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Properties

Label 32.32.2318482735...0625.2
Degree $32$
Signature $[32, 0]$
Discriminant $5^{24}\cdot 97^{31}$
Root discriminant $281.13$
Ramified primes $5, 97$
Class number Not computed
Class group Not computed
Galois group $C_{32}$ (as 32T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-7522269649, -15465730875, 244081450784, -309785246453, -1005718811728, 2162678325954, 922086899405, -4492953890543, 675907939625, 4578752971493, -1767203506112, -2722189078308, 1394173531832, 1031625598042, -606028450275, -262233944896, 165619980768, 45970229245, -30036394961, -5615084090, 3700993657, 475350824, -311391664, -27276312, 17695139, 1014228, -659185, -22502, 15176, 254, -192, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - x^31 - 192*x^30 + 254*x^29 + 15176*x^28 - 22502*x^27 - 659185*x^26 + 1014228*x^25 + 17695139*x^24 - 27276312*x^23 - 311391664*x^22 + 475350824*x^21 + 3700993657*x^20 - 5615084090*x^19 - 30036394961*x^18 + 45970229245*x^17 + 165619980768*x^16 - 262233944896*x^15 - 606028450275*x^14 + 1031625598042*x^13 + 1394173531832*x^12 - 2722189078308*x^11 - 1767203506112*x^10 + 4578752971493*x^9 + 675907939625*x^8 - 4492953890543*x^7 + 922086899405*x^6 + 2162678325954*x^5 - 1005718811728*x^4 - 309785246453*x^3 + 244081450784*x^2 - 15465730875*x - 7522269649)
 
gp: K = bnfinit(x^32 - x^31 - 192*x^30 + 254*x^29 + 15176*x^28 - 22502*x^27 - 659185*x^26 + 1014228*x^25 + 17695139*x^24 - 27276312*x^23 - 311391664*x^22 + 475350824*x^21 + 3700993657*x^20 - 5615084090*x^19 - 30036394961*x^18 + 45970229245*x^17 + 165619980768*x^16 - 262233944896*x^15 - 606028450275*x^14 + 1031625598042*x^13 + 1394173531832*x^12 - 2722189078308*x^11 - 1767203506112*x^10 + 4578752971493*x^9 + 675907939625*x^8 - 4492953890543*x^7 + 922086899405*x^6 + 2162678325954*x^5 - 1005718811728*x^4 - 309785246453*x^3 + 244081450784*x^2 - 15465730875*x - 7522269649, 1)
 

Normalized defining polynomial

\( x^{32} - x^{31} - 192 x^{30} + 254 x^{29} + 15176 x^{28} - 22502 x^{27} - 659185 x^{26} + 1014228 x^{25} + 17695139 x^{24} - 27276312 x^{23} - 311391664 x^{22} + 475350824 x^{21} + 3700993657 x^{20} - 5615084090 x^{19} - 30036394961 x^{18} + 45970229245 x^{17} + 165619980768 x^{16} - 262233944896 x^{15} - 606028450275 x^{14} + 1031625598042 x^{13} + 1394173531832 x^{12} - 2722189078308 x^{11} - 1767203506112 x^{10} + 4578752971493 x^{9} + 675907939625 x^{8} - 4492953890543 x^{7} + 922086899405 x^{6} + 2162678325954 x^{5} - 1005718811728 x^{4} - 309785246453 x^{3} + 244081450784 x^{2} - 15465730875 x - 7522269649 \)

