Properties

 Label 32.0.30634913242...4097.1 Degree $32$ Signature $[0, 16]$ Discriminant $3^{16}\cdot 193^{31}$ Root discriminant $283.59$ Ramified primes $3, 193$ Class number Not computed Class group Not computed Galois group $C_{32}$ (as 32T33)

Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![38241539870119, -218583702854457, 562130389554400, -862139564170835, 894386134923647, -694285600401985, 455392827342446, -284753440308162, 174984915122456, -99818909359402, 50683905711002, -23194926103486, 9985317638552, -4173402576653, 1688255459920, -647554500300, 232272715410, -77412047603, 23969357883, -6919013614, 1855235314, -459438417, 104962263, -22080636, 4491744, -871109, 146725, -25796, 4310, -369, 100, -1, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - x^31 + 100*x^30 - 369*x^29 + 4310*x^28 - 25796*x^27 + 146725*x^26 - 871109*x^25 + 4491744*x^24 - 22080636*x^23 + 104962263*x^22 - 459438417*x^21 + 1855235314*x^20 - 6919013614*x^19 + 23969357883*x^18 - 77412047603*x^17 + 232272715410*x^16 - 647554500300*x^15 + 1688255459920*x^14 - 4173402576653*x^13 + 9985317638552*x^12 - 23194926103486*x^11 + 50683905711002*x^10 - 99818909359402*x^9 + 174984915122456*x^8 - 284753440308162*x^7 + 455392827342446*x^6 - 694285600401985*x^5 + 894386134923647*x^4 - 862139564170835*x^3 + 562130389554400*x^2 - 218583702854457*x + 38241539870119)

gp: K = bnfinit(x^32 - x^31 + 100*x^30 - 369*x^29 + 4310*x^28 - 25796*x^27 + 146725*x^26 - 871109*x^25 + 4491744*x^24 - 22080636*x^23 + 104962263*x^22 - 459438417*x^21 + 1855235314*x^20 - 6919013614*x^19 + 23969357883*x^18 - 77412047603*x^17 + 232272715410*x^16 - 647554500300*x^15 + 1688255459920*x^14 - 4173402576653*x^13 + 9985317638552*x^12 - 23194926103486*x^11 + 50683905711002*x^10 - 99818909359402*x^9 + 174984915122456*x^8 - 284753440308162*x^7 + 455392827342446*x^6 - 694285600401985*x^5 + 894386134923647*x^4 - 862139564170835*x^3 + 562130389554400*x^2 - 218583702854457*x + 38241539870119, 1)

Normalizeddefining polynomial

$$x^{32} - x^{31} + 100 x^{30} - 369 x^{29} + 4310 x^{28} - 25796 x^{27} + 146725 x^{26} - 871109 x^{25} + 4491744 x^{24} - 22080636 x^{23} + 104962263 x^{22} - 459438417 x^{21} + 1855235314 x^{20} - 6919013614 x^{19} + 23969357883 x^{18} - 77412047603 x^{17} + 232272715410 x^{16} - 647554500300 x^{15} + 1688255459920 x^{14} - 4173402576653 x^{13} + 9985317638552 x^{12} - 23194926103486 x^{11} + 50683905711002 x^{10} - 99818909359402 x^{9} + 174984915122456 x^{8} - 284753440308162 x^{7} + 455392827342446 x^{6} - 694285600401985 x^{5} + 894386134923647 x^{4} - 862139564170835 x^{3} + 562130389554400 x^{2} - 218583702854457 x + 38241539870119$$

