# Properties

 Label 29.29.1931816133...7041.1 Degree $29$ Signature $[29, 0]$ Discriminant $233^{28}$ Root discriminant $193.07$ Ramified prime $233$ Class number $1$ (GRH) Class group Trivial (GRH) Galois group $C_{29}$ (as 29T1)

# Learn more about

Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^29 - x^28 - 112*x^27 + 91*x^26 + 5198*x^25 - 3644*x^24 - 132219*x^23 + 83238*x^22 + 2053518*x^21 - 1187959*x^20 - 20553532*x^19 + 11071128*x^18 + 136460842*x^17 - 69042962*x^16 - 609473492*x^15 + 292259011*x^14 + 1836592125*x^13 - 845018358*x^12 - 3706016039*x^11 + 1661552324*x^10 + 4906886664*x^9 - 2177019390*x^8 - 4095369839*x^7 + 1819962089*x^6 + 1998032360*x^5 - 895362174*x^4 - 490947342*x^3 + 221892059*x^2 + 42927079*x - 19524467)

gp: K = bnfinit(x^29 - x^28 - 112*x^27 + 91*x^26 + 5198*x^25 - 3644*x^24 - 132219*x^23 + 83238*x^22 + 2053518*x^21 - 1187959*x^20 - 20553532*x^19 + 11071128*x^18 + 136460842*x^17 - 69042962*x^16 - 609473492*x^15 + 292259011*x^14 + 1836592125*x^13 - 845018358*x^12 - 3706016039*x^11 + 1661552324*x^10 + 4906886664*x^9 - 2177019390*x^8 - 4095369839*x^7 + 1819962089*x^6 + 1998032360*x^5 - 895362174*x^4 - 490947342*x^3 + 221892059*x^2 + 42927079*x - 19524467, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-19524467, 42927079, 221892059, -490947342, -895362174, 1998032360, 1819962089, -4095369839, -2177019390, 4906886664, 1661552324, -3706016039, -845018358, 1836592125, 292259011, -609473492, -69042962, 136460842, 11071128, -20553532, -1187959, 2053518, 83238, -132219, -3644, 5198, 91, -112, -1, 1]);

## Normalizeddefining polynomial

$$x^{29} - x^{28} - 112 x^{27} + 91 x^{26} + 5198 x^{25} - 3644 x^{24} - 132219 x^{23} + 83238 x^{22} + 2053518 x^{21} - 1187959 x^{20} - 20553532 x^{19} + 11071128 x^{18} + 136460842 x^{17} - 69042962 x^{16} - 609473492 x^{15} + 292259011 x^{14} + 1836592125 x^{13} - 845018358 x^{12} - 3706016039 x^{11} + 1661552324 x^{10} + 4906886664 x^{9} - 2177019390 x^{8} - 4095369839 x^{7} + 1819962089 x^{6} + 1998032360 x^{5} - 895362174 x^{4} - 490947342 x^{3} + 221892059 x^{2} + 42927079 x - 19524467$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $29$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[29, 0]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$1931816133436230496253440348173909042983780087233421135736506627041=233^{28}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $193.07$ 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: $233$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Gal(K/\Q)|$: $29$ This field is Galois and abelian over $\Q$. Conductor: $$233$$ Dirichlet character group: $\lbrace$$\chi_{233}(128,·), \chi_{233}(1,·), \chi_{233}(2,·), \chi_{233}(4,·), \chi_{233}(135,·), \chi_{233}(8,·), \chi_{233}(204,·), \chi_{233}(74,·), \chi_{233}(76,·), \chi_{233}(142,·), \chi_{233}(16,·), \chi_{233}(19,·), \chi_{233}(148,·), \chi_{233}(23,·), \chi_{233}(152,·), \chi_{233}(92,·), \chi_{233}(32,·), \chi_{233}(37,·), \chi_{233}(38,·), \chi_{233}(64,·), \chi_{233}(71,·), \chi_{233}(46,·), \chi_{233}(175,·), \chi_{233}(51,·), \chi_{233}(117,·), \chi_{233}(184,·), \chi_{233}(102,·), \chi_{233}(126,·), \chi_{233}(63,·)$$\rbrace$ This is not a CM field.