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:  \(2318482735674757180308401186646582923236113207344277714859125196933746337890625=5^{24}\cdot 97^{31}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $281.13$
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:  $5, 97$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(485=5\cdot 97\)
Dirichlet character group:    $\lbrace$$\chi_{485}(1,·)$, $\chi_{485}(263,·)$, $\chi_{485}(264,·)$, $\chi_{485}(407,·)$, $\chi_{485}(142,·)$, $\chi_{485}(143,·)$, $\chi_{485}(272,·)$, $\chi_{485}(148,·)$, $\chi_{485}(279,·)$, $\chi_{485}(28,·)$, $\chi_{485}(161,·)$, $\chi_{485}(421,·)$, $\chi_{485}(299,·)$, $\chi_{485}(52,·)$, $\chi_{485}(309,·)$, $\chi_{485}(443,·)$, $\chi_{485}(63,·)$, $\chi_{485}(67,·)$, $\chi_{485}(77,·)$, $\chi_{485}(333,·)$, $\chi_{485}(341,·)$, $\chi_{485}(216,·)$, $\chi_{485}(89,·)$, $\chi_{485}(79,·)$, $\chi_{485}(96,·)$, $\chi_{485}(228,·)$, $\chi_{485}(109,·)$, $\chi_{485}(366,·)$, $\chi_{485}(241,·)$, $\chi_{485}(117,·)$, $\chi_{485}(124,·)$, $\chi_{485}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $\frac{1}{61} a^{26} - \frac{21}{61} a^{25} - \frac{19}{61} a^{24} + \frac{7}{61} a^{23} + \frac{7}{61} a^{22} - \frac{19}{61} a^{21} - \frac{21}{61} a^{20} - \frac{15}{61} a^{18} + \frac{3}{61} a^{17} + \frac{6}{61} a^{16} - \frac{29}{61} a^{15} + \frac{21}{61} a^{14} + \frac{22}{61} a^{13} - \frac{28}{61} a^{12} + \frac{7}{61} a^{11} - \frac{11}{61} a^{10} - \frac{28}{61} a^{9} + \frac{30}{61} a^{8} + \frac{4}{61} a^{7} + \frac{25}{61} a^{6} + \frac{2}{61} a^{5} + \frac{9}{61} a^{4} + \frac{13}{61} a^{3} + \frac{25}{61} a^{2} - \frac{23}{61} a + \frac{21}{61}$, $\frac{1}{61} a^{27} + \frac{28}{61} a^{25} - \frac{26}{61} a^{24} - \frac{29}{61} a^{23} + \frac{6}{61} a^{22} + \frac{7}{61} a^{21} - \frac{14}{61} a^{20} - \frac{15}{61} a^{19} - \frac{7}{61} a^{18} + \frac{8}{61} a^{17} - \frac{25}{61} a^{16} + \frac{22}{61} a^{15} - \frac{25}{61} a^{14} + \frac{7}{61} a^{13} + \frac{29}{61} a^{12} + \frac{14}{61} a^{11} - \frac{15}{61} a^{10} - \frac{9}{61} a^{9} + \frac{24}{61} a^{8} - \frac{13}{61} a^{7} - \frac{22}{61} a^{6} - \frac{10}{61} a^{5} + \frac{19}{61} a^{4} - \frac{7}{61} a^{3} + \frac{14}{61} a^{2} + \frac{26}{61} a + \frac{14}{61}$, $\frac{1}{61} a^{28} + \frac{13}{61} a^{25} + \frac{15}{61} a^{24} - \frac{7}{61} a^{23} - \frac{6}{61} a^{22} + \frac{30}{61} a^{21} + \frac{24}{61} a^{20} - \frac{7}{61} a^{19} + \frac{1}{61} a^{18} + \frac{13}{61} a^{17} - \frac{24}{61} a^{16} - \frac{6}{61} a^{15} + \frac{29}{61} a^{14} + \frac{23}{61} a^{13} + \frac{5}{61} a^{12} - \frac{28}{61} a^{11} - \frac{6}{61} a^{10} + \frac{15}{61} a^{9} + \frac{1}{61} a^{8} - \frac{12}{61} a^{7} + \frac{22}{61} a^{6} + \frac{24}{61} a^{5} - \frac{15}{61} a^{4} + \frac{16}{61} a^{3} - \frac{3}{61} a^{2} - \frac{13}{61} a + \frac{22}{61}$, $\frac{1}{61} a^{29} - \frac{17}{61} a^{25} - \frac{4}{61} a^{24} + \frac{25}{61} a^{23} + \frac{27}{61} a^{21} + \frac{22}{61} a^{20} + \frac{1}{61} a^{19} + \frac{25}{61} a^{18} - \frac{2}{61} a^{17} - \frac{23}{61} a^{16} - \frac{21}{61} a^{15} - \frac{6}{61} a^{14} + \frac{24}{61} a^{13} - \frac{30}{61} a^{12} + \frac{25}{61} a^{11} - \frac{25}{61} a^{10} - \frac{1}{61} a^{9} + \frac{25}{61} a^{8} - \frac{30}{61} a^{7} + \frac{4}{61} a^{6} + \frac{20}{61} a^{5} + \frac{21}{61} a^{4} + \frac{11}{61} a^{3} + \frac{28}{61} a^{2} + \frac{16}{61} a - \frac{29}{61}$, $\frac{1}{68869} a^{30} - \frac{219}{68869} a^{29} + \frac{137}{68869} a^{28} - \frac{224}{68869} a^{27} + \frac{340}{68869} a^{26} + \frac{8384}{68869} a^{25} + \frac{17796}{68869} a^{24} - \frac{13299}{68869} a^{23} + \frac{7924}{68869} a^{22} + \frac{19575}{68869} a^{21} - \frac{10831}{68869} a^{20} + \frac{24960}{68869} a^{19} + \frac{30584}{68869} a^{18} - \frac{11579}{68869} a^{17} - \frac{6756}{68869} a^{16} + \frac{6058}{68869} a^{15} - \frac{23011}{68869} a^{14} - \frac{1644}{68869} a^{13} + \frac{28120}{68869} a^{12} - \frac{31750}{68869} a^{11} - \frac{24585}{68869} a^{10} + \frac{17438}{68869} a^{9} - \frac{28765}{68869} a^{8} - \frac{22694}{68869} a^{7} + \frac{5275}{68869} a^{6} - \frac{15197}{68869} a^{5} - \frac{49}{1129} a^{4} + \frac{8033}{68869} a^{3} - \frac{3239}{68869} a^{2} - \frac{19166}{68869} a + \frac{16044}{68869}$, $\frac{1}{83620588305290694065806019200195179360861982993866260972209999730792522060828755375983057245707875986289314178486384859902673413830280042746703497565432868548434508444587} a^{31} - \frac{586878558988711996651653332177189636740804516256688341765886367373701815060834018954517820336803794190001387015332786326666828407961362117701509166343918613549031540}{83620588305290694065806019200195179360861982993866260972209999730792522060828755375983057245707875986289314178486384859902673413830280042746703497565432868548434508444587} a^{30} - \frac{113446684530911340737405858532697614221740129758310871189546566221339578552981918235316304505908474968474260531233590894189524606131233497846012166651291214971125486433}{83620588305290694065806019200195179360861982993866260972209999730792522060828755375983057245707875986289314178486384859902673413830280042746703497565432868548434508444587} a^{29} + \frac{354339981209362789622595002201448136130231425844730620375038487457622234041093402334499954274702060576683431755729726469307033275190969979266147757251456824490179109717}{83620588305290694065806019200195179360861982993866260972209999730792522060828755375983057245707875986289314178486384859902673413830280042746703497565432868548434508444587} a^{28} - \frac{415324635853989548227881018396467419902966478196695774489954230818234111230225494709032245936092021583194507228768171336225926287745087110010045558827884862056008753128}{83620588305290694065806019200195179360861982993866260972209999730792522060828755375983057245707875986289314178486384859902673413830280042746703497565432868548434508444587} a^{27} + \frac{553455124078046382337853856884392767216047613234775920143123538454214277329563417679730869515401746643301023584012018600924922340910625336157543728519870847748429409448}{83620588305290694065806019200195179360861982993866260972209999730792522060828755375983057245707875986289314178486384859902673413830280042746703497565432868548434508444587} a^{26} - \frac{31698437266929296825485964923055925157181655598483966541573656943045279397776954823113893641831968267427884853864061532737474270489218478085885084277827707611273098138791}{83620588305290694065806019200195179360861982993866260972209999730792522060828755375983057245707875986289314178486384859902673413830280042746703497565432868548434508444587} a^{25} + \frac{2416375157643328530564330949848986038612043418850238485118910887710840835560099863268101650561560962112909600934412980758923517582618282283030440004965630350260647605880}{83620588305290694065806019200195179360861982993866260972209999730792522060828755375983057245707875986289314178486384859902673413830280042746703497565432868548434508444587} a^{24} - \frac{17319499477689489312579526940617072737848330910133913961091260461636703734917054269729431751974580256045571648420815061778969456407930319422669448733434598345981702418865}{83620588305290694065806019200195179360861982993866260972209999730792522060828755375983057245707875986289314178486384859902673413830280042746703497565432868548434508444587} a^{23} - \frac{12508245356770578649276729073529399037128842663865705065939520002736987046168175943811516360352722073822019947009388544479937351385966567549654159997479567122598516194789}{83620588305290694065806019200195179360861982993866260972209999730792522060828755375983057245707875986289314178486384859902673413830280042746703497565432868548434508444587} a^{22} + \frac{10658192619385523494991363938800277083683994869735068581288549926361534714819774736802386396493472918257755572811292460036407236941915999537304471049308616161044498958657}{83620588305290694065806019200195179360861982993866260972209999730792522060828755375983057245707875986289314178486384859902673413830280042746703497565432868548434508444587} a^{21} + \frac{15548243230910096825736357427280540458092359779553508574