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Properties

Label 32.0.15388994837...2801.1
Degree $32$
Signature $[0, 16]$
Discriminant $13^{24}\cdot 17^{28}$
Root discriminant $81.68$
Ramified primes $13, 17$
Class number $97920$ (GRH)
Class group $[2, 2, 2, 2, 6, 1020]$ (GRH)
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![34431289, -14835712, 50770223, 47363710, 111572666, -165678188, 842388, 175428026, 33195458, -17714304, 13433300, -16856174, 10151814, 9810462, 2685146, 5502716, 1621632, 815952, 698130, -5090, 119832, 55176, -6246, 6274, -1596, 1208, 1218, -558, -46, 90, -9, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 4*x^31 - 9*x^30 + 90*x^29 - 46*x^28 - 558*x^27 + 1218*x^26 + 1208*x^25 - 1596*x^24 + 6274*x^23 - 6246*x^22 + 55176*x^21 + 119832*x^20 - 5090*x^19 + 698130*x^18 + 815952*x^17 + 1621632*x^16 + 5502716*x^15 + 2685146*x^14 + 9810462*x^13 + 10151814*x^12 - 16856174*x^11 + 13433300*x^10 - 17714304*x^9 + 33195458*x^8 + 175428026*x^7 + 842388*x^6 - 165678188*x^5 + 111572666*x^4 + 47363710*x^3 + 50770223*x^2 - 14835712*x + 34431289)
 
gp: K = bnfinit(x^32 - 4*x^31 - 9*x^30 + 90*x^29 - 46*x^28 - 558*x^27 + 1218*x^26 + 1208*x^25 - 1596*x^24 + 6274*x^23 - 6246*x^22 + 55176*x^21 + 119832*x^20 - 5090*x^19 + 698130*x^18 + 815952*x^17 + 1621632*x^16 + 5502716*x^15 + 2685146*x^14 + 9810462*x^13 + 10151814*x^12 - 16856174*x^11 + 13433300*x^10 - 17714304*x^9 + 33195458*x^8 + 175428026*x^7 + 842388*x^6 - 165678188*x^5 + 111572666*x^4 + 47363710*x^3 + 50770223*x^2 - 14835712*x + 34431289, 1)
 

Normalized defining polynomial

\( x^{32} - 4 x^{31} - 9 x^{30} + 90 x^{29} - 46 x^{28} - 558 x^{27} + 1218 x^{26} + 1208 x^{25} - 1596 x^{24} + 6274 x^{23} - 6246 x^{22} + 55176 x^{21} + 119832 x^{20} - 5090 x^{19} + 698130 x^{18} + 815952 x^{17} + 1621632 x^{16} + 5502716 x^{15} + 2685146 x^{14} + 9810462 x^{13} + 10151814 x^{12} - 16856174 x^{11} + 13433300 x^{10} - 17714304 x^{9} + 33195458 x^{8} + 175428026 x^{7} + 842388 x^{6} - 165678188 x^{5} + 111572666 x^{4} + 47363710 x^{3} + 50770223 x^{2} - 14835712 x + 34431289 \)

