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Properties

Label 28.0.40608979500...0761.1
Degree $28$
Signature $[0, 14]$
Discriminant $3^{14}\cdot 7^{14}\cdot 29^{24}$
Root discriminant $82.15$
Ramified primes $3, 7, 29$
Class number $448$ (GRH)
Class group $[4, 4, 28]$ (GRH)
Galois group $C_2\times C_{14}$ (as 28T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![495997441, -302462451, -282222973, 172023768, 26652928, -10418664, -81474689, 42898388, 40409501, -26399240, 4348963, 177615, 1443969, -1442414, -1650291, 1277344, -344330, 73977, 74988, -59815, 20952, -5213, 1715, -307, 206, -77, 26, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 5*x^27 + 26*x^26 - 77*x^25 + 206*x^24 - 307*x^23 + 1715*x^22 - 5213*x^21 + 20952*x^20 - 59815*x^19 + 74988*x^18 + 73977*x^17 - 344330*x^16 + 1277344*x^15 - 1650291*x^14 - 1442414*x^13 + 1443969*x^12 + 177615*x^11 + 4348963*x^10 - 26399240*x^9 + 40409501*x^8 + 42898388*x^7 - 81474689*x^6 - 10418664*x^5 + 26652928*x^4 + 172023768*x^3 - 282222973*x^2 - 302462451*x + 495997441)
 
gp: K = bnfinit(x^28 - 5*x^27 + 26*x^26 - 77*x^25 + 206*x^24 - 307*x^23 + 1715*x^22 - 5213*x^21 + 20952*x^20 - 59815*x^19 + 74988*x^18 + 73977*x^17 - 344330*x^16 + 1277344*x^15 - 1650291*x^14 - 1442414*x^13 + 1443969*x^12 + 177615*x^11 + 4348963*x^10 - 26399240*x^9 + 40409501*x^8 + 42898388*x^7 - 81474689*x^6 - 10418664*x^5 + 26652928*x^4 + 172023768*x^3 - 282222973*x^2 - 302462451*x + 495997441, 1)
 

Normalized defining polynomial

\( x^{28} - 5 x^{27} + 26 x^{26} - 77 x^{25} + 206 x^{24} - 307 x^{23} + 1715 x^{22} - 5213 x^{21} + 20952 x^{20} - 59815 x^{19} + 74988 x^{18} + 73977 x^{17} - 344330 x^{16} + 1277344 x^{15} - 1650291 x^{14} - 1442414 x^{13} + 1443969 x^{12} + 177615 x^{11} + 4348963 x^{10} - 26399240 x^{9} + 40409501 x^{8} + 42898388 x^{7} - 81474689 x^{6} - 10418664 x^{5} + 26652928 x^{4} + 172023768 x^{3} - 282222973 x^{2} - 302462451 x + 495997441 \)

