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Properties

Label 28.0.69632758551...0625.2
Degree $28$
Signature $[0, 14]$
Discriminant $3^{14}\cdot 5^{21}\cdot 29^{27}$
Root discriminant $148.92$
Ramified primes $3, 5, 29$
Class number Not computed
Class group Not computed
Galois group $C_{28}$ (as 28T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![24797115841, 25984132239, 119652295787, -47326599591, 163068261008, -93383072609, 96931321441, -39322573172, 39843676753, -12321166201, 11671511826, -2087317747, 2653721996, -288729656, 503292507, -38454407, 78712410, -5999927, 10659821, -790193, 1087588, -67948, 75865, -3394, 3394, -88, 88, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 + 88*x^26 - 88*x^25 + 3394*x^24 - 3394*x^23 + 75865*x^22 - 67948*x^21 + 1087588*x^20 - 790193*x^19 + 10659821*x^18 - 5999927*x^17 + 78712410*x^16 - 38454407*x^15 + 503292507*x^14 - 288729656*x^13 + 2653721996*x^12 - 2087317747*x^11 + 11671511826*x^10 - 12321166201*x^9 + 39843676753*x^8 - 39322573172*x^7 + 96931321441*x^6 - 93383072609*x^5 + 163068261008*x^4 - 47326599591*x^3 + 119652295787*x^2 + 25984132239*x + 24797115841)
 
gp: K = bnfinit(x^28 - x^27 + 88*x^26 - 88*x^25 + 3394*x^24 - 3394*x^23 + 75865*x^22 - 67948*x^21 + 1087588*x^20 - 790193*x^19 + 10659821*x^18 - 5999927*x^17 + 78712410*x^16 - 38454407*x^15 + 503292507*x^14 - 288729656*x^13 + 2653721996*x^12 - 2087317747*x^11 + 11671511826*x^10 - 12321166201*x^9 + 39843676753*x^8 - 39322573172*x^7 + 96931321441*x^6 - 93383072609*x^5 + 163068261008*x^4 - 47326599591*x^3 + 119652295787*x^2 + 25984132239*x + 24797115841, 1)
 

Normalized defining polynomial

\( x^{28} - x^{27} + 88 x^{26} - 88 x^{25} + 3394 x^{24} - 3394 x^{23} + 75865 x^{22} - 67948 x^{21} + 1087588 x^{20} - 790193 x^{19} + 10659821 x^{18} - 5999927 x^{17} + 78712410 x^{16} - 38454407 x^{15} + 503292507 x^{14} - 288729656 x^{13} + 2653721996 x^{12} - 2087317747 x^{11} + 11671511826 x^{10} - 12321166201 x^{9} + 39843676753 x^{8} - 39322573172 x^{7} + 96931321441 x^{6} - 93383072609 x^{5} + 163068261008 x^{4} - 47326599591 x^{3} + 119652295787 x^{2} + 25984132239 x + 24797115841 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $28$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 14]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6963275855162665256600064094162323796066183958530426025390625=3^{14}\cdot 5^{21}\cdot 29^{27}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $148.92$
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:  $3, 5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(435=3\cdot 5\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{435}(256,·)$, $\chi_{435}(1,·)$, $\chi_{435}(4,·)$, $\chi_{435}(98,·)$, $\chi_{435}(263,·)$, $\chi_{435}(136,·)$, $\chi_{435}(226,·)$, $\chi_{435}(398,·)$, $\chi_{435}(109,·)$, $\chi_{435}(16,·)$, $\chi_{435}(274,·)$, $\chi_{435}(278,·)$, $\chi_{435}(154,·)$, $\chi_{435}(286,·)$, $\chi_{435}(287,·)$, $\chi_{435}(289,·)$, $\chi_{435}(34,·)$, $\chi_{435}(293,·)$, $\chi_{435}(64,·)$, $\chi_{435}(338,·)$, $\chi_{435}(302,·)$, $\chi_{435}(47,·)$, $\chi_{435}(392,·)$, $\chi_{435}(242,·)$, $\chi_{435}(181,·)$, $\chi_{435}(182,·)$, $\chi_{435}(188,·)$, $\chi_{435}(317,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{17} a^{14} - \frac{6}{17} a^{13} + \frac{2}{17} a^{12} - \frac{4}{17} a^{11} - \frac{7}{17} a^{10} + \frac{7}{17} a^{9} - \frac{6}{17} a^{8} - \frac{3}{17} a^{7} + \frac{6}{17} a^{6} + \frac{1}{17} a^{5} + \frac{4}{17} a^{4} + \frac{7}{17} a^{3} - \frac{2}{17} a$, $\frac{1}{17} a^{15} + \frac{8}{17} a^{12} + \frac{3}{17} a^{11} - \frac{1}{17} a^{10} + \frac{2}{17} a^{9} - \frac{5}{17} a^{8} + \frac{5}{17} a^{7} + \frac{3}{17} a^{6} - \frac{7}{17} a^{5} - \frac{3}{17} a^{4} + \frac{8}{17} a^{3} - \frac{2}{17} a^{2} + \frac{5}{17} a$, $\frac{1}{17} a^{16} + \frac{8}{17} a^{13} + \frac{3}{17} a^{12} - \frac{1}{17} a^{11} + \frac{2}{17} a^{10} - \frac{5}{17} a^{9} + \frac{5}{17} a^{8} + \frac{3}{17} a^{7} - \frac{7}{17} a^{6} - \frac{3}{17} a^{5} + \frac{8}{17} a^{4} - \frac{2}{17} a^{3} + \frac{5}{17} a^{2}$, $\frac{1}{17} a^{