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

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 16]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(15388994837398808258189466602517183151497694747133163269152801=13^{24}\cdot 17^{28}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.68$
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:  $13, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(221=13\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{221}(1,·)$, $\chi_{221}(135,·)$, $\chi_{221}(8,·)$, $\chi_{221}(138,·)$, $\chi_{221}(144,·)$, $\chi_{221}(18,·)$, $\chi_{221}(21,·)$, $\chi_{221}(151,·)$, $\chi_{221}(25,·)$, $\chi_{221}(155,·)$, $\chi_{221}(157,·)$, $\chi_{221}(161,·)$, $\chi_{221}(38,·)$, $\chi_{221}(168,·)$, $\chi_{221}(174,·)$, $\chi_{221}(47,·)$, $\chi_{221}(53,·)$, $\chi_{221}(183,·)$, $\chi_{221}(60,·)$, $\chi_{221}(64,·)$, $\chi_{221}(66,·)$, $\chi_{221}(196,·)$, $\chi_{221}(70,·)$, $\chi_{221}(200,·)$, $\chi_{221}(203,·)$, $\chi_{221}(77,·)$, $\chi_{221}(83,·)$, $\chi_{221}(213,·)$, $\chi_{221}(86,·)$, $\chi_{221}(220,·)$, $\chi_{221}(103,·)$, $\chi_{221}(118,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{8}$, $\frac{1}{6} a^{24} + \frac{1}{6} a^{22} + \frac{1}{6} a^{20} - \frac{1}{6} a^{18} + \frac{1}{6} a^{16} - \frac{1}{3} a^{14} - \frac{1}{2} a^{9} - \frac{1}{3} a^{8} - \frac{1}{2} a^{7} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} + \frac{1}{3} a^{4} - \frac{1}{6} a^{3} - \frac{1}{3} a^{2} + \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{6} a^{25} + \frac{1}{6} a^{23} + \frac{1}{6} a^{21} - \frac{1}{6} a^{19} + \frac{1}{6} a^{17} + \frac{1}{6} a^{15} - \frac{1}{2} a^{10} - \frac{1}{3} a^{9} - \frac{1}{2} a^{8} - \frac{1}{3} a^{7} - \frac{1}{2} a^{6} + \frac{1}{3} a^{5} - \frac{1}{6} a^{4} - \frac{1}{3} a^{3} + \frac{1}{6} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{6} a^{26} + \frac{1}{6} a^{20} - \frac{1}{6} a^{18} + \frac{1}{3} a^{14} - \frac{1}{2} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{6} - \frac{1}{6} a^{5} + \frac{1}{3} a^{4} - \frac{1}{6} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{6} a^{27} + \frac{1}{6} a^{21} - \frac{1}{6} a^{19} - \frac{1}{6} a^{15} - \frac{1}{2} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{7} - \frac{1}{6} a^{6} + \frac{1}{3} a^{5} - \frac{1}{6} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{6} a^{28} + \frac{1}{6} a^{22} - \frac{1}{6} a^{20} - \frac{1}{6} a^{16} - \frac{1}{2} a^{13} - \frac{1}{3} a^{12} - \frac{1}{3} a^{8} - \frac{1}{6} a^{7} + \frac{1}{3} a^{6} - \frac{1}{6} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{29} + \frac{1}{6} a^{23} - \frac{1}{6} a^{21} - \frac{1}{6} a^{17} - \frac{1}{2} a^{14} - \frac{1}{3} a^{13} - \frac{1}{3} a^{9} - \frac{1}{6} a^{8} + \frac{1}{3} a^{7} - \frac{1}{6} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{24} a^{30} - \frac{1}{12} a^{27} - \frac{1}{12} a^{26} + \frac{1}{6} a^{22} - \frac{1}{12} a^{21} - \frac{1}{4} a^{20} + \frac{1}{12} a^{19} + \frac{1}{12} a^{18} - \frac{1}{6} a^{16} - \frac{1}{6} a^{15} - \frac{1}{6} a^{14} - \frac{1}{4} a^{12} + \frac{5}{12} a^{11} + \frac{1}{3} a^{10} - \frac{1}{6} a^{9} + \frac{1}{6} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{12} a^{3} - \frac{1}{6} a^{2} - \frac{1}{6} a - \frac{1}{24}$, $\frac{1}{518190946522178006289558680795084829973912190943869150839848985815969875718633070689023146927831690452430655881546411032818964557965924303765968} a^{31} + \frac{3122503064926619402632573744213922287395156775592524021883167920424168679074012471870678557608894962934344782947785626770831526359894876688665}{172730315507392668763186226931694943324637396981289716946616328605323291906211023563007715642610563484143551960515470344272988185988641434588656} a^{30} + \frac{2186599558524042782629888807705480728151578243498437994269389504451964408100317934957134872995188478059210004127670931124410845662572680106991}{32386934157636125393097417549692801873369511933991821927490561613498117232414566918063946682989480653276915992596650689551185284872870268985373} a^{29} - \frac{1500821981076102230210107002519381997128065546450995171105983706759126809650915450589783659147456793863117783943103430780620256765220965896333}{86365157753696334381593113465847471662318698490644858473308164302661645953105511781503857821305281742071775980257735172136494092994320717294328} a^{28} - \frac{935578922945752288428464645998356597688010540579859167113654757418987832841885753924993415178646672113334841360102868237243398678212680234304}{32386934157636125393097417549692801873369511933991821927490561613498117232414566918063946