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:  \(406089795003640616506784472740838252236679434560990761=3^{14}\cdot 7^{14}\cdot 29^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $82.15$
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:  $3, 7, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(609=3\cdot 7\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{609}(1,·)$, $\chi_{609}(587,·)$, $\chi_{609}(517,·)$, $\chi_{609}(139,·)$, $\chi_{609}(335,·)$, $\chi_{609}(400,·)$, $\chi_{609}(146,·)$, $\chi_{609}(83,·)$, $\chi_{609}(20,·)$, $\chi_{609}(407,·)$, $\chi_{609}(344,·)$, $\chi_{609}(281,·)$, $\chi_{609}(538,·)$, $\chi_{609}(239,·)$, $\chi_{609}(314,·)$, $\chi_{609}(286,·)$, $\chi_{609}(197,·)$, $\chi_{609}(545,·)$, $\chi_{609}(547,·)$, $\chi_{609}(484,·)$, $\chi_{609}(169,·)$, $\chi_{609}(349,·)$, $\chi_{609}(181,·)$, $\chi_{609}(596,·)$, $\chi_{609}(442,·)$, $\chi_{609}(223,·)$, $\chi_{609}(190,·)$, $\chi_{609}(575,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{17} a^{23} + \frac{2}{17} a^{22} - \frac{2}{17} a^{21} + \frac{6}{17} a^{20} + \frac{5}{17} a^{19} - \frac{3}{17} a^{17} - \frac{3}{17} a^{16} + \frac{1}{17} a^{15} + \frac{4}{17} a^{14} - \frac{2}{17} a^{13} - \frac{7}{17} a^{12} - \frac{2}{17} a^{11} + \frac{2}{17} a^{10} + \frac{1}{17} a^{9} - \frac{6}{17} a^{8} - \frac{8}{17} a^{7} - \frac{5}{17} a^{6} + \frac{1}{17} a^{4} - \frac{3}{17} a^{3} + \frac{2}{17} a^{2} + \frac{4}{17} a - \frac{6}{17}$, $\frac{1}{17} a^{24} - \frac{6}{17} a^{22} - \frac{7}{17} a^{21} - \frac{7}{17} a^{20} + \frac{7}{17} a^{19} - \frac{3}{17} a^{18} + \frac{3}{17} a^{17} + \frac{7}{17} a^{16} + \frac{2}{17} a^{15} + \frac{7}{17} a^{14} - \frac{3}{17} a^{13} - \frac{5}{17} a^{12} + \frac{6}{17} a^{11} - \frac{3}{17} a^{10} - \frac{8}{17} a^{9} + \frac{4}{17} a^{8} - \frac{6}{17} a^{7} - \frac{7}{17} a^{6} + \frac{1}{17} a^{5} - \frac{5}{17} a^{4} + \frac{8}{17} a^{3} + \frac{3}{17} a - \frac{5}{17}$, $\frac{1}{17} a^{25} + \frac{5}{17} a^{22} - \frac{2}{17} a^{21} - \frac{8}{17} a^{20} - \frac{7}{17} a^{19} + \frac{3}{17} a^{18} + \frac{6}{17} a^{17} + \frac{1}{17} a^{16} - \frac{4}{17} a^{15} + \frac{4}{17} a^{14} - \frac{2}{17} a^{12} + \frac{2}{17} a^{11} + \frac{4}{17} a^{10} - \frac{7}{17} a^{9} - \frac{8}{17} a^{8} - \frac{4}{17} a^{7} + \frac{5}{17} a^{6} - \frac{5}{17} a^{5} - \frac{3}{17} a^{4} - \frac{1}{17} a^{3} - \frac{2}{17} a^{2} + \frac{2}{17} a - \frac{2}{17}$, $\frac{1}{121186832295498329819649523} a^{26} + \frac{1640170667482962959155211}{121186832295498329819649523} a^{25} + \frac{1443773454268443286756607}{121186832295498329819649523} a^{24} + \frac{595303439056516519356684}{121186832295498329819649523} a^{23} - \frac{1347818458708297282398405}{7128637193852842930567619} a^{22} + \frac{1589965003246119845308846}{121186832295498329819649523} a^{21} - \frac{49959129480514879260152289}{121186832295498329819649523} a^{20} + \frac{28960391423986772347279688}{121186832295498329819649523} a^{19} + \frac{10837757857333619288487736}{121186832295498329819649523} a^{18} - \frac{8287502352269467212989729}{121186832295498329819649523} a^{17} - \frac{31677032857414480384087930}{121186832295498329819649523} a^{16} + \frac{17424240279665087131968199}{121186832295498329819649523} a^{15} - \frac{28028883474768792733408838}{121186832295498329819649523} a^{14} - \frac{21645292085973343945077247}{121186832295498329819649523} a^{13} - \frac{49914459913637959302158344}{121186832295498329819649523} a^{12} - \frac{55788759588780235014755204}{121186832295498329819649523} a^{11} + \frac{35517010855434548974576305}{121186832295498329819649523} a^{10} - \frac{33104336187398055738060843}{121186832295498329819649523} a^{9} - \frac{58944654261796672488290926}{121186832295498329819649523} a^{8} - \frac{52787628653571675946182709}{121186832295498329819649523} a^{7} - \frac{236252211382941793876881}{634486032960724239893453} a^{6} + \frac{52789075807991440231292291}{121186832295498329819649523} a^{5} + \frac{29894540120680461835589572}{121186832295498329819649523} a^{4} + \frac{18910784434681478185159151}{121186832295498329819649523} a^{3} - \frac{30528043662809296109523213}{121186832295498329819649523} a^{2} + \frac{31378791322178187010186187}{121186832295498329819649523} a + \frac{11765923381412869741361369}{121186832295498329819649523}$, $\frac{1}{3616739060232963747654619655675581965471019304628410414122802690070020585925101179205862684876973} a^{27} + \frac{9009143161942805965318956144480698675913810864400385285739244427710333}{3616739060232963747654619655675581965471019304628410414122802690070020585925101179205862684876973} a^{26} + \frac{55978106427795892666758556515001854476191988977179568829132390035724518348608057311094308080025}{3616739060232963747654619655675581965471019304628410414122802690070020585925101179205862684876973} a^{25} - \frac{13938145978992583603162210059091086396895945726745082316675150836407348554266241447204096355387}{3616739060232963747654619655675581965471019304628410414122802690070020585925101179205862684876973} a^{24} - \frac{2699380459240927235699832930232191499293801446