17} - \frac{1}{17} a$, $\frac{1}{17} a^{18} - \frac{1}{17} a^{2}$, $\frac{1}{17} a^{19} - \frac{1}{17} a^{3}$, $\frac{1}{17} a^{20} - \frac{1}{17} a^{4}$, $\frac{1}{17} a^{21} - \frac{1}{17} a^{5}$, $\frac{1}{17051} a^{22} - \frac{405}{17051} a^{21} + \frac{1}{1003} a^{20} - \frac{284}{17051} a^{19} - \frac{161}{17051} a^{18} - \frac{208}{17051} a^{17} + \frac{264}{17051} a^{16} - \frac{200}{17051} a^{15} + \frac{420}{17051} a^{14} + \frac{406}{1003} a^{13} - \frac{1702}{17051} a^{12} + \frac{4749}{17051} a^{11} - \frac{2365}{17051} a^{10} - \frac{6158}{17051} a^{9} - \frac{64}{17051} a^{8} - \frac{6126}{17051} a^{7} - \frac{1884}{17051} a^{6} - \frac{4432}{17051} a^{5} + \frac{8149}{17051} a^{4} - \frac{740}{17051} a^{3} - \frac{6347}{17051} a^{2} + \frac{367}{1003} a$, $\frac{1}{17051} a^{23} + \frac{484}{17051} a^{21} - \frac{420}{17051} a^{20} + \frac{164}{17051} a^{19} - \frac{218}{17051} a^{18} + \frac{276}{17051} a^{17} + \frac{402}{17051} a^{16} - \frac{20}{1003} a^{15} + \frac{474}{17051} a^{14} + \frac{1256}{17051} a^{13} + \frac{1491}{17051} a^{12} + \frac{7256}{17051} a^{11} - \frac{4112}{17051} a^{10} - \frac{2602}{17051} a^{9} - \frac{7974}{17051} a^{8} + \frac{4526}{17051} a^{7} - \frac{6175}{17051} a^{6} - \frac{468}{17051} a^{5} - \frac{8289}{17051} a^{4} - \frac{1135}{17051} a^{3} + \frac{3384}{17051} a^{2} + \frac{486}{1003} a$, $\frac{1}{17051} a^{24} + \frac{15}{17051} a^{21} - \frac{40}{17051} a^{20} - \frac{173}{17051} a^{19} - \frac{2}{1003} a^{18} - \frac{229}{17051} a^{17} + \frac{268}{17051} a^{16} - \frac{1}{1003} a^{15} - \frac{421}{17051} a^{14} + \frac{7934}{17051} a^{13} + \frac{5555}{17051} a^{12} + \frac{260}{17051} a^{11} - \frac{359}{17051} a^{10} + \frac{609}{17051} a^{9} + \frac{5412}{17051} a^{8} + \frac{1965}{17051} a^{7} - \frac{7360}{17051} a^{6} - \frac{594}{17051} a^{5} - \frac{3461}{17051} a^{4} + \frac{5479}{17051} a^{3} + \frac{5012}{17051} a^{2} - \frac{333}{1003} a$, $\frac{1}{45509119} a^{25} + \frac{1038}{45509119} a^{24} - \frac{1077}{45509119} a^{23} + \frac{1211}{45509119} a^{22} - \frac{253916}{45509119} a^{21} - \frac{812741}{45509119} a^{20} + \frac{547820}{45509119} a^{19} + \frac{412924}{45509119} a^{18} - \frac{843634}{45509119} a^{17} - \frac{1183063}{45509119} a^{16} - \frac{599205}{45509119} a^{15} - \frac{1105240}{45509119} a^{14} - \frac{10220737}{45509119} a^{13} + \frac{6744769}{45509119} a^{12} - \frac{14688742}{45509119} a^{11} + \frac{19065403}{45509119} a^{10} - \frac{17432461}{45509119} a^{9} + \frac{2324762}{45509119} a^{8} - \frac{12361446}{45509119} a^{7} + \frac{19539079}{45509119} a^{6} + \frac{30317}{289867} a^{5} - \frac{205185}{771341} a^{4} + \frac{19861942}{45509119} a^{3} + \frac{21688147}{45509119} a^{2} - \frac{1279449}{2677007} a + \frac{8}{17}$, $\frac{1}{2685038021} a^{26} + \frac{28}{2685038021} a^{25} + \frac{39495}{2685038021} a^{24} + \frac{34726}{2685038021} a^{23} - \frac{57118}{2685038021} a^{22} + \frac{464667}{2685038021} a^{21} + \frac{10683459}{2685038021} a^{20} + \frac{69263962}{2685038021} a^{19} + \frac{62963511}{2685038021} a^{18} + \frac{52680123}{2685038021} a^{17} - \frac{48560122}{2685038021} a^{16} - \frac{33300087}{2685038021} a^{15} + \frac{2837973}{157943413} a^{14} + \frac{75775475}{2685038021} a^{13} - \frac{1326400698}{2685038021} a^{12} - \frac{917891458}{2685038021} a^{11} + \frac{16838790}{2685038021} a^{10} - \frac{302241693}{2685038021} a^{9} + \frac{1049746178}{2685038021} a^{8} - \frac{520069750}{2685038021} a^{7} + \frac{167522170}{2685038021} a^{6} + \frac{175576909}{2685038021} a^{5} - \frac{1272588396}{2685038021} a^{4} - \frac{1280476692}{2685038021} a^{3} + \frac{1072976089}{2685038021} a^{2} + \frac{797880}{2677007} a - \frac{7}{17}$, $\frac{1}{174881990908972258656356840962789703185063470287714535186231492388265087998178960194119200877056722688484488977457977562556680399} a^{27} - \frac{28577247436615013711732511149208700461919637228312386814527000529440864472634729719958593546078652090197141487344494525}{174881990908972258656356840962789703185063470287714535186231492388265087998178960194119200877056722688484488977457977562556680399} a^{26} + \frac{1082818223081331708750560133515221847766963802467992314167466901281957783108014330875719263639816711740431899830241141252}{174881990908972258656356840962789703185063470287714535186231492388265087998178960194119200877056722688484488977457977562556680399} a^{25} + \frac{2614385775869013725836470152705852617353636724193672169077498932117638322702020026050439695520542519245724151415650275808817}{174881990908972258656356840962789703185063470287714535186231492388265087998178960194119200877056722688484488977457977562556680399} a^{24} - \frac{2348551122959002794882053315853980912854485597002194686397660130579256255895681286145478179350749581661300896650506268083119}{174881990908972258656356840962789703185063470287714535186231492388265087998178960194119200877056722688484488977457977562556680399} a^{23} + \frac{2305098302880855914930147473584232635809913664934942123922528157763633786286549651680982831235364517828504403387646692684798}{174881990908972258656356840962789703185063470287714535186231492388265087998178960194119200877056722688484488977457977562556680399} a^{22} - \frac{1570741864746111244662115027144564234872658149463261485733129357046672095137690099477431620107204638787533605674307442015129942}{174881990908972258656356840962789703185063470287714535186231492388265087998178960194119200877056722688484488977457977562556680399} a^{21} + \frac{4625892826163114281993305942362256691591407239244974974791132843928407583902576048333074346281013284235370681696005313955261576}{174881990908972258656356840962789703185063470287714535186231492388265087998178960194119200877056722688484488977457977562556680399} a^{20} - \frac{3000360377169538974443640455228981700874164015657632481107374364535229280388299975055886115975275755884732511897515373480171770}{174881990908972258656356840962789703185063470287714535186231492388265087998178960194119200877056722688484488977457977562556680399} a^{19} - \frac{895808073781076655057658301085845277570145862640933314461141899741499168620476261719182037543989194449880099605095930694431354}{174881990908972258656356840962789703185063470287714535186231492388265087998178960194119200877056722688484488977457977562556680399} a^{18} - \frac{4239659565952632034273260639671589497394055881611200309899265212917751376570476325088407298528567290133254219417766735864894543}{174881990908972258656356840962789703185063470287714535186231492388265087998178960194119200877056722688484488977457977562556680399} a^{17} + \frac{4943717945025674245876595466465089378706564807495083375183126180161854469757721343214894322082769447666152017361049177236237197}{174881990908972258656356840962789703185063470287714535186231492388265087998178960194119200877056722688484488977457977562556680399} a^{16} - \frac{1827096808870776214194212659898403902050788575960175572014853672228699419824880239985293298375921351183307979592542484939074759}{174881990908972258656356840962789703185063470287714535186231492388265087998178960194119200877056722688484488977457977562556680399} a^{15} + \frac{1016298196635888785972555370971818368281621068151910585436695121316311858934591044237208734376250426358265159550278607991616541}{174881990908972258656356840962789703185063470287714535186231492388265087998178960194119200877056722688484488977457977562556680399} a^{14} - \frac{1275576096620657663607107744385826901593569673171010581472240566543819260563071991574725360716598305356305952041823008405850196}{2964101540830038282311132897674401748899380852334144664173415125224831999969134918544393235204351232008211677584033518009435261} a^{13} - \frac{56714191823929064446204200994679401115375162327268611650342978484119038878960213227382753150892357708591224408136910414204736073}{174881990908972258656356840962789703185063470287714535186231492388265087998178960194119200877056722688484488977457977562556680399} a^{12} - \frac{49069048874472313009858104542044110889336025748349496151334202943082916740125560443284068523304638327327563510106533822747605006}{174881990908972258656356840962789703185063470287714535186231492388265