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Properties

Label 32.32.4756524023...0000.1
Degree $32$
Signature $[32, 0]$
Discriminant $2^{48}\cdot 5^{24}\cdot 17^{28}$
Root discriminant $112.83$
Ramified primes $2, 5, 17$
Class number Not computed
Class group Not computed
Galois group $C_4\times C_8$ (as 32T43)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-522239, 14755336, -5956104, -555932464, 538535816, 5285519244, -3540548500, -19452177308, 9895657258, 36075446764, -14997610308, -38328647620, 13793063056, 25146964908, -8173761298, -10699434056, 3236345879, 3049830256, -876219412, -594024912, 164374860, 79690868, -21441958, -7327884, 1929761, 451980, -117028, -17812, 4549, 404, -102, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 4*x^31 - 102*x^30 + 404*x^29 + 4549*x^28 - 17812*x^27 - 117028*x^26 + 451980*x^25 + 1929761*x^24 - 7327884*x^23 - 21441958*x^22 + 79690868*x^21 + 164374860*x^20 - 594024912*x^19 - 876219412*x^18 + 3049830256*x^17 + 3236345879*x^16 - 10699434056*x^15 - 8173761298*x^14 + 25146964908*x^13 + 13793063056*x^12 - 38328647620*x^11 - 14997610308*x^10 + 36075446764*x^9 + 9895657258*x^8 - 19452177308*x^7 - 3540548500*x^6 + 5285519244*x^5 + 538535816*x^4 - 555932464*x^3 - 5956104*x^2 + 14755336*x - 522239)
 
gp: K = bnfinit(x^32 - 4*x^31 - 102*x^30 + 404*x^29 + 4549*x^28 - 17812*x^27 - 117028*x^26 + 451980*x^25 + 1929761*x^24 - 7327884*x^23 - 21441958*x^22 + 79690868*x^21 + 164374860*x^20 - 594024912*x^19 - 876219412*x^18 + 3049830256*x^17 + 3236345879*x^16 - 10699434056*x^15 - 8173761298*x^14 + 25146964908*x^13 + 13793063056*x^12 - 38328647620*x^11 - 14997610308*x^10 + 36075446764*x^9 + 9895657258*x^8 - 19452177308*x^7 - 3540548500*x^6 + 5285519244*x^5 + 538535816*x^4 - 555932464*x^3 - 5956104*x^2 + 14755336*x - 522239, 1)
 

Normalized defining polynomial

\( x^{32} - 4 x^{31} - 102 x^{30} + 404 x^{29} + 4549 x^{28} - 17812 x^{27} - 117028 x^{26} + 451980 x^{25} + 1929761 x^{24} - 7327884 x^{23} - 21441958 x^{22} + 79690868 x^{21} + 164374860 x^{20} - 594024912 x^{19} - 876219412 x^{18} + 3049830256 x^{17} + 3236345879 x^{16} - 10699434056 x^{15} - 8173761298 x^{14} + 25146964908 x^{13} + 13793063056 x^{12} - 38328647620 x^{11} - 14997610308 x^{10} + 36075446764 x^{9} + 9895657258 x^{8} - 19452177308 x^{7} - 3540548500 x^{6} + 5285519244 x^{5} + 538535816 x^{4} - 555932464 x^{3} - 5956104 x^{2} + 14755336 x - 522239 \)