797841514674653785879992897850767173656009641699}{3616739060232963747654619655675581965471019304628410414122802690070020585925101179205862684876973} a^{23} + \frac{242276573581684594771711578790783879165249733208666430056016583998776819881117602638267340528143}{3616739060232963747654619655675581965471019304628410414122802690070020585925101179205862684876973} a^{22} + \frac{1515908424052645931029059565312962350623154581276865109839113594247626368277821240225622653206315}{3616739060232963747654619655675581965471019304628410414122802690070020585925101179205862684876973} a^{21} + \frac{1573521462202125791639150898220392143261517158668200063020312762160058145736710238974314438588944}{3616739060232963747654619655675581965471019304628410414122802690070020585925101179205862684876973} a^{20} - \frac{78703345275487381129537122864047541404066413851490022948257262110961275137908446435596443602178}{212749356484291985156154097392681292086530547331082965536635452357060034466182422306227216757469} a^{19} + \frac{205011056384985707645455596501493235103449792724094389681793346566860590556411385585080970608746}{3616739060232963747654619655675581965471019304628410414122802690070020585925101179205862684876973} a^{18} + \frac{62550642572202225588992889045292152459562137239430120420802500422107088552237715807630750946422}{3616739060232963747654619655675581965471019304628410414122802690070020585925101179205862684876973} a^{17} + \frac{347751703002927724434233406559240344564343174910366461638719809659812265492593152345006822505515}{3616739060232963747654619655675581965471019304628410414122802690070020585925101179205862684876973} a^{16} + \frac{67241304393072810981346656661967252903698493132429948613679540859446584592860708222561358289458}{212749356484291985156154097392681292086530547331082965536635452357060034466182422306227216757469} a^{15} - \frac{1176913496964371480027884354320946144694291909353644446376798620412910655190285300162098232815665}{3616739060232963747654619655675581965471019304628410414122802690070020585925101179205862684876973} a^{14} - \frac{841285537390229787200614960691149351548860689876536364975471925623170975719440984523436689406533}{3616739060232963747654619655675581965471019304628410414122802690070020585925101179205862684876973} a^{13} - \frac{1345189935397182832765325923942738755265768085686567893892811957274881062885485671990563256941260}{3616739060232963747654619655675581965471019304628410414122802690070020585925101179205862684876973} a^{12} + \frac{1009844616605039897984604868410081322039926612750547417572851926086194878521713203767651957554835}{3616739060232963747654619655675581965471019304628410414122802690070020585925101179205862684876973} a^{11} - \frac{658086302314423296589151471053268416504314815856215447229371878715690464045863339612096469343437}{3616739060232963747654619655675581965471019304628410414122802690070020585925101179205862684876973} a^{10} - \frac{376969532162759894421637924246819230180774462053002305010315296893628985316709987331275388146510}{3616739060232963747654619655675581965471019304628410414122802690070020585925101179205862684876973} a^{9} - \frac{750125209345212441814875932897408219144943609726131795748015505847313451164815500455079967794973}{3616739060232963747654619655675581965471019304628410414122802690070020585925101179205862684876973} a^{8} + \frac{799599660892953307561717505089421561245948517629692719695894719581519197289922234922009645030872}{3616739060232963747654619655675581965471019304628410414122802690070020585925101179205862684876973} a^{7} - \frac{994562338927123313235592081348404295302809490618991247287949011874314154092713604692614156470592}{3616739060232963747654619655675581965471019304628410414122802690070020585925101179205862684876973} a^{6} + \frac{1523283053858315109304745185670125412331196585443449760878340460030376953599492308085095288948671}{3616739060232963747654619655675581965471019304628410414122802690070020585925101179205862684876973} a^{5} + \frac{360847228902594901025064410786337446027230197909191039070155035461542969408508831509262911130104}{3616739060232963747654619655675581965471019304628410414122802690070020585925101179205862684876973} a^{4} + \frac{728490167396328284676286738508730457569167120630589926033011924310546023040799073486531976421177}{3616739060232963747654619655675581965471019304628410414122802690070020585925101179205862684876973} a^{3} + \frac{209890918987901372615818264566842012564664580703728304375516738949329994420828820326490339646143}{3616739060232963747654619655675581965471019304628410414122802690070020585925101179205862684876973} a^{2} - \frac{1323833890956745725912318556513307346213357919209170973570856704691968701903629479003470585474281}{3616739060232963747654619655675581965471019304628410414122802690070020585925101179205862684876973} a + \frac{57563686186308645175556249681161032146281681512153550802530880532277754421393264636705376905}{162396796741635478768560893344509989020296318289632724804580067804320442994257158601134330963}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{28}$, which has order $448$ (assuming GRH)

