}{15} a^{14} - \frac{1}{15} a^{12} + \frac{1}{15} a^{11} - \frac{1}{15} a^{10} + \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{7}{15} a^{6} - \frac{2}{5} a^{5} + \frac{7}{15} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{7}{15} a + \frac{1}{5}$, $\frac{1}{45} a^{18} + \frac{2}{15} a^{15} - \frac{2}{15} a^{13} + \frac{2}{15} a^{12} - \frac{2}{15} a^{11} - \frac{1}{9} a^{10} + \frac{1}{15} a^{9} - \frac{1}{5} a^{8} + \frac{1}{15} a^{7} + \frac{1}{5} a^{6} - \frac{1}{15} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{13}{45} a^{2} - \frac{4}{15} a$, $\frac{1}{45} a^{19} - \frac{2}{15} a^{15} + \frac{1}{15} a^{14} - \frac{2}{15} a^{13} - \frac{1}{15} a^{12} + \frac{4}{45} a^{11} + \frac{2}{15} a^{10} - \frac{2}{5} a^{8} + \frac{1}{3} a^{7} - \frac{1}{15} a^{6} - \frac{1}{15} a^{5} + \frac{1}{15} a^{4} + \frac{4}{45} a^{3} - \frac{2}{15} a^{2} - \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{45} a^{20} - \frac{2}{15} a^{15} - \frac{2}{15} a^{13} + \frac{1}{45} a^{12} - \frac{1}{15} a^{11} - \frac{1}{15} a^{10} + \frac{1}{15} a^{9} - \frac{1}{5} a^{8} + \frac{2}{15} a^{7} - \frac{2}{5} a^{6} + \frac{2}{15} a^{5} + \frac{16}{45} a^{4} + \frac{1}{15} a^{3} - \frac{1}{3} a^{2} - \frac{1}{15} a - \frac{2}{5}$, $\frac{1}{45} a^{21} + \frac{2}{15} a^{15} - \frac{2}{45} a^{13} - \frac{2}{15} a^{12} + \frac{1}{15} a^{11} - \frac{1}{15} a^{9} - \frac{2}{5} a^{8} + \frac{7}{15} a^{7} - \frac{1}{5} a^{6} + \frac{19}{45} a^{5} + \frac{1}{3} a^{4} - \frac{7}{15} a^{3} - \frac{1}{5} a^{2} + \frac{7}{15} a - \frac{2}{5}$, $\frac{1}{45} a^{22} - \frac{2}{15} a^{15} + \frac{7}{45} a^{14} - \frac{1}{15} a^{13} + \frac{2}{15} a^{12} - \frac{2}{15} a^{11} + \frac{2}{15} a^{9} - \frac{1}{15} a^{7} + \frac{19}{45} a^{6} + \frac{4}{15} a^{5} + \frac{4}{15} a^{4} - \frac{1}{15} a^{3} - \frac{2}{5} a^{2} + \frac{1}{15} a + \frac{2}{5}$, $\frac{1}{135} a^{23} + \frac{1}{135} a^{22} + \frac{1}{135} a^{21} - \frac{1}{135} a^{20} + \frac{1}{135} a^{19} + \frac{1}{45} a^{17} + \frac{1}{45} a^{16} - \frac{14}{135} a^{15} + \frac{19}{135} a^{14} - \frac{11}{135} a^{13} + \frac{4}{27} a^{12} + \frac{16}{135} a^{11} - \frac{1}{9} a^{10} - \frac{7}{45} a^{9} - \frac{19}{45} a^{8} + \frac{8}{27} a^{7} + \frac{61}{135} a^{6} - \frac{17}{135} a^{5} + \frac{7}{27} a^{4} + \frac{2}{27} a^{3} - \frac{13}{45} a^{2} - \frac{1}{15} a - \frac{1}{5}$, $\frac{1}{2025} a^{24} + \frac{1}{675} a^{23} + \frac{4}{675} a^{22} + \frac{1}{2025} a^{21} - \frac{22}{2025} a^{20} - \frac{7}{2025} a^{19} + \frac{2}{675} a^{18} - \frac{1}{225} a^{17} + \frac{46}{2025} a^{16} - \frac{29}{225} a^{15} - \frac{4}{225} a^{14} + \frac{61}{2025} a^{13} + \frac{43}{405} a^{12} + \frac{278}{2025} a^{11} - \frac{112}{675} a^{10} - \frac{7}{45} a^{9} + \frac{37}{81} a^{8} + \frac{46}{135} a^{7} - \frac{307}{675} a^{6} + \frac{181}{2025} a^{5} + \frac{77}{2025} a^{4} - \frac{379}{2025} a^{3} + \frac{22}{135} a^{2} - \frac{34}{75} a + \frac{14}{75}$, $\frac{1}{6075} a^{25} + \frac{2}{675} a^{23} - \frac{4}{1215} a^{22} - \frac{2}{1215} a^{21} - \frac{1}{6075} a^{20} - \frac{1}{2025} a^{19} + \frac{7}{675} a^{18} - \frac{152}{6075} a^{17} - \frac{28}{2025} a^{16} - \frac{271}{2025} a^{15} - \frac{356}{6075} a^{14} + \frac{407}{6075} a^{13} + \frac{23}{6075} a^{12} - \frac{11}{405} a^{11} - \frac{73}{675} a^{10} - \frac{148}{1215} a^{9} - \frac{5}{81} a^{8} - \frac{77}{2025} a^{7} - \frac{596}{6075} a^{6} + \frac{1979}{6075} a^{5} + \frac{217}{1215} a^{4} - \frac{106}{2025} a^{3} - \frac{8}{75} a^{2} + \frac{101}{225} a + \frac{12}{25}$, $\frac{1}{1038825} a^{26} + \frac{7}{207765} a^{25} + \frac{64}{346275} a^{24} + \frac{2257}{1038825} a^{23} + \frac{5203}{1038825} a^{22} - \frac{2969}{346275} a^{21} - \frac{1616}{1038825} a^{20} - \frac{1}{95} a^{19} + \frac{568}{54675} a^{18} - \frac{2582}{207765} a^{17} + \frac{11452}{346275} a^{16} + \frac{3794}{207765} a^{15} - \frac{136802}{1038825} a^{14} - \frac{7756}{346275} a^{13} + \frac{271}{2187} a^{12} - \frac{458}{7695} a^{11} - \frac{68699}{1038825} a^{10} - \frac{29771}{207765} a^{9} - \frac{86387}{346275} a^{8} - \frac{399686}{1038825} a^{7} - \frac{61981}{207765} a^{6} + \frac{2516}{115425} a^{5} - \frac{35833}{207765} a^{4} - \frac{181}{1425} a^{3} - \frac{1873}{6075} a^{2} - \frac{2108}{38475} a - \frac{598}{12825}$, $\frac{1}{3116475} a^{27} + \frac{1}{3116475} a^{26} + \frac{28}{3116475} a^{25} - \frac{167}{3116475} a^{24} - \frac{8}{10935} a^{23} - \frac{26266}{3116475} a^{22} + \frac{33436}{3116475} a^{21} + \frac{1312}{623295} a^{20} - \frac{26279}{3116475} a^{19} + \frac{32614}{3116475} a^{18} - \frac{88952}{3116475} a^{17} + \frac{38461}{3116475} a^{16} + \frac{102097}{1038825} a^{15} + \frac{9161}{124659} a^{14} - \frac{382157}{3116475} a^{13} + \frac{445793}{3116475} a^{12} + \frac{157993}{3116475} a^{11} + \frac{72947}{623295} a^{10} + \frac{72049}{3116475} a^{9} - \frac{65537}{3116475} a^{8} - \frac{96683}{346275} a^{7} + \frac{158339}{3116475} a^{6} + \frac{1147057}{3116475} a^{5} - \frac{1421896}{3116475} a^{4} + \frac{127271}{346275} a^{3} + \frac{18262}{346275} a^{2} + \frac{39098}{115425} a - \frac{3184}{7695}$, $\frac{1}{311114306734387642916404128248722589483923959792296489925} a^{28} + \frac{2080979614479176471838724174088017700289203956256}{103704768911462547638801376082907529827974653264098829975} a^{27} - \frac{28835628902313537514487788217070375071614732139914}{103704768911462547638801376082907529827974653264098829975} a^{26} - \frac{2242267982325825546573370654154949750061793586542863}{34568256303820849212933792027635843275991551088032943325} a^{25} + \frac{1617400211422543727423588492367322147023620075355281}{16374437196546718048231796223616978393890734725910341575} a^{24} - \frac{835884936172259402989269869819162406570009736045590472}{311114306734387642916404128248722589483923959792296489925} a^{23} - \frac{2641052221601323946663054519322377775918351820740411913}{311114306734387642916404128248722589483923959792296489925} a^{22} + \frac{938981111298739540752695591030743108992418802435019433}{311114306734387642916404128248722589483923959792296489925} a^{21} - \frac{128797600524548157123377398971170517574400033691951617}{62222861346877528583280825649744517896784791958459297985} a^{20} + \frac{219339389911764306739768521569293538436569536347266996}{20740953782292509527760275216581505965594930652819765995} a^{19} + \frac{521707711485952428641233906018624780479342564499145242}{103704768911462547638801376082907529827974653264098829975} a^{18} - \frac{1695615901675732349521351202665276909495109046304122883}{103704768911462547638801376082907529827974653264098829975} a^{17} + \frac{8311416130569545592755790616827882650786763330345219407}{311114306734387642916404128248722589483923959792296489925} a^{16} - \frac{40859504723004769009834818587061497343414174312480162592}{311114306734387642916404128248722589483923959792296489925} a^{15} - \frac{243389569735337309630511608123256542843481517366108699}{10035945378528633642464649298345889983352385799751499675} a^{14} + \frac{1929050082498633626986377248956237974189919249073483566}{12444572269375505716656165129948903579356958391691859597} a^{13} - \frac{2791326725836421760438084384271081134580569550141542649}{311114306734387642916404128248722589483923959792296489925} a^{12} - \frac{4776946970318572119829239901750356832493270263199423237}{34568256303820849212933792027635843275991551088032943325} a^{11} - \frac{4158833950457136241625799291511828546246253980920328027}{34568256303820849212933792027635843275991551088032943325} a^{10} + \frac{10840803174636016728633496054202030640344497360566764782}{103704768911462547638801376082907529827974653264098829975} a^{9} - \frac{131706670828140004721436907908084382570970349071974072864}{311