064544755598874768218239583002079786445345800881051708144675150387453208742101202836357585579239059731433953542273}{83620588305290694065806019200195179360861982993866260972209999730792522060828755375983057245707875986289314178486384859902673413830280042746703497565432868548434508444587} a^{20} + \frac{40495132099189688084287816957209067379368660286425124867283335483197557905697791035602607584453501245543415554087071490335267714506265808569867582076753114737145844787069}{83620588305290694065806019200195179360861982993866260972209999730792522060828755375983057245707875986289314178486384859902673413830280042746703497565432868548434508444587} a^{19} - \frac{21835775381904228463342276171525257532234504744707516208283385939362811289915964880638620720835829683060564132814702924456424157408272185762772211575809993628012406381993}{83620588305290694065806019200195179360861982993866260972209999730792522060828755375983057245707875986289314178486384859902673413830280042746703497565432868548434508444587} a^{18} + \frac{9078580101588717190887958013691057353856420477658874025259637993750060732196810844826511226735592397292313696500058800666247764616948297333944343480899434421335440277520}{83620588305290694065806019200195179360861982993866260972209999730792522060828755375983057245707875986289314178486384859902673413830280042746703497565432868548434508444587} a^{17} - \frac{2346909001508823877870490701752413625123439672753245901660927033838984774263573338288312740971471755407029124217901731091894986957817620095803021154498031250889977018276}{83620588305290694065806019200195179360861982993866260972209999730792522060828755375983057245707875986289314178486384859902673413830280042746703497565432868548434508444587} a^{16} - \frac{22749864196004546400443281189534419739180572161124432029745454783469903943962895392803225328811251903762220172668019476542076016659180377580924348162920177482563363639345}{83620588305290694065806019200195179360861982993866260972209999730792522060828755375983057245707875986289314178486384859902673413830280042746703497565432868548434508444587} a^{15} + \frac{35607218774532785426963498881802408376667629557787502683373820137338744648798553307040300777934659352747723122680490050959293570506919913964205443335594984620651965319690}{83620588305290694065806019200195179360861982993866260972209999730792522060828755375983057245707875986289314178486384859902673413830280042746703497565432868548434508444587} a^{14} + \frac{18719309804500267944629231180162843164941809230288539865708763352958542197114093239205998586700891745377470598831246972319361509088676539286622136912410195435140871844384}{83620588305290694065806019200195179360861982993866260972209999730792522060828755375983057245707875986289314178486384859902673413830280042746703497565432868548434508444587} a^{13} + \frac{41696620933828369589107930464338774416697514406033364140164724348779061424488844127691984341602002388995929528258679532840085649345819254814558847576084203855688566112240}{83620588305290694065806019200195179360861982993866260972209999730792522060828755375983057245707875986289314178486384859902673413830280042746703497565432868548434508444587} a^{12} + \frac{566516428313837505676221274353831107799902573765246923170280083268298595777997847280565854075031022703281539276337158657898593774073714269761041385158328240766628346044}{1370829316480175312554197036068773432145278409735512474954262290668729869849651727475132085995211081742447773417809587867256941210332459717159073730580866697515319810567} a^{11} - \frac{38978755383628611992089256745583299827637201982976440428630771418035636152589409885306061548299062797378118453222685294836226492393949830034995971043456272287027415230509}{83620588305290694065806019200195179360861982993866260972209999730792522060828755375983057245707875986289314178486384859902673413830280042746703497565432868548434508444587} a^{10} - \frac{3926106468898902585533642986371404462683557010291875797042218888000238452163122554176515692642144379791462913804461362969739837061231925538012861803932743827988017870