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:  \( -\frac{4969228538477118899246639352955567717990564847117917922139614174552}{8625009570830687283031087617103475374419052610221389278607087042785800639} a^{27} + \frac{17327852385012449024491734525037066179448374016125717343250246818707}{8625009570830687283031087617103475374419052610221389278607087042785800639} a^{26} - \frac{102811283776260091725162350601462525520679618658928486707917594994317}{8625009570830687283031087617103475374419052610221389278607087042785800639} a^{25} + \frac{226223507837717545334745405289375971113977743920253660704752257968925}{8625009570830687283031087617103475374419052610221389278607087042785800639} a^{24} - \frac{676685866214359799120252558467031159494818659164317217128856908547580}{8625009570830687283031087617103475374419052610221389278607087042785800639} a^{23} + \frac{487375245077691283274350454057582372721989456585012144519854835293158}{8625009570830687283031087617103475374419052610221389278607087042785800639} a^{22} - \frac{7742944629627320097641744418943934997701445933736916509937647971510286}{8625009570830687283031087617103475374419052610221389278607087042785800639} a^{21} + \frac{14117443967275957328071418756994163423937056258749366012613582150317901}{8625009570830687283031087617103475374419052610221389278607087042785800639} a^{20} - \frac{4847244463327057679776867946783827518644116835893228245679952779053533}{507353504166511016648887506888439727907003094718905251682769826046223567} a^{19} + \frac{171572932998667585738022775977430962476407561170076256289823346545836238}{8625009570830687283031087617103475374419052610221389278607087042785800639} a^{18} - \frac{108969919075791807398494404971358992901233960214011110342324213587410812}{8625009570830687283031087617103475374419052610221389278607087042785800639} a^{17} - \frac{544033913537777279947355700581370695689842319579456237170413499624610128}{8625009570830687283031087617103475374419052610221389278607087042785800639} a^{16} + \frac{53312386133230241753008033741139987092518432371247744603685516881004757}{507353504166511016648887506888439727907003094718905251682769826046223567} a^{15} - \frac{4979111833733609545821467295222751229596066058212399891689576135529072155}{8625009570830687283031087617103475374419052610221389278607087042785800639} a^{14} + \frac{632653665333443079843892310256226464526465483545681661545756588216772463}{8625009570830687283031087617103475374419052610221389278607087042785800639} a^{13} + \frac{8344804392506758826012568302530197373656506851943344146660345487513495097}{8625009570830687283031087617103475374419052610221389278607087042785800639} a^{12} + \frac{5106471101522655631663321923572816261545597083105964820603546670898867742}{8625009570830687283031087617103475374419052610221389278607087042785800639} a^{11} + \frac{6943460898648156432362923175368257028240915863004296987333390736503913474}{8625009570830687283031087617103475374419052610221389278607087042785800639} a^{10} - \frac{11410409463119296275502419056452486855949062400751203107294612346659065184}{8625009570830687283031087617103475374419052610221389278607087042785800639} a^{9} + \frac{114172784437592850121800256818538102213051938573646671990991638981356940603}{8625009570830687283031087617103475374419052610221389278607087042785800639} a^{8} - \frac{27838377438809340001995005798928781793652054572074396290877912638026545135}{8625009570830687283031087617103475374419052610221389278607087042785800639} a^{7} - \frac{259679108781297018840069243876973806261738132749518861602181613209736109687}{8625009570830687283031087617103475374419052610221389278607087042785800639} a^{6} + \frac{19537310886495635007619287587257777733289223437862367510246141304808056508}{8625009570830687283031087617103475374419052610221389278607087042785800639} a^{5} + \frac{82036291839558606553418337866950511873687853425518874140759943747802561855}{8625009570830687283031087617103475374419052610221389278607087042785800639} a^{4} - \frac{10473495326753830394865679315900191447229764048523875546298469913697231249}{8625009570830687283031087617103475374419052610221389278607087042785800639} a^{3} - \frac{875220905265276034941452769166140488835180512855264541238255795451580007772}{8625009570830687283031087617103475374419052610221389278607087042785800639} a^{2} + \frac{85009051932320360804305291244781880004670800104829779788078790699582082359}{8625009570830687283031087617103475374419052610221389278607087042785800639} a + \frac{74714008849459725510516336379822444121395121762056217102197593833065117}{387275361269394606574966890445129332963003574613685477913299225126209} \) (order $6$)
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:  \( 5378062490297.565 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{14}$ (as 28T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 28
The 28 conjugacy class representatives for $C_2\times C_{14}$
Character table for $C_2\times C_{14}$ is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{-7})\), 7.7.594823321.1, 14.0.291381688005381590432263.1, 14.14.637251751667769538275359181.1, 14.0.773792930870360792667.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/5.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/31.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$

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
3Data not computed
$7$7.14.7.1$x^{14} - 117649 x^{2} + 1647086$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
7.14.7.1$x^{14} - 117649 x^{2} + 1647086$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$29$29.14.12.1$x^{14} + 2407 x^{7} + 1839267$$7$$2$$12$$C_{14}$$[\ ]_{7}^{2}$
29.14.12.1$x^{14} + 2407 x^{7} + 1839267$$7$$2$$12$$C_{14}$$[\ ]_{7}^{2}$