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: $$3063491324247901158147277924095178749079926649475132560984974419980910475284097=3^{16}\cdot 193^{31}$$ magma: Discriminant(K);  sage: K.disc()  gp: K.disc Root discriminant: $283.59$ magma: Abs(Discriminant(K))^(1/Degree(K));  sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $3, 193$ magma: PrimeDivisors(Discriminant(K));  sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~ This field is Galois and abelian over $\Q$. Conductor: $$579=3\cdot 193$$ Dirichlet character group: $\lbrace$$\chi_{579}(512,·), \chi_{579}(1,·), \chi_{579}(260,·), \chi_{579}(385,·), \chi_{579}(8,·), \chi_{579}(14,·), \chi_{579}(529,·), \chi_{579}(274,·), \chi_{579}(23,·), \chi_{579}(202,·), \chi_{579}(410,·), \chi_{579}(166,·), \chi_{579}(170,·), \chi_{579}(43,·), \chi_{579}(428,·), \chi_{579}(179,·), \chi_{579}(436,·), \chi_{579}(184,·), \chi_{579}(185,·), \chi_{579}(314,·), \chi_{579}(317,·), \chi_{579}(190,·), \chi_{579}(64,·), \chi_{579}(322,·), \chi_{579}(196,·), \chi_{579}(455,·), \chi_{579}(458,·), \chi_{579}(343,·), \chi_{579}(344,·), \chi_{579}(220,·), \chi_{579}(362,·), \chi_{579}(112,·)$$\rbrace$ This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $\frac{1}{277} a^{29} - \frac{97}{277} a^{28} - \frac{89}{277} a^{27} + \frac{17}{277} a^{26} + \frac{114}{277} a^{25} + \frac{60}{277} a^{24} - \frac{10}{277} a^{23} - \frac{45}{277} a^{22} + \frac{83}{277} a^{21} + \frac{109}{277} a^{20} - \frac{77}{277} a^{19} - \frac{74}{277} a^{18} - \frac{39}{277} a^{17} - \frac{16}{277} a^{16} + \frac{97}{277} a^{15} + \frac{135}{277} a^{14} - \frac{20}{277} a^{13} + \frac{18}{277} a^{12} - \frac{21}{277} a^{11} + \frac{120}{277} a^{10} - \frac{20}{277} a^{9} + \frac{103}{277} a^{8} - \frac{75}{277} a^{7} - \frac{53}{277} a^{6} - \frac{115}{277} a^{5} + \frac{59}{277} a^{4} + \frac{116}{277} a^{3} - \frac{1}{277} a^{2} - \frac{40}{277} a - \frac{114}{277}$, $\frac{1}{26065423} a^{30} - \frac{26533}{26065423} a^{29} + \frac{8571502}{26065423} a^{28} + \frac{11800460}{26065423} a^{27} - \frac{3293257}{26065423} a^{26} + \frac{5323225}{26065423} a^{25} + \frac{6346556}{26065423} a^{24} + \frac{3964204}{26065423} a^{23} - \frac{9061790}{26065423} a^{22} - \frac{2725088}{26065423} a^{21} + \frac{9532708}{26065423} a^{20} + \frac{10996725}{26065423} a^{19} + \frac{404194}{26065423} a^{18} - \frac{4064981}{26065423} a^{17} - \frac{9084675}{26065423} a^{16} - \frac{9206617}{26065423} a^{15} - \frac{11655895}{26065423} a^{14} + \frac{6283690}{26065423} a^{13} + \frac{245716}{26065423} a^{12} - \frac{5215742}{26065423} a^{11} + \frac{11480683}{26065423} a^{10} - \frac{9115763}{26065423} a^{9} - \frac{4919039}{26065423} a^{8} - \frac{8534766}{26065423} a^{7} + \frac{7422696}{26065423} a^{6} + \frac{4874216}{26065423} a^{5} + \frac{4006430}{26065423} a^{4} - \frac{11799556}{26065423} a^{3} + \frac{9743279}{26065423} a^{2} + \frac{7727486}{26065423} a - \frac{11410240}{26065423}$, $\frac{1}{339044391111247788987061614145488452561244058632576507206839339147120199550494524059204042111120834802998623244270780044535544055957766186294012016422289013656587565072761413868224275928398105950467210508267835732325380549} a^{31} - \frac{1773949834581028973100507162751599753765455984003044149508495079494457060563866020738339514086772328789202855792312026052785900560004174727407228360104036473083294268294428724560911876825062105131319022724993128335}{339044391111247788987061614145488452561244058632576507206839339147120199550494524059204042111120834802998623244270780044535544055957766186294012016422289013656587565072761413868224275928398105950467210508267835732325380549} a^{30} + \frac{359911576540605663826150732082448034580515933186286407701801385526480231812679678675456290979411397464626003741758553218226728114045101749797087001770597402980190167816804874815066782176474848730902455776212972647926667}{339044391111247788987061614145488452561244058632576507206839339147120199550494524059204042111120834802998623244270780044535544055957766186294012016422289013656587565072761413868224275928398105950467210508267835732325380549} a^{29} - \frac{79944915558311709456076028448286308314939628084373238895284474822331555698030384628109198899158094782603241696972569092203611868530762474540666800518347747321553868029392008741716494390842138223897581406801308185986217176}{339044391111247788987061614145488452561244058632576507206839339147120199550494524059204042111120834802998623244270780044535544055957766186294012016422289013656587565072761413868224275928398105950467210508267835732325380549} a