087998178960194119200877056722688484488977457977562556680399} a^{11} - \frac{3270589464010066873258071327700735635668117605092822568657822366632246710609219621607186884941424494318225837482312795801553796}{174881990908972258656356840962789703185063470287714535186231492388265087998178960194119200877056722688484488977457977562556680399} a^{10} + \frac{28351601015617265830537491784268132462981211280540088553510134811666934231473958747822676884235373562673872273768129337645407461}{174881990908972258656356840962789703185063470287714535186231492388265087998178960194119200877056722688484488977457977562556680399} a^{9} - \frac{57911324071144549796731873290574117620932829340878712960241673342510483244336455011922696330773980560327624606937511696835729332}{174881990908972258656356840962789703185063470287714535186231492388265087998178960194119200877056722688484488977457977562556680399} a^{8} + \frac{9371796159540948150130644193908301983225840093788249855190505313756776591651327710710236716788101835541891394880279625998253841}{174881990908972258656356840962789703185063470287714535186231492388265087998178960194119200877056722688484488977457977562556680399} a^{7} + \frac{10782751446878488881346703017663626684974571262029586287825627119498768085357493930988278293049698381849020748701196355812873087}{174881990908972258656356840962789703185063470287714535186231492388265087998178960194119200877056722688484488977457977562556680399} a^{6} + \frac{3256001370067160140042160947701489090587382140454586118205724691156170042151988631969104764399587033450277171903918033469522094}{10287175935821897568020990644869982540297851193394972658013617199309711058716409423183482404532748393440264057497528091915098847} a^{5} - \frac{36074925655091224546980630789271422788981751864110793940001300023020087472736091259748352835276899882802909541390887929013135193}{174881990908972258656356840962789703185063470287714535186231492388265087998178960194119200877056722688484488977457977562556680399} a^{4} - \frac{2835488234119592025157239079321721192612701885946463533482353738163296517196940157370918721239673593476594567248668533641005326}{174881990908972258656356840962789703185063470287714535186231492388265087998178960194119200877056722688484488977457977562556680399} a^{3} + \frac{38188322007189115850361816429865440947272436717536230386694030827676443640570214002751578405987534777624889433058447745759967064}{174881990908972258656356840962789703185063470287714535186231492388265087998178960194119200877056722688484488977457977562556680399} a^{2} - \frac{521717139065400234572425318591837498897859035123477321550605981773068106463653452744597894258184020499837346449234425216305}{1110566332270527644178019069941701666878748914325269638131665464677718995867041932762979855827782402400978522886486893221969} a - \frac{2026230341001752353749831836927954811041099043160262987274865146969485821733486498133530914838678650183272484455870486}{7052513366083454376856812174569931396122136230323485836323294223556839010783204099567411496896459680836335089549738639}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $13$
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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{28}$ (as 28T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 28
The 28 conjugacy class representatives for $C_{28}$
Character table for $C_{28}$ is not computed

Intermediate fields

\(\Q(\sqrt{145}) \), 4.0.27437625.2, 7.7.594823321.1, 14.14.801611618199890796015625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{4}$ R R $28$ $28$ $28$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{28}$ $28$ $28$ R $28$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{7}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ $28$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{28}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.14.7.2$x^{14} + 243 x^{4} - 729 x^{2} + 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
3.14.7.2$x^{14} + 243 x^{4} - 729 x^{2} + 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
5Data not computed
29Data not computed