682989480653276915992596650689551185284872870268985373} a^{27} + \frac{7147806071217124945927379125436133260545573043154257862505782695911181492645272867499546783289454054407419741934832321245470500051501331819091}{86365157753696334381593113465847471662318698490644858473308164302661645953105511781503857821305281742071775980257735172136494092994320717294328} a^{26} + \frac{785623987205942972722276305297204414708201196608754326033785267585696480689006756063513225312161719782989905998454008040372323447963366564253}{21591289438424083595398278366461867915579674622661214618327041075665411488276377945375964455326320435517943995064433793034123523248580179323582} a^{25} + \frac{2131814161040819196905613566171014485106722217556549262103061683330498794079155441987088340064486279496631173258340894805299354688978756883291}{32386934157636125393097417549692801873369511933991821927490561613498117232414566918063946682989480653276915992596650689551185284872870268985373} a^{24} - \frac{2830706614077449955509834226205348178913725393702077285851598799263218116898380754583399000514927708552798830633512657486549472407009758217895}{43182578876848167190796556732923735831159349245322429236654082151330822976552755890751928910652640871035887990128867586068247046497160358647164} a^{23} - \frac{32930582251165075005533140475077615366740578607243924544213812139849394575738123860083738585423068726626573506480590525015821486588920806704517}{259095473261089003144779340397542414986956095471934575419924492907984937859316535344511573463915845226215327940773205516409482278982962151882984} a^{22} - \frac{25718552979757797322532727334913283510562370938220026950593674313307743104366649907307256019301338246410465170244120241244505389531898312328137}{129547736630544501572389670198771207493478047735967287709962246453992468929658267672255786731957922613107663970386602758204741139491481075941492} a^{21} + \frac{20642578182941258441381961378287214374579336015828185306499829181277084920733116857054317073870272701238511002430201106453616093896505984986807}{129547736630544501572389670198771207493478047735967287709962246453992468929658267672255786731957922613107663970386602758204741139491481075941492} a^{20} - \frac{17602259035599023761915524878241101286210773109760071491200228744590409365162461885184810087079308556924097442532499647264861961636821677988351}{129547736630544501572389670198771207493478047735967287709962246453992468929658267672255786731957922613107663970386602758204741139491481075941492} a^{19} - \frac{15383516725991278258648978158619847728313546066426859416416841249281698139633781840675535533609281609145702693058693088947712413028497675444503}{259095473261089003144779340397542414986956095471934575419924492907984937859316535344511573463915845226215327940773205516409482278982962151882984} a^{18} - \frac{4531065706320536677089358548391621283422453181314867663363076661546375063155861359201868731151435249621024092044698888003397493365317811644919}{21591289438424083595398278366461867915579674622661214618327041075665411488276377945375964455326320435517943995064433793034123523248580179323582} a^{17} - \frac{2484671521641269751779733571594950187101789634358182739665419137586782705267855294265614342840544308290906518578854864495698665211165721332799}{64773868315272250786194835099385603746739023867983643854981123226996234464829133836127893365978961306553831985193301379102370569745740537970746} a^{16} - \frac{7067050530460860779876573634097239052076768354429632506006509994150449127169056645100843573915716984512203997468699326118612341766077339137691}{32386934157636125393097417549692801873369511933991821927490561613498117232414566918063946682989480653276915992596650689551185284872870268985373} a^{15} - \frac{36236318474701415896137264998147442271824070996899935513309761272936152553742040839178556150344948449451894303915336084213987993179966605996197}{129547736630544501572389670198771207493478047735967287709962246453992468929658267672255786731957922613107663970386602758204741139491481075941492} a^{14} + \frac{67279886051612300064141072089529326574894621925805714286666245450760574629846198844249123187113762389152115194185442165353157766966406062062563}{259095473261089003144779340397542414986956095471934575419924492907984937859316535344511573463915845226215327940773205516409482278982962151882984} a^{13} + \frac{5525354532258807333619896026254952271275879959312315081944327508934637632633467662681427351181183167376263461301505694000317484464262247211039}{21591289438424083595398278366461867915579674622661214618327041075665411488276377945375964455326320435517943995064433793034123523248