^{28} - \frac{54773245734304791530073741350958689408458012595576576850207838422124025049188962481798210214376269856753323972690567100648237508842330312257123349687984272942279948124679755116843368177232676135933515644642879590552699512}{339044391111247788987061614145488452561244058632576507206839339147120199550494524059204042111120834802998623244270780044535544055957766186294012016422289013656587565072761413868224275928398105950467210508267835732325380549} a^{27} + \frac{51623852735947264525886307818413356006511283600221512570474930257602233446303356310949772620136294505124556926050408148834095104471726628725008753993551337497177242308530537020941032581329910318522836290574534851076456750}{339044391111247788987061614145488452561244058632576507206839339147120199550494524059204042111120834802998623244270780044535544055957766186294012016422289013656587565072761413868224275928398105950467210508267835732325380549} a^{26} + \frac{152677843757820544385113080801985093059040674545079803278493051687787577296129217834999636101694842073768430678872399978117142474132535089897239602227119184628207845871415250533387434720817400005947512481836643048301640075}{339044391111247788987061614145488452561244058632576507206839339147120199550494524059204042111120834802998623244270780044535544055957766186294012016422289013656587565072761413868224275928398105950467210508267835732325380549} a^{25} - \frac{14047468087004867105089138789920514219870683612219581023447494980009620866037334605377308370126429150794049569919635766737435051115323950754284329042157014440366322714604163107537791853191596007616885343823266790718217347}{339044391111247788987061614145488452561244058632576507206839339147120199550494524059204042111120834802998623244270780044535544055957766186294012016422289013656587565072761413868224275928398105950467210508267835732325380549} a^{24} - \frac{91001965336338052511954895584992070099382483133825097903830082791295146683435321535689225119546591229152626798820979774707179445111640258930801870589878245591231549647213758373138352114447474453187457885610361112400575581}{339044391111247788987061614145488452561244058632576507206839339147120199550494524059204042111120834802998623244270780044535544055957766186294012016422289013656587565072761413868224275928398105950467210508267835732325380549} a^{23} + \frac{155654494125096237331914000673468341766368537136707576608584161883625448161776088901603082786002574214421423955360386646546195363272724097104423036968556559839925687038025087275905357734599339021464252528841884640097639421}{339044391111247788987061614145488452561244058632576507206839339147120199550494524059204042111120834802998623244270780044535544055957766186294012016422289013656587565072761413868224275928398105950467210508267835732325380549} a^{22} - \frac{153383315742361063170019976010612089215746945759387752004330373107971724609761622507880850414779630626270619875616814349651136542658419282061469215846322858735391499856896950222906860571976783300875866953632465534194590704}{339044391111247788987061614145488452561244058632576507206839339147120199550494524059204042111120834802998623244270780044535544055957766186294012016422289013656587565072761413868224275928398105950467210508267835732325380549} a^{21} + \frac{161453581619438914684840749041949558789776341046246349336535700383317703937788531819276592733002048818895837973687883963495923069401538888328419314670828849563404381737135564762678478931034455779971616278033462249221054842}{339044391111247788987061614145488452561244058632576507206839339147120199550494524059204042111120834802998623244270780044535544055957766186294012016422289013656587565072761413868224275928398105950467210508267835732325380549} a^{20} - \frac{167772201975719006342808184433973916652858764822415890886498810418441582977551130262271835829022113177944880809806862703832645383677172990578824874280523356265348482760557131611246156644088114447057648472327076774109363471}{339044391111247788987061614145488452561244058632576507206839339147120199550494524059204042111120834802998623244270780044535544055957766186294012016422289013656587565072761413868224275928398105950467210508267835732325380549} a^{19} - \frac{136262953719221314626873563726466844478767751191342583128230019549849246704191027080358595144977663034955792809627079693503600755971465400794671126385717971273752367763089785888615243322336965621074564459625183060177669991}{339044391111247788987061614145488452561244058632576507206839339147120199550494524059204042111120834802998623244270780044535544055957766186294012016422289013656587565072761