## Integral basis (with respect to field generator $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{467} a^{25} + \frac{58}{467} a^{24} + \frac{199}{467} a^{23} - \frac{110}{467} a^{22} + \frac{233}{467} a^{21} - \frac{210}{467} a^{20} + \frac{157}{467} a^{19} + \frac{113}{467} a^{18} + \frac{210}{467} a^{17} - \frac{2}{467} a^{16} - \frac{194}{467} a^{15} - \frac{44}{467} a^{14} - \frac{33}{467} a^{13} - \frac{156}{467} a^{12} - \frac{117}{467} a^{11} - \frac{182}{467} a^{10} + \frac{166}{467} a^{9} - \frac{173}{467} a^{8} + \frac{91}{467} a^{7} - \frac{66}{467} a^{6} - \frac{33}{467} a^{5} - \frac{223}{467} a^{4} - \frac{101}{467} a^{3} + \frac{228}{467} a^{2} + \frac{27}{467} a - \frac{3}{467}$, $\frac{1}{4031611} a^{26} + \frac{3580}{4031611} a^{25} + \frac{1191713}{4031611} a^{24} + \frac{666677}{4031611} a^{23} + \frac{1386012}{4031611} a^{22} - \frac{1509914}{4031611} a^{21} + \frac{530777}{4031611} a^{20} + \frac{578285}{4031611} a^{19} + \frac{695208}{4031611} a^{18} - \frac{182707}{4031611} a^{17} - \frac{831026}{4031611} a^{16} + \frac{90507}{4031611} a^{15} + \frac{1463621}{4031611} a^{14} + \frac{1987920}{4031611} a^{13} + \frac{1234858}{4031611} a^{12} + \frac{818289}{4031611} a^{11} - \frac{1999808}{4031611} a^{10} + \frac{734386}{4031611} a^{9} + \frac{84280}{4031611} a^{8} + \frac{85068}{4031611} a^{7} - \frac{253033}{4031611} a^{6} - \frac{1799517}{4031611} a^{5} + \frac{601016}{4031611} a^{4} - \frac{72025}{4031611} a^{3} + \frac{387413}{4031611} a^{2} - \frac{305128}{4031611} a + \frac{624554}{4031611}$, $\frac{1}{4031611} a^{27} + \frac{3964}{4031611} a^{25} + \frac{961296}{4031611} a^{24} + \frac{2897}{8633} a^{23} - \frac{1650656}{4031611} a^{22} - \frac{134363}{4031611} a^{21} + \frac{424962}{4031611} a^{20} - \frac{1970225}{4031611} a^{19} - \frac{1790983}{4031611} a^{18} - \frac{1000504}{4031611} a^{17} + \frac{767033}{4031611} a^{16} + \frac{1721307}{4031611} a^{15} - \frac{358618}{4031611} a^{14} - \frac{467765}{4031611} a^{13} - \frac{1172301}{4031611} a^{12} + \frac{1639120}{4031611} a^{11} + \frac{87183}{4031611} a^{10} - \frac{1192831}{4031611} a^{9} + \frac{750759}{4031611} a^{8} - \frac{500489}{4031611} a^{7} - \frac{507117}{4031611} a^{6} - \frac{384940}{4031611} a^{5} + \frac{307669}{4031611} a^{4} - \frac{1081141}{4031611} a^{3} - \frac{1837094}{4031611} a^{2} - \frac{75868}{4031611} a - \frac{992280}{4031611}$, $\frac{1}{2529915280015368143345128825903699191752559690247794805012287971330397679909527379728191789951018167} a^{28} + \frac{145085650481152602155968129322297735078305544800236082531418181469402090310580758585431686238}{2529915280015368143345128825903699191752559690247794805012287971330397679909527379728191789951018167} a^{27} + \frac{69293157558821546107377658280872921246102454107094834895844377887724889282754591633293586618}{2529915280015368143345128825903699191752559690247794805012287971330397679909527379728191789951018167} a^{26} - \frac{1672144207060240107745504240661381186728894314851222090958759227864019917186441230430336145117969}{2529915280015368143345128825903699191752559690247794805012287971330397679909527379728191789951018167} a^{25} - \frac{699771556466158433700689268198019179678229305050006530734892861846342497793700432402100107060998542}{2529915280015368143345128825903699191752559690247794805012287971330397679909527379728191789951018167} a^{24} - \frac{463001076116526843661892575046654517487454301050780399957049189698481633302542631738944314987684535}{2529915280015368143345128825903699191752559690247794805012287971330397679909527379728191789951018167} a^{23} + \frac{790757354602371172544032331114653289865863637503358877831789453034023531949075648543136414637995631}{2529915280015368143345128825903699191752559690247794805012287971330397679909527379728191789951018167} a^{22} - \frac{168752494562776047035005419955274174519677397900846877093648094037584788972167489877917087938147157}{2529915280015368143345128825903699191752559690247794805012287971330397679909527379728191789951018167} a^{21} - \frac{675807095576379036354706629872156936720166622964616174767751463266291750431744003654600881479269002}{2529915280015368143345128825903699191752559690247794805012287971330397679909527379728191789951018167} a^{20} - \frac{599821261526082616910309561237199538182006812767671837005056922726654690230844548908705399202395483}{2529915280015368143345128825903699191752559690247794805012287971330397679909527379728191789951018167} a^{19} - \frac{478984825311350639928030027218097619362024110954005870776913509765396544607078783435195739162767397}{2529915280015368143345128825903699191752