Normalized defining polynomial
\( x^{29} - x^{28} - 112 x^{27} + 91 x^{26} + 5198 x^{25} - 3644 x^{24} - 132219 x^{23} + 83238 x^{22} + 2053518 x^{21} - 1187959 x^{20} - 20553532 x^{19} + 11071128 x^{18} + 136460842 x^{17} - 69042962 x^{16} - 609473492 x^{15} + 292259011 x^{14} + 1836592125 x^{13} - 845018358 x^{12} - 3706016039 x^{11} + 1661552324 x^{10} + 4906886664 x^{9} - 2177019390 x^{8} - 4095369839 x^{7} + 1819962089 x^{6} + 1998032360 x^{5} - 895362174 x^{4} - 490947342 x^{3} + 221892059 x^{2} + 42927079 x - 19524467 \)
Invariants
| Degree: | $29$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[29, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(1931816133436230496253440348173909042983780087233421135736506627041=233^{28}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $193.07$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $233$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| 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}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $28$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| 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) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 900609177445488600000000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| 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
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $29$ | $29$ | $29$ | $29$ | $29$ | $29$ | $29$ | $29$ | $29$ | $29$ | $29$ | $29$ | $29$ | $29$ | $29$ | $29$ | $29$ |
Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| 233 | Data not computed | ||||||