413868224275928398105950467210508267835732325380549} a^{18} - \frac{92903154210948693754036937801840095792999349538325239166767969577641007401705150825733903931495683323659070439198324715116410413349954632218363811418166239149706215181137945386203977407432747718867694189526277058272480441}{339044391111247788987061614145488452561244058632576507206839339147120199550494524059204042111120834802998623244270780044535544055957766186294012016422289013656587565072761413868224275928398105950467210508267835732325380549} a^{17} - \frac{159687699299763002968650182478683699683514773194763353211578933176734379138997960710580660240810665844397866960017110507521521330027952095815148762513964930426896047126348893344501536737756715605313893640800718603585190988}{339044391111247788987061614145488452561244058632576507206839339147120199550494524059204042111120834802998623244270780044535544055957766186294012016422289013656587565072761413868224275928398105950467210508267835732325380549} a^{16} - \frac{9899900331441399607409266430851243782196218455005820711940408051246781744682514505375334957665668385586131663904452225009452831223033458794195515295173074413597524838228811990786609878543464149990909103200088890806858475}{339044391111247788987061614145488452561244058632576507206839339147120199550494524059204042111120834802998623244270780044535544055957766186294012016422289013656587565072761413868224275928398105950467210508267835732325380549} a^{15} - \frac{126782672633141927903959490542244508145935291948027373862035694347107639224328727297763161502907097209320562641162602698394932761943513908386465538296290086628137804670047918682355383820680003501332679630221123350525932775}{339044391111247788987061614145488452561244058632576507206839339147120199550494524059204042111120834802998623244270780044535544055957766186294012016422289013656587565072761413868224275928398105950467210508267835732325380549} a^{14} + \frac{132991316118777539144942600817242132636031386363051014000557171387504943551356533263994576455066495610719934132696788722253565788536737019269018136356052800362938799884269915894026051938820383898791315177093840551953542685}{339044391111247788987061614145488452561244058632576507206839339147120199550494524059204042111120834802998623244270780044535544055957766186294012016422289013656587565072761413868224275928398105950467210508267835732325380549} a^{13} + \frac{156735532289606484169525480938450224381257380272555377920950743212223420747238149244556189417837388121385583673860871024105446029505304225303489285791930956639726462502950085385538807907281429337712796217553496102267769665}{339044391111247788987061614145488452561244058632576507206839339147120199550494524059204042111120834802998623244270780044535544055957766186294012016422289013656587565072761413868224275928398105950467210508267835732325380549} a^{12} - \frac{130681001397960034714065860400825664365133215439891762679697958670685756619662455013788617967591993890517712324419264458264615851630334967542119757147381971683913535010138076301790477803980614345385960278997022090100245624}{339044391111247788987061614145488452561244058632576507206839339147120199550494524059204042111120834802998623244270780044535544055957766186294012016422289013656587565072761413868224275928398105950467210508267835732325380549} a^{11} + \frac{57555297294184163489145212277945983112259465484606228370419640218782445744949562470551881864975056114196058246581633586268397261298594752708662478969612881810313908710673945311427407847355100763602819154657097986639776090}{339044391111247788987061614145488452561244058632576507206839339147120199550494524059204042111120834802998623244270780044535544055957766186294012016422289013656587565072761413868224275928398105950467210508267835732325380549} a^{10} + \frac{128283026936545830800472391766689622318923206811389710254487183637469637285323520173322174695874676109317201947439211058935258397777040077129818580924248071089896220001023515706271598955258217283568298109794907134259368455}{339044391111247788987061614145488452561244058632576507206839339147120199550494524059204042111120834802998623244270780044535544055957766186294012016422289013656587565072761413868224275928398105950467210508267835732325380549} a^{9} - \frac{127051102359680169187777536274220572530559420574678205105516250757413017478128613301447053881661800024093119514866623945898314887054544388942261665178431668899352822217996522904297600376571214658206367188582131884834896644}{339044391111247788987061614145488452561244058632576507206839339147120199550494524059204042111120834802998623244270