Properties

Label 29.1.506...201.1
Degree $29$
Signature $[1, 14]$
Discriminant $5.069\times 10^{49}$
Root discriminant $51.76$
Ramified primes $53, 67$
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 - 11*x^28 + 45*x^27 - 70*x^26 + 66*x^25 - 800*x^24 + 4130*x^23 - 11632*x^22 + 26627*x^21 - 43970*x^20 + 56823*x^19 - 114622*x^18 + 216543*x^17 - 400317*x^16 + 743290*x^15 - 1619073*x^14 + 3038554*x^13 - 4110770*x^12 + 3834768*x^11 - 1791942*x^10 - 198548*x^9 + 1096532*x^8 - 913725*x^7 + 101442*x^6 + 212195*x^5 - 352802*x^4 + 186129*x^3 - 93243*x^2 + 23048*x - 4757)
 
gp: K = bnfinit(x^29 - 11*x^28 + 45*x^27 - 70*x^26 + 66*x^25 - 800*x^24 + 4130*x^23 - 11632*x^22 + 26627*x^21 - 43970*x^20 + 56823*x^19 - 114622*x^18 + 216543*x^17 - 400317*x^16 + 743290*x^15 - 1619073*x^14 + 3038554*x^13 - 4110770*x^12 + 3834768*x^11 - 1791942*x^10 - 198548*x^9 + 1096532*x^8 - 913725*x^7 + 101442*x^6 + 212195*x^5 - 352802*x^4 + 186129*x^3 - 93243*x^2 + 23048*x - 4757, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4757, 23048, -93243, 186129, -352802, 212195, 101442, -913725, 1096532, -198548, -1791942, 3834768, -4110770, 3038554, -1619073, 743290, -400317, 216543, -114622, 56823, -43970, 26627, -11632, 4130, -800, 66, -70, 45, -11, 1]);
 

\( x^{29} - 11 x^{28} + 45 x^{27} - 70 x^{26} + 66 x^{25} - 800 x^{24} + 4130 x^{23} - 11632 x^{22} + 26627 x^{21} - 43970 x^{20} + 56823 x^{19} - 114622 x^{18} + 216543 x^{17} - 400317 x^{16} + 743290 x^{15} - 1619073 x^{14} + 3038554 x^{13} - 4110770 x^{12} + 3834768 x^{11} - 1791942 x^{10} - 198548 x^{9} + 1096532 x^{8} - 913725 x^{7} + 101442 x^{6} + 212195 x^{5} - 352802 x^{4} + 186129 x^{3} - 93243 x^{2} + 23048 x - 4757 \)