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

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[32, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(475652402320384421216839760983630708473856000000000000000000000000=2^{48}\cdot 5^{24}\cdot 17^{28}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $112.83$
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:  $2, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(680=2^{3}\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{680}(1,·)$, $\chi_{680}(387,·)$, $\chi_{680}(9,·)$, $\chi_{680}(523,·)$, $\chi_{680}(529,·)$, $\chi_{680}(627,·)$, $\chi_{680}(281,·)$, $\chi_{680}(409,·)$, $\chi_{680}(667,·)$, $\chi_{680}(161,·)$, $\chi_{680}(169,·)$, $\chi_{680}(427,·)$, $\chi_{680}(563,·)$, $\chi_{680}(49,·)$, $\chi_{680}(307,·)$, $\chi_{680}(569,·)$, $\chi_{680}(443,·)$, $\chi_{680}(321,·)$, $\chi_{680}(67,·)$, $\chi_{680}(203,·)$, $\chi_{680}(81,·)$, $\chi_{680}(467,·)$, $\chi_{680}(43,·)$, $\chi_{680}(441,·)$, $\chi_{680}(89,·)$, $\chi_{680}(603,·)$, $\chi_{680}(587,·)$, $\chi_{680}(361,·)$, $\chi_{680}(83,·)$, $\chi_{680}(489,·)$, $\chi_{680}(121,·)$, $\chi_{680}(123,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{24} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{25} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{26} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{27} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{28} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{29} - \frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7}$, $\frac{1}{3160945496842240579712748812353170240862} a^{30} - \frac{332494247392677558423818272980492880004}{1580472748421120289856374406176585120431} a^{29} + \frac{353399939383766975516789465134622212887}{1580472748421120289856374406176585120431} a^{28} - \frac{349708402470809355023670539623040447249}{1580472748421120289856374406176585120431} a^{27} - \frac{81807362084729094769356882677298005371}{3160945496842240579712748812353170240862} a^{26} + \frac{222672096687077836121967914336140638614}{1580472748421120289856374406176585120431} a^{25} + \frac{21385974421675662289475946106070046592}{1580472748421120289856374406176585120431} a^{24} - \frac{641976788382835725902611842951942808671}{3160945496842240579712748812353170240862} a^{23} - \frac{415304349599832839630286512363449032241}{3160945496842240579712748812353170240862} a^{22} - \frac{190093947800335655369180732982387828576}{1580472748421120289856374406176585120431} a^{21} - \frac{121992511229812463302191139239534944217}{3160945496842240579712748812353170240862} a^{20} - \frac{288077930334599026414734191569254810277}{3160945496842240579712748812353170240862} a^{19} + \frac{4836688616300553284605681421297010593}{35516241537553264940592683284867081358} a^{18} + \frac{223745448605028307370043809584982729015}{3160945496842240579712748812353170240862} a^{17} + \frac{38355610067940453661185338512793672994}{1580472748421120289856374406176585120431} a^{16} - \frac{1468941293734863664753204972471005363739}{3160945496842240579712748812353170240862} a^{15} + \frac{829534207999247947761211076554423395145}{3160945496842240579712748812353170240862} a^{14} + \frac{6380041412954020920639077862141661543}{17758120768776632470296341642433540679} a^{13} + \frac{951966716419380329407694551953807804989}{3160945496842240579712748812353170240862} a^{12} + \frac{712301858538988731622529091397337912221}{3160945496842240579712748812353170240862} a^{11} - \frac{603273142386309438464284477360309061897}{3160945496842240579712748812353170240862} a^{10} + \frac{815040507593089001677803510382611775829}{3160945496842240579712748812353170240862} a^{9} - \frac{183720384088325794824416852327428184045}{3160945496842240579712748812353170240862} a^{8} + \frac{1222451373694568974924433558681368262611}{3160945496842240579712748812353170240862} a^{7} - \frac{497745238913326506737628077115690353111}{3160945496842240579712748812353170240862} a^{6} - \frac{409245074672813360218297611711015273651}{3160945496842240579712748812353170240862} a^{5} - \frac{380005381746611400006083781755005115745}{3160945496842240579712748812353170240862} a^{4} - \frac{1574697876638488020075212373674790353203}{3160945496842240579712748812353170240862} a^{3} + \frac{322656301813598484190743456308879822772}{1580472748421120289856374406176585120431} a^{2} + \frac{199758177060620340916523294813412613325}{3160945496842240579712748812353170240862} a - \frac{163659316382872863772562995144860149071}{3160945496842240579712748812353170240862}$, $\frac{1}{7908700365953114688527714162425475740318545385403695474140939976946101298005460850391326649387803842} a^{31} + \frac{593801861149711459363848426842384863187007702930145984279531}{3954350182976557344263857081212737870159272692701847737070469988473050649002730425195663324693901921} a^{30} - \frac{75262140597784731755807539035858209470236344043900368979805876505344499449704738508572756716114951}{3954350182976557344263857081212737870159272692701847737070469988473050649002730425195663324693901921} a^{29} + \frac{768148866396985736773729496132726732128006216504915730690424856421664352944048557690569991127480337}{3954350182976557344263857081212737870159272692701847737070469988473050649002730425195663324693901921} a^{28} + \frac{1893882137497782402379527859326923064278786273786914148853431012071790434424001331644790000228712445}{7908700365953114688527714162425475740318545385403695474140939976946101298005460850391326649387803842} a^{27} + \frac{408346749758365499255616894739770628130003609513359246700024141637821634082025990460836087149753823}{3954350182976557344263857081212737870159272692701847737070469988473050649002730425195663324693901921} a^{26} + \frac{22659240144894958841348875281389051831180531855699227532820776163645473252128567050120565346204363}{3954350182976557344263857081212737870159272692701847737070469988473050649002730425195663324693901921} a^{25} + \frac{79627418872548699728112360217726512811740236479843400166404704896968704518782380992811427268839324}{3954350182976557344263857081212737870159272692701847737070469988473050649002730425195663324693901921} a^{24} - \frac{514974188528