559690247794805012287971330397679909527379728191789951018167} a^{18} - \frac{621386074512339566977476973441303398902686602605041899055304387675273092590980409428147925897587778}{2529915280015368143345128825903699191752559690247794805012287971330397679909527379728191789951018167} a^{17} + \frac{686518679731055109968232694885170487718099946461454221285106553473803882428852859366191198250697601}{2529915280015368143345128825903699191752559690247794805012287971330397679909527379728191789951018167} a^{16} - \frac{697226637726456845437556231626925026961639162054481416432018755096501250774365338519574590857021861}{2529915280015368143345128825903699191752559690247794805012287971330397679909527379728191789951018167} a^{15} - \frac{438831094034924893915705511505525587148192834066739442895106098885411217057987733005117104548556421}{2529915280015368143345128825903699191752559690247794805012287971330397679909527379728191789951018167} a^{14} - \frac{731574325928059940914595843872938215445373633736377054754583119534562481058216465072540541237319226}{2529915280015368143345128825903699191752559690247794805012287971330397679909527379728191789951018167} a^{13} - \frac{706657627546836297753673771793694898903797297183168535185671619421241149767695235099166073490004060}{2529915280015368143345128825903699191752559690247794805012287971330397679909527379728191789951018167} a^{12} + \frac{1078165443071075039983817048161447951357812652051626813270594983584509380305533640058133430578150439}{2529915280015368143345128825903699191752559690247794805012287971330397679909527379728191789951018167} a^{11} + \frac{263256046036600063692548222097372389761077956803176118069904921922339215608116979769258689513619810}{2529915280015368143345128825903699191752559690247794805012287971330397679909527379728191789951018167} a^{10} + \frac{840562216158587744209496578502291745341768504793955116247286260520120485963836780328031891935804559}{2529915280015368143345128825903699191752559690247794805012287971330397679909527379728191789951018167} a^{9} + \frac{1072023456745969965452361159719651204128011274884078583550468367026641748909461805488157298614394489}{2529915280015368143345128825903699191752559690247794805012287971330397679909527379728191789951018167} a^{8} + \frac{725537680715542189535508617615800026234955067988912223518225395793864951395951158864634065397880690}{2529915280015368143345128825903699191752559690247794805012287971330397679909527379728191789951018167} a^{7} + \frac{693199124790903451330075872317501202655334498534504788773978994087266816721186645517558924576189988}{2529915280015368143345128825903699191752559690247794805012287971330397679909527379728191789951018167} a^{6} + \frac{1243203002359064174538755243714677007844900830903438772903605839749612955071778124785188718602846796}{2529915280015368143345128825903699191752559690247794805012287971330397679909527379728191789951018167} a^{5} - \frac{528056665134996150055417451166791703071597895774807908079604139104457615433666449551561268169761950}{2529915280015368143345128825903699191752559690247794805012287971330397679909527379728191789951018167} a^{4} + \frac{1210345042407190680924813070654967089273230710946966255357519513993375388894564210832287658269500059}{2529915280015368143345128825903699191752559690247794805012287971330397679909527379728191789951018167} a^{3} - \frac{1178769742100916918710503255007851076115103954990919223170204263624152714253035020970442665786757748}{2529915280015368143345128825903699191752559690247794805012287971330397679909527379728191789951018167} a^{2} - \frac{756381129826289412621676192379164025407619600798376221080608331782685669602692023582531261049971403}{2529915280015368143345128825903699191752559690247794805012287971330397679909527379728191789951018167} a + \frac{94824176098276618177757183837539952870004805152187461113123764771275037338575909714961540614205753}{2529915280015368143345128825903699191752559690247794805012287971330397679909527379728191789951018167}$

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: $28$ 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: $$900609177445488600000000$$ (assuming GRH) sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Galois group

$C_{29}$ (as 29T1):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A cyclic group of order 29 The 29 conjugacy class representatives for $C_{29}$ Character table for $C_{29}$ is not computed

## Intermediate fields

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

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$

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
233Data not computed