453}{83620588305290694065806019200195179360861982993866260972209999730792522060828755375983057245707875986289314178486384859902673413830280042746703497565432868548434508444587} a^{9} + \frac{16612720649396539066515151499801777073654376175091753448558574762496779593553177363238206952578142263027826770769233751052731455972461614152095844581729595335331510315125}{83620588305290694065806019200195179360861982993866260972209999730792522060828755375983057245707875986289314178486384859902673413830280042746703497565432868548434508444587} a^{8} - \frac{7595769212824263662174973114020139238814394207534328962553569251973790437963507550049824052484386900258679607004912292058590072557115891034673605981853789010835336844091}{83620588305290694065806019200195179360861982993866260972209999730792522060828755375983057245707875986289314178486384859902673413830280042746703497565432868548434508444587} a^{7} - \frac{3733011742499036126931291615195930830620883791657928598707568041721454065122185133653919681940734929636107896796496788950298187731513848204539267657273390739358770242633}{83620588305290694065806019200195179360861982993866260972209999730792522060828755375983057245707875986289314178486384859902673413830280042746703497565432868548434508444587} a^{6} + \frac{31965536271778437527165955296441938498082035749733598119359850490515887763344203189051563360811633824208691025314679322549714344307342467458203624608700765739225097300165}{83620588305290694065806019200195179360861982993866260972209999730792522060828755375983057245707875986289314178486384859902673413830280042746703497565432868548434508444587} a^{5} + \frac{15889760795126954134746131209684156061405702923277095369201559319443520422668376491123250038673693811169017720006814551258503949032324110977161574809803428729843730872282}{83620588305290694065806019200195179360861982993866260972209999730792522060828755375983057245707875986289314178486384859902673413830280042746703497565432868548434508444587} a^{4} - \frac{34768757174503672884007877071646668933052160229709310700991426705802256030040143855357124873113533769424988343065847486120775299327993784581876040598673821562476319348553}{83620588305290694065806019200195179360861982993866260972209999730792522060828755375983057245707875986289314178486384859902673413830280042746703497565432868548434508444587} a^{3} + \frac{13474424170250526600581689408220411122198488012032107720890354482766841139298596805881643499514210826200297442915657571157065197588200537731262409361226166023487119141993}{83620588305290694065806019200195179360861982993866260972209999730792522060828755375983057245707875986289314178486384859902673413830280042746703497565432868548434508444587} a^{2} + \frac{39629515434304850819204377271664087272646291162436551783820965735241180284985248379146816016309117501500120350845775718053568217335111295345357327345329794256264296899950}{83620588305290694065806019200195179360861982993866260972209999730792522060828755375983057245707875986289314178486384859902673413830280042746703497565432868548434508444587} a - \frac{2529688685805054643475473721311173090382065268775282091135481596872454247802076594694159402058710401088149162654432715529985274251445631493066804178764159415043605644}{9927649092400652269477148189504354667085596936230115276292294874841804827357088374211451649733809329964301813900793643583363815010124663747679389477078578718797875869}$

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

$C_{32}$ (as 32T33):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 32
The 32 conjugacy class representatives for $C_{32}$
Character table for $C_{32}$ is not computed

Intermediate fields

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.80798284478113.1, 16.16.247363745756558353923154669762890625.1

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 $16^{2}$ $16^{2}$ R $32$ $16^{2}$ $32$ $32$ $32$ $32$ $32$ $16^{2}$ $32$ $32$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{4}$ $16^{2}$ $32$

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
5Data not computed
97Data not computed