114306734387642916404128248722589483923959792296489925} a^{8} + \frac{153843972573105240105566803888743049131168343578867522533}{311114306734387642916404128248722589483923959792296489925} a^{7} - \frac{86173528626907681353202774526977525405317500490644368531}{311114306734387642916404128248722589483923959792296489925} a^{6} + \frac{4110054301736820856618573446667788330083617364369058058}{311114306734387642916404128248722589483923959792296489925} a^{5} + \frac{683067289487347769492739857827526517694369012167056074}{16374437196546718048231796223616978393890734725910341575} a^{4} - \frac{10672612389515614026494747930698819333825046833715809639}{34568256303820849212933792027635843275991551088032943325} a^{3} - \frac{97366878692602579747314594596228849997229165772367262}{803912937298159284021716093665949843627710490419370775} a^{2} + \frac{5675419338422537288140584882524360573806179671865217393}{11522752101273616404311264009211947758663850362677647775} a - \frac{1677731407353694185361918414348265779687167889287169969}{3840917367091205468103754669737315919554616787559215925}$

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:  \( 133207857013622.31 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{14}\cdot 133207857013622.31 \cdot 1}{2\sqrt{57471868152223727924656865491755923007187161136129}}\approx 2.62615491650965$ (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 $29$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ $29$ $29$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ $29$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $29$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $29$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $29$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ $29$

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
3583Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.3583.2t1.a.a$1$ $ 3583 $ \(\Q(\sqrt{-3583}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.3583.29t2.a.e$2$ $ 3583 $ 29.1.57471868152223727924656865491755923007187161136129.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3583.29t2.a.f$2$ $ 3583 $ 29.1.57471868152223727924656865491755923007187161136129.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3583.29t2.a.c$2$ $ 3583 $ 29.1.57471868152223727924656865491755923007187161136129.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3583.29t2.a.n$2$ $ 3583 $ 29.1.57471868152223727924656865491755923007187161136129.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3583.29t2.a.m$2$ $ 3583 $ 29.1.57471868152223727924656865491755923007187161136129.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3583.29t2.a.h$2$ $ 3583 $ 29.1.57471868152223727924656865491755923007187161136129.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3583.29t2.a.d$2$ $ 3583 $ 29.1.57471868152223727924656865491755923007187161136129.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3583.29t2.a.b$2$ $ 3583 $ 29.1.57471868152223727924656865491755923007187161136129.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3583.29t2.a.k$2$ $ 3583 $ 29.1.57471868152223727924656865491755923007187161136129.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3583.29t2.a.g$2$ $ 3583 $ 29.1.57471868152223727924656865491755923007187161136129.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3583.29t2.a.a$2$ $ 3583 $ 29.1.57471868152223727924656865491755923007187161136129.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3583.29t2.a.l$2$ $ 3583 $ 29.1.57471868152223727924656865491755923007187161136129.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3583.29t2.a.i$2$ $ 3583 $ 29.1.57471868152223727924656865491755923007187161136129.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3583.29t2.a.j$2$ $ 3583 $ 29.1.57471868152223727924656865491755923007187161136129.1 $D_{29}$ (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 *.