Properties

Label 32.32.2318482735...0625.1
Degree $32$
Signature $[32, 0]$
Discriminant $5^{24}\cdot 97^{31}$
Root discriminant $281.13$
Ramified primes $5, 97$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
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![-18277845959, -11758954445, 420651733214, -588593534703, -1244875523883, 2597013247154, 1274517894230, -4692807095098, -46102330825, 4674809722638, -1010392771187, -2853145220348, 989059812617, 1120910038107, -484526195470, -291596165966, 143975452663, 51265014015, -27704655126, -6168097580, 3552738857, 508700394, -306303044, -28355437, 17622874, 1028778, -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 + 1028778*x^25 + 17622874*x^24 - 28355437*x^23 - 306303044*x^22 + 508700394*x^21 + 3552738857*x^20 - 6168097580*x^19 - 27704655126*x^18 + 51265014015*x^17 + 143975452663*x^16 - 291596165966*x^15 - 484526195470*x^14 + 1120910038107*x^13 + 989059812617*x^12 - 2853145220348*x^11 - 1010392771187*x^10 + 4674809722638*x^9 - 46102330825*x^8 - 4692807095098*x^7 + 1274517894230*x^6 + 2597013247154*x^5 - 1244875523883*x^4 - 588593534703*x^3 + 420651733214*x^2 - 11758954445*x - 18277845959)
 
gp: K = bnfinit(x^32 - x^31 - 192*x^30 + 254*x^29 + 15176*x^28 - 22502*x^27 - 659185*x^26 + 1028778*x^25 + 17622874*x^24 - 28355437*x^23 - 306303044*x^22 + 508700394*x^21 + 3552738857*x^20 - 6168097580*x^19 - 27704655126*x^18 + 51265014015*x^17 + 143975452663*x^16 - 291596165966*x^15 - 484526195470*x^14 + 1120910038107*x^13 + 989059812617*x^12 - 2853145220348*x^11 - 1010392771187*x^10 + 4674809722638*x^9 - 46102330825*x^8 - 4692807095098*x^7 + 1274517894230*x^6 + 2597013247154*x^5 - 1244875523883*x^4 - 588593534703*x^3 + 420651733214*x^2 - 11758954445*x - 18277845959, 1)
 

Normalized defining polynomial

\( x^{32} - x^{31} - 192 x^{30} + 254 x^{29} + 15176 x^{28} - 22502 x^{27} - 659185 x^{26} + 1028778 x^{25} + 17622874 x^{24} - 28355437 x^{23} - 306303044 x^{22} + 508700394 x^{21} + 3552738857 x^{20} - 6168097580 x^{19} - 27704655126 x^{18} + 51265014015 x^{17} + 143975452663 x^{16} - 291596165966 x^{15} - 484526195470 x^{14} + 1120910038107 x^{13} + 989059812617 x^{12} - 2853145220348 x^{11} - 1010392771187 x^{10} + 4674809722638 x^{9} - 46102330825 x^{8} - 4692807095098 x^{7} + 1274517894230 x^{6} + 2597013247154 x^{5} - 1244875523883 x^{4} - 588593534703 x^{3} + 420651733214 x^{2} - 11758954445 x - 18277845959 \)