838524625107106205462213355146084408020412160054760924981171457785498203986793248875925}{7908700365953114688527714162425475740318545385403695474140939976946101298005460850391326649387803842} a^{23} + \frac{325547569289786935343965551292278699648673533023580857361402485855828131716984239779271936060621699}{3954350182976557344263857081212737870159272692701847737070469988473050649002730425195663324693901921} a^{22} - \frac{709547014536774878407529751635554954857477094401749694541817770358646333212493394931287707163202302}{3954350182976557344263857081212737870159272692701847737070469988473050649002730425195663324693901921} a^{21} + \frac{873698565974835926452396281933092052561522037158149078929470938570382075536121508149906041904569943}{3954350182976557344263857081212737870159272692701847737070469988473050649002730425195663324693901921} a^{20} - \frac{511715343327552126072536830679501397606868564042364416321493676566682752765813779824159934274664253}{3954350182976557344263857081212737870159272692701847737070469988473050649002730425195663324693901921} a^{19} + \frac{863168062549688712024878870183710195942055738932401971064845672256550136777925944688462987659849217}{7908700365953114688527714162425475740318545385403695474140939976946101298005460850391326649387803842} a^{18} - \frac{192421707970140919430607118275840263265519615115608321500647832435749565005451759536435172370827867}{7908700365953114688527714162425475740318545385403695474140939976946101298005460850391326649387803842} a^{17} + \frac{711291491188972272000226921562988729120623190481709357090930039546028156166777657492599183222538545}{7908700365953114688527714162425475740318545385403695474140939976946101298005460850391326649387803842} a^{16} + \frac{1551259389662960371706565633007576982953908414716399951151644394754420950585079655807208105171281875}{7908700365953114688527714162425475740318545385403695474140939976946101298005460850391326649387803842} a^{15} - \frac{537137822988404796993284276106375951571909745098935953415187712756676405620513243272645588269521669}{7908700365953114688527714162425475740318545385403695474140939976946101298005460850391326649387803842} a^{14} + \frac{632739996623312587443138235320628515554558840628270660453684849819653150443440475613774701691905469}{3954350182976557344263857081212737870159272692701847737070469988473050649002730425195663324693901921} a^{13} - \frac{1164493723660067561869455387367920370932341054061087118362911719563139831345001245319225344103818374}{3954350182976557344263857081212737870159272692701847737070469988473050649002730425195663324693901921} a^{12} - \frac{2904133360327146985689119190658526940992173605948591270917263591357624047944970220952291854501920305}{7908700365953114688527714162425475740318545385403695474140939976946101298005460850391326649387803842} a^{11} + \frac{1365761657423887692621607835226424072480563605396012162712150367977513867243803034112158776036879517}{7908700365953114688527714162425475740318545385403695474140939976946101298005460850391326649387803842} a^{10} - \frac{1891922468773758318820446067605837950880050817939240922458067229473594227945074115194792339093994299}{7908700365953114688527714162425475740318545385403695474140939976946101298005460850391326649387803842} a^{9} - \frac{362425163303408395394861890638721093589613516428519151864251498918555096778936252644351303272919207}{3954350182976557344263857081212737870159272692701847737070469988473050649002730425195663324693901921} a^{8} + \frac{2540790985882963557444477297415667778740177755976705318854385944582904052305374566059825780630807241}{7908700365953114688527714162425475740318545385403695474140939976946101298005460850391326649387803842} a^{7} + \frac{1782923721512131445325311157346863673030834356444056935581239763271094687231065364492325358271712201}{3954350182976557344263857081212737870159272692701847737070469988473050649002730425195663324693901921} a^{6} + \frac{1008622750689472685978297008919690940897632705923908531684466350338982416636676177921814742878667136}{3954350182976557344263857081212737870159272692701847737070469988473050649002730425195663324693901921} a^{5} - \frac{785031137483050964820713973180845613912157808254103542506612785644467809898102384661066603680716114}{3954350182976557344263857081212737870159272692701847737070469988473050649002730425195663324693901921} a^{4} - \frac{599324532825031585542083408146149058740096472322544867955657320121907696107346603296330577864310392}{3954350182976557344263857081212737870159272692701847737070469988473050649002730425195663324693901921} a^{3} - \frac{3270624323141478235088996188467849156939929612765673611741043945191038745190264835202074872924461423}{7908700365953114688527714162425475740318545385403695474140939976946101298005460850391326649387803842} a^{2} + \frac{2889052181422577665424648368146635009807876406151046307798735276153880013317483339619634526671652953}{7908700365953114688527714162425475740318545385403695474140939976946101298005460850391326649387803842} a - \frac{653888787683790226313531598312926945934134711351608364343543723095089257507633916018718419180273957}{3954350182976557344263857081212737870159272692701847737070469988473050649002730425195663324693901921}$

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:  $31$
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

$C_4\times C_8$ (as 32T43):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_4\times C_8$
Character table for $C_4\times C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{85}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{5}) \), 4.4.39304000.2, \(\Q(\sqrt{5}, \sqrt{17})\), 4.4.39304000.1, 4.4.122825.1, 4.4.4913.1, 4.4.8000.1, 4.4.2312000.1, 8.8.1544804416000000.3, 8.8.15085980625.1, 8.8.5345344000000.2, 8.8.26261675072000000.1, 8.8.26261675072000000.2, \(\Q(\zeta_{17})^+\), 8.8.256461670625.1, 16.16.2386420683693101056000000000000.1, 16.16.689675577587306205184000000000000.1, 16.16.65772588499765987890625.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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
$2$2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
5Data not computed
17Data not computed