580179323582} a^{12} - \frac{73415908836087925591580544682120670258860046301140231321194682276527647737824341157271933956206965669755332326229626155679345084653183357216673}{259095473261089003144779340397542414986956095471934575419924492907984937859316535344511573463915845226215327940773205516409482278982962151882984} a^{11} + \frac{48391433668315600875950128570623049882705196129997037810170775241363379139941866556601420566845268050354857950522422028981838549484246130471455}{129547736630544501572389670198771207493478047735967287709962246453992468929658267672255786731957922613107663970386602758204741139491481075941492} a^{10} - \frac{9883054562414678935382264575028428582841717592862418762076357814820710221322019377067360542495092271814064306765575936427436587927032030865787}{21591289438424083595398278366461867915579674622661214618327041075665411488276377945375964455326320435517943995064433793034123523248580179323582} a^{9} - \frac{3986317268179153721062956834916938517773526288334256711051230835571814626930924249203137631689947647387139865142954032885725818423899306479721}{32386934157636125393097417549692801873369511933991821927490561613498117232414566918063946682989480653276915992596650689551185284872870268985373} a^{8} + \frac{62520714860903129539010000437510457872853279791780059882132069031357369598598756694355490876815978874686825224485019087503193847064675525839377}{259095473261089003144779340397542414986956095471934575419924492907984937859316535344511573463915845226215327940773205516409482278982962151882984} a^{7} + \frac{2614493106713206350429621328299589460220147846665364020939615355615122248510356914629369947009424552647566091159183964089202252054268866129519}{10795644719212041797699139183230933957789837311330607309163520537832705744138188972687982227663160217758971997532216896517061761624290089661791} a^{6} - \frac{5529722100860860881165655443380742524016678752483342403289502102443697961104343222582565070722684703138203465272738406038959370806129736354669}{43182578876848167190796556732923735831159349245322429236654082151330822976552755890751928910652640871035887990128867586068247046497160358647164} a^{5} - \frac{1240471253342642575114317627407476151519330433086782707883427152288509431403799544348038804134591606998131239162653133321482462607698892118307}{21591289438424083595398278366461867915579674622661214618327041075665411488276377945375964455326320435517943995064433793034123523248580179323582} a^{4} + \frac{32477038385374903985348094783721553159147771832849672349024353858356168513608216040504052116501172093561126249480198088117869567736645794690437}{259095473261089003144779340397542414986956095471934575419924492907984937859316535344511573463915845226215327940773205516409482278982962151882984} a^{3} + \frac{30053510571925766155653175460894152369628263259452116292807615782210873109674182942104718619454655630799350388857671690724157678237738652043963}{129547736630544501572389670198771207493478047735967287709962246453992468929658267672255786731957922613107663970386602758204741139491481075941492} a^{2} + \frac{24203454634110214807746841308741745957476970456785223994052228372917048076038554818648745676946378216368393903539907525464445613200907621961921}{172730315507392668763186226931694943324637396981289716946616328605323291906211023563007715642610563484143551960515470344272988185988641434588656} a - \frac{317085007315294369589442259878109009027806255811512759907818139028989383323251383784538708176094126580329546299053529872073696678846840362209}{1169731256257738163181848037912155372401607654500833297606882586492031322163957270178381821507520745942281390251797767568440100582315856216176}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{6}\times C_{1020}$, which has order $97920$ (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:  $15$
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:  \( 2356853515467.5938 \) (assuming GRH)
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

\(\Q(\sqrt{221}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{13}) \), 4.0.10793861.1, \(\Q(\sqrt{13}, \sqrt{17})\), 4.0.10793861.2, 4.4.830297.1, 4.4.4913.1, 4.0.634933.1, 4.0.2197.1, 8.0.116507435287321.1, 8.8.689393108209.1, 8.0.403139914489.1, 8.0.1980626399884457.2, 8.0.1980626399884457.1, 8.8.11719682839553.1, \(\Q(\zeta_{17})^+\), 16.0.13573982477229290545823357041.2, 16.0.3922880935919264967742950184849.4, 16.16.137350965859713069141239809.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ R 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
$13$13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17Data not computed