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:  \(50688496810580930109918950679078495889944103627201\)\(\medspace = 53^{14}\cdot 67^{14}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $51.76$
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:  $53, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{7} a^{23} - \frac{3}{7} a^{22} - \frac{3}{7} a^{21} + \frac{2}{7} a^{20} + \frac{2}{7} a^{17} - \frac{1}{7} a^{16} + \frac{1}{7} a^{14} - \frac{1}{7} a^{13} + \frac{1}{7} a^{12} - \frac{2}{7} a^{11} - \frac{1}{7} a^{10} + \frac{3}{7} a^{9} - \frac{2}{7} a^{6} - \frac{1}{7} a^{4} + \frac{3}{7} a^{3} - \frac{2}{7} a^{2} + \frac{2}{7} a - \frac{3}{7}$, $\frac{1}{7} a^{24} + \frac{2}{7} a^{22} - \frac{1}{7} a^{20} + \frac{2}{7} a^{18} - \frac{2}{7} a^{17} - \frac{3}{7} a^{16} + \frac{1}{7} a^{15} + \frac{2}{7} a^{14} - \frac{2}{7} a^{13} + \frac{1}{7} a^{12} + \frac{2}{7} a^{9} - \frac{2}{7} a^{7} + \frac{1}{7} a^{6} - \frac{1}{7} a^{5} + \frac{3}{7} a^{2} + \frac{3}{7} a - \frac{2}{7}$, $\frac{1}{91} a^{25} + \frac{1}{91} a^{24} - \frac{2}{91} a^{23} + \frac{18}{91} a^{21} + \frac{2}{7} a^{20} - \frac{12}{91} a^{19} - \frac{1}{7} a^{17} - \frac{33}{91} a^{16} - \frac{3}{7} a^{15} - \frac{4}{91} a^{14} - \frac{25}{91} a^{13} - \frac{10}{91} a^{12} + \frac{43}{91} a^{11} - \frac{1}{91} a^{10} - \frac{31}{91} a^{9} - \frac{9}{91} a^{8} + \frac{34}{91} a^{7} + \frac{29}{91} a^{6} - \frac{22}{91} a^{5} - \frac{17}{91} a^{4} - \frac{44}{91} a^{3} + \frac{6}{13} a^{2} - \frac{2}{13} a - \frac{11}{91}$, $\frac{1}{74347} a^{26} + \frac{358}{74347} a^{25} + \frac{303}{74347} a^{24} + \frac{3485}{74347} a^{23} - \frac{24513}{74347} a^{22} + \frac{15149}{74347} a^{21} - \frac{19954}{74347} a^{20} + \frac{184}{559} a^{19} - \frac{359}{5719} a^{18} + \frac{3373}{74347} a^{17} - \frac{1030}{74347} a^{16} + \frac{4295}{10621} a^{15} - \frac{28935}{74347} a^{14} - \frac{927}{74347} a^{13} - \frac{8298}{74347} a^{12} - \frac{6152}{74347} a^{11} - \frac{27}{91} a^{10} + \frac{2013}{5719} a^{9} + \frac{23211}{74347} a^{8} - \frac{2849}{10621} a^{7} - \frac{485}{74347} a^{6} - \frac{2664}{10621} a^{5} + \frac{18171}{74347} a^{4} + \frac{19226}{74347} a^{3} - \frac{3190}{10621} a^{2} - \frac{9689}{74347} a - \frac{14509}{74347}$, $\frac{1}{996918923} a^{27} + \frac{4663}{996918923} a^{26} + \frac{3490855}{996918923} a^{25} - \frac{43082161}{996918923} a^{24} + \frac{2104943}{996918923} a^{23} - \frac{7536941}{43344301} a^{22} - \frac{246830515}{996918923} a^{21} - \frac{403148010}{996918923} a^{20} - \frac{47929260}{996918923} a^{19} + \frac{55703394}{996918923} a^{18} + \frac{8571183}{43344301} a^{17} - \frac{220507121}{996918923} a^{16} + \frac{44619298}{90628993} a^{15} + \frac{482241857}{996918923} a^{14} + \frac{193852}{995923} a^{13} - \frac{41896354}{90628993} a^{12} - \frac{104877144}{996918923} a^{11} - \frac{494664880}{996918923} a^{10} + \frac{8630087}{996918923} a^{9} - \frac{231013451}{996918923} a^{8} - \frac{303019419}{996918923} a^{7} - \frac{425697380}{996918923} a^{6} + \frac{208387082}{996918923} a^{5} + \frac{1029263}{43344301} a^{4} + \frac{351523894}{996918923} a^{3} + \frac{3246544}{142416989} a^{2} - \frac{42347391}{996918923} a - \frac{1697687}{996918923}$, $\frac{1}{887444673614865564455515702910011971603537525489245118254531434181984440340375285220647} a^{28} - \frac{89772279035935884643974005082234165133634478088090913571588466338534249009336}{887444673614865564455515702910011971603537525489245118254531434181984440340375285220647} a^{27} - \frac{3246184359876809695792655663577476916402996781540919926722471498298287201386381928}{887444673614865564455515702910011971603537525489245118254531434181984440340375285220647} a^{26} - \frac{2602549737731727582684578139391354179853263828441493711184386894572102712910514743707}{887444673614865564455515702910011971603537525489245118254531434181984440340375285220647} a^{25} + \frac{38971513456737912475089424060860595101383160617433314805048246585238904509227031200110}{887444673614865564455515702910011971603537525489245118254531434181984440340375285220647} a^{24} - \frac{1182236315084952964568138416703867774684624863304271501833575755685481130920473042646}{68264974893451197265808900223847074738733655806865009096502418013998803103105791170819} a^{23} + \frac{313207075300701158163101836186050686963199816365820976485544721161300651499002560991232}{887444673614865564455515702910011971603537525489245118254531434181984440340375285220647} a^{22} - \frac{1768327262144077636251617761311971656360279942659627419856826083253291950046483700866}{11525255501491760577344359778052103527318669162197988548760148495869927796628250457411} a^{21} + \frac{18785771251107037121615897684466821311836096302516254725704507468454804690953990747186}{38584551026733285411109378387391824852327718499532396445849192790521062623494577618289} a^{20} + \frac{279024443366417307535254427936025806402476861199848076301108122941810821666990945095353}{887444673614865564455515702910011971603537525489245118254531434181984440340375285220647} a^{19} - \frac{303541910365548991149150798629637186675169823292610766257138289168482000564404886860279}{887444673614865564455515702910011971603537525489245118254531434181984440340375285220647} a^{18} + \frac{254518503338091819036111513349422499183135774728182531823982622095554129866318150136605}{887444673614865564455515702910011971603537525489245118254531434181984440340375285220647} a^{17} + \frac{119680149712989403691670372877928482797962153975190655911208651671719487177803507697371}{887444673614865564455515702910011971603537525489245118254531434181984440340375285220647} a^{16} + \frac{51009960481350179663848943303402200557752429057348238458642195969823989938170600468445}{887444673614865564455515702910011971603537525489245118254531434181984440340375285220647} a^{15} - \frac{422789435480419662481467473655092480199382767054579622760763327731403973907287902938850}{887444673614865564455515702910011971603537525489245118254531434181984440340375285220647} a^{14} - \frac{13356945230653430263541885898263524345960772266689696261847263364830323166465909482595}{80676788510442324041410518446364724691230684135385919841321039471089494576397753201877} a^{13} + \frac{293799541274320484676704601166110798847752079817131778682955505536437646751214522445709}{887444673614865564455515702910011971603537525489245118254531434181984440340375285220647} a^{12} + \frac{89714920109667422597006676519748377517117881155228846854371027635728090895865197935284}{887444673614865564455515702910011971603537525489245118254531434181984440340375285220647} a^{11} + \frac{25573358768688088593437596202578946085767995450445123011570565437368449079781970242269}{68264974893451197265808900223847074738733655806865009096502418013998803103105791170819} a^{10} + \frac{5386156353042649443663323489002396649795093531351893720534541939885914841604797377902}{68264974893451197265808900223847074738733655806865009096502418013998803103105791170819} a^{9} + \frac{11229490092361779976327095332919928177773494211406989538693822231352586382691860085821}{68264974893451197265808900223847074738733655806865009096502418013998803103105791170819} a^{8} + \frac{400537379119074926200051350725333756023502871453270112700117159088488774768520358341390}{887444673614865564455515702910011971603537525489245118254531434181984440340375285220647} a^{7} + \frac{340665648113497595503196298177864534425355797872712963264523742537393695463336302764351}{887444673614865564455515702910011971603537525489245118254531434181984440340375285220647} a^{6} + \frac{85539236089468151801933377913726663293722390716574639365044774221564003282427484506104}{887444673614865564455515702910011971603537525489245118254531434181984440340375285220647} a^{5} - \frac{561784233676944867629591951391794480904157768482624138118377183294441548989997776927}{2392034160686969176429961463369304505669912467625997623327577989708852938922844434557} a^{4} - \frac{84267622569244538634090384702872046434139479088702303205294278748104768536602211602054}{887444673614865564455515702910011971603537525489245118254531434181984440340375285220647} a^{3} - \frac{229107525731110242693741538868296018702338518952569597320393206085718664895365900265045}{887444673614865564455515702910011971603537525489245118254531434181984440340375285220647} a^{2} + \frac{5382652169611336977853596901757234231444022318453390119783003451088448910116941755298}{13245442889774112902321129894179283158261754111779777884395991554954991646871272913741} a - \frac{1715286474537772894826301150655390486836445181546692090955943555039075867899746791823}{13245442889774112902321129894179283158261754111779777884395991554954991646871272913741}$