780044535544055957766186294012016422289013656587565072761413868224275928398105950467210508267835732325380549} a^{8} + \frac{122367122268553988991549333567020547630957557025834903504842700133315258439863307863562169164177059723761948499409861824410167636278789128788892972181129330719460255684353031246948147256846172723384212638553726084627958435}{339044391111247788987061614145488452561244058632576507206839339147120199550494524059204042111120834802998623244270780044535544055957766186294012016422289013656587565072761413868224275928398105950467210508267835732325380549} a^{7} - \frac{8482019838234204927040188675427571475583086122752462791434348470120029259987038230864870087142907221640718866005061460376430358536900939133767575846221282522685305578828426870612204999673371936735138398012310442699503382}{339044391111247788987061614145488452561244058632576507206839339147120199550494524059204042111120834802998623244270780044535544055957766186294012016422289013656587565072761413868224275928398105950467210508267835732325380549} a^{6} - \frac{50298156453796253230785378525123985146718265140703552032696066470755997681098482108129752938206246404466823285357010051457074803998521001993726757728713841088615031132497262142154268509215141174340822066757427388832937512}{339044391111247788987061614145488452561244058632576507206839339147120199550494524059204042111120834802998623244270780044535544055957766186294012016422289013656587565072761413868224275928398105950467210508267835732325380549} a^{5} + \frac{136597862599744320872344358488589008928976346246466285920989482181389072618023774517139005753524658779286351699966402174407208629530826665781203270593494136923953885916525430705038114110081313042444775556920922253391083934}{339044391111247788987061614145488452561244058632576507206839339147120199550494524059204042111120834802998623244270780044535544055957766186294012016422289013656587565072761413868224275928398105950467210508267835732325380549} a^{4} + \frac{6301268937620886110308061260822601063242568960406545173354301962243829024513508976313110678697295201120848764124449854264845456579900034743115221363233502643209386266195955745941214360241673144757676434001545020004097609}{339044391111247788987061614145488452561244058632576507206839339147120199550494524059204042111120834802998623244270780044535544055957766186294012016422289013656587565072761413868224275928398105950467210508267835732325380549} a^{3} - \frac{73252079692605987073827065493302102512590047496474717394606845260371864301945942003917800736511046944586750134871482179207635163801245019842109130599046411321729807864232595666931884884667065472181126040693026029631481429}{339044391111247788987061614145488452561244058632576507206839339147120199550494524059204042111120834802998623244270780044535544055957766186294012016422289013656587565072761413868224275928398105950467210508267835732325380549} a^{2} + \frac{88723649872669647682172923988035635761658894206487545125832697228496210976847246822725389743041568554838448693752156073990350688080004828688925018128686012736974623181049089551282712535126471864726458504854202812166457824}{339044391111247788987061614145488452561244058632576507206839339147120199550494524059204042111120834802998623244270780044535544055957766186294012016422289013656587565072761413868224275928398105950467210508267835732325380549} a + \frac{73948674467976641532207063116218287407277891455543820283962226141765243963861015694131506150339389163420963105058165352624747057511534692499331802404216221173256532478949677104729406136775337587412943419788842816947615904}{339044391111247788987061614145488452561244058632576507206839339147120199550494524059204042111120834802998623244270780044535544055957766186294012016422289013656587565072761413868224275928398105950467210508267835732325380549}$

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: $15$ 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: 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

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

Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 $16^{2}$ R $32$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ $32$ $32$ $32$ $32$ $16^{2}$ $32$ $16^{2}$ $32$ $32$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ $32$ $32$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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$3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4} 3.8.4.2x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4} 3.8.4.2x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
193Data not computed