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:  \( 21909985755609.77 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{14}\cdot 21909985755609.77 \cdot 1}{2\sqrt{50688496810580930109918950679078495889944103627201}}\approx 0.459944623028265$ (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 $29$ $29$ $29$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ $29$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $29$ $29$ $29$ $29$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $29$ R $29$

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
$53$$\Q_{53}$$x + 2$$1$$1$$0$Trivial$[\ ]$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
$67$$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.3551.2t1.a.a$1$ $ 53 \cdot 67 $ \(\Q(\sqrt{-3551}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.3551.29t2.a.i$2$ $ 53 \cdot 67 $ 29.1.50688496810580930109918950679078495889944103627201.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3551.29t2.a.c$2$ $ 53 \cdot 67 $ 29.1.50688496810580930109918950679078495889944103627201.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3551.29t2.a.b$2$ $ 53 \cdot 67 $ 29.1.50688496810580930109918950679078495889944103627201.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3551.29t2.a.d$2$ $ 53 \cdot 67 $ 29.1.50688496810580930109918950679078495889944103627201.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3551.29t2.a.h$2$ $ 53 \cdot 67 $ 29.1.50688496810580930109918950679078495889944103627201.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3551.29t2.a.e$2$ $ 53 \cdot 67 $ 29.1.50688496810580930109918950679078495889944103627201.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3551.29t2.a.j$2$ $ 53 \cdot 67 $ 29.1.50688496810580930109918950679078495889944103627201.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3551.29t2.a.m$2$ $ 53 \cdot 67 $ 29.1.50688496810580930109918950679078495889944103627201.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3551.29t2.a.a$2$ $ 53 \cdot 67 $ 29.1.50688496810580930109918950679078495889944103627201.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3551.29t2.a.n$2$ $ 53 \cdot 67 $ 29.1.50688496810580930109918950679078495889944103627201.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3551.29t2.a.f$2$ $ 53 \cdot 67 $ 29.1.50688496810580930109918950679078495889944103627201.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3551.29t2.a.k$2$ $ 53 \cdot 67 $ 29.1.50688496810580930109918950679078495889944103627201.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3551.29t2.a.g$2$ $ 53 \cdot 67 $ 29.1.50688496810580930109918950679078495889944103627201.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3551.29t2.a.l$2$ $ 53 \cdot 67 $ 29.1.50688496810580930109918950679078495889944103627201.1 $D_{29}$ (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 *.