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(K);
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $281.13$
magma: Abs(Discriminant(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(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$$\chi_{485}(1,·)$, $\chi_{485}(257,·)$, $\chi_{485}(264,·)$, $\chi_{485}(408,·)$, $\chi_{485}(279,·)$, $\chi_{485}(152,·)$, $\chi_{485}(161,·)$, $\chi_{485}(418,·)$, $\chi_{485}(421,·)$, $\chi_{485}(422,·)$, $\chi_{485}(42,·)$, $\chi_{485}(299,·)$, $\chi_{485}(433,·)$, $\chi_{485}(309,·)$, $\chi_{485}(457,·)$, $\chi_{485}(78,·)$, $\chi_{485}(79,·)$, $\chi_{485}(337,·)$, $\chi_{485}(341,·)$, $\chi_{485}(342,·)$, $\chi_{485}(343,·)$, $\chi_{485}(216,·)$, $\chi_{485}(89,·)$, $\chi_{485}(222,·)$, $\chi_{485}(96,·)$, $\chi_{485}(358,·)$, $\chi_{485}(109,·)$, $\chi_{485}(366,·)$, $\chi_{485}(368,·)$, $\chi_{485}(241,·)$, $\chi_{485}(124,·)$, $\chi_{485}(213,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $\frac{1}{6283} a^{27} + \frac{944}{6283} a^{26} - \frac{5}{61} a^{25} - \frac{451}{6283} a^{24} + \frac{2988}{6283} a^{23} + \frac{2207}{6283} a^{22} + \frac{1945}{6283} a^{21} + \frac{1346}{6283} a^{20} - \frac{555}{6283} a^{19} - \frac{2046}{6283} a^{18} + \frac{2643}{6283} a^{17} + \frac{473}{6283} a^{16} + \frac{2391}{6283} a^{15} + \frac{24}{103} a^{14} + \frac{879}{6283} a^{13} - \frac{393}{6283} a^{12} - \frac{2777}{6283} a^{11} + \frac{159}{6283} a^{10} + \frac{2583}{6283} a^{9} + \frac{2443}{6283} a^{8} + \frac{2343}{6283} a^{7} - \frac{1415}{6283} a^{6} + \frac{767}{6283} a^{5} + \frac{1656}{6283} a^{4} + \frac{1409}{6283} a^{3} + \frac{1190}{6283} a^{2} - \frac{1610}{6283} a - \frac{20}{103}$, $\frac{1}{6283} a^{28} + \frac{535}{6283} a^{26} + \frac{1918}{6283} a^{25} + \frac{1488}{6283} a^{24} + \frac{2602}{6283} a^{23} - \frac{1790}{6283} a^{22} - \frac{98}{6283} a^{21} - \frac{33}{103} a^{20} + \frac{385}{6283} a^{19} - \frac{1097}{6283} a^{18} - \frac{168}{6283} a^{17} + \frac{1972}{6283} a^{16} - \frac{43}{6283} a^{15} + \frac{1123}{6283} a^{14} - \frac{813}{6283} a^{13} - \frac{2482}{6283} a^{12} + \frac{1636}{6283} a^{11} - \frac{3004}{6283} a^{10} + \frac{1895}{6283} a^{9} + \frac{2012}{6283} a^{8} - \frac{1591}{6283} a^{7} - \frac{1752}{6283} a^{6} + \frac{153}{6283} a^{5} + \frac{2612}{6283} a^{4} + \frac{30}{61} a^{3} - \frac{313}{6283} a^{2} - \frac{1866}{6283} a + \frac{31}{103}$, $\frac{1}{6283} a^{29} - \frac{482}{6283} a^{26} + \frac{561}{6283} a^{25} - \frac{1150}{6283} a^{24} + \frac{1795}{6283} a^{23} + \frac{361}{6283} a^{22} + \frac{390}{6283} a^{21} + \frac{2820}{6283} a^{20} + \frac{527}{6283} a^{19} + \frac{1200}{6283} a^{18} + \frac{1642}{6283} a^{17} - \frac{1778}{6283} a^{16} - \frac{2613}{6283} a^{15} + \frac{1322}{6283} a^{14} - \frac{1522}{6283} a^{13} - \frac{1731}{6283} a^{12} - \frac{97}{6283} a^{11} - \frac{1491}{6283} a^{10} + \frac{2367}{6283} a^{9} - \frac{1732}{6283} a^{8} + \frac{1343}{6283} a^{7} - \frac{3065}{6283} a^{6} + \frac{662}{6283} a^{5} + \frac{3033}{6283} a^{4} - \frac{168}{6283} a^{3} + \frac{2350}{6283} a^{2} + \frac{2470}{6283} a - \frac{12}{103}$, $\frac{1}{6283} a^{30} - \frac{30}{61} a^{26} + \frac{1940}{6283} a^{25} - \frac{1965}{6283} a^{24} + \frac{1770}{6283} a^{23} + \frac{2337}{6283} a^{22} - \frac{2140}{6283} a^{21} + \frac{2150}{6283} a^{20} - \frac{2424}{6283} a^{19} + \frac{1901}{6283} a^{18} + \frac{2982}{6283} a^{17} - \frac{815}{6283} a^{16} - \frac{2288}{6283} a^{15} + \frac{430}{6283} a^{14} + \frac{986}{6283} a^{13} - \frac{1033}{6283} a^{12} - \frac{1726}{6283} a^{11} - \frac{2674}{6283} a^{10} - \frac{760}{6283} a^{9} - \frac{2335}{6283} a^{8} + \frac{1604}{6283} a^{7} - \frac{2804}{6283} a^{6} + \frac{2030}{6283} a^{5} + \frac{83}{6283} a^{4} + \frac{2924}{6283} a^{3} - \frac{1986}{6283} a^{2} + \frac{2340}{6283} a + \frac{42}{103}$, $\frac{1}{1105391053730327536508814873113847107658704537314039123850383966168015112308881425390700565568379774788165815609153461287117472223704614249259656308965064869476759} a^{31} + \frac{1905961929910505615037976140772051705380376392230370897308001713857455624664589234826871662253750151734891521946223201649050305772224927235654804136420877093}{1105391053730327536508814873113847107658704537314039123850383966168015112308881425390700565568379774788165815609153461287117472223704614249259656308965064869476759} a^{30} - \frac{44058814922749340046573194313394384802577736949267506518960996493896284016541397335518667101071831543713004230366647518366357888549882609962863168196205104623}{1105391053730327536508814873113847107658704537314039123850383966168015112308881425390700565568379774788165815609153461287117472223704614249259656308965064869476759} a^{29} + \frac{56636610311113192641743311086983929696165444074807427426406471986877627859953869103095389682346093952720614603975794495421361390285796209065235732517558253199}{1105391053730327536508814873113847107658704537314039123850383966168015112308881425390700565568379774788165815609153461287117472223704614249259656308965064869476759} a^{28} + \frac{14319153646645883829784760322706944697967356519104233280138408605120596111316865332753558807837079817178213638685260274521729452476847765899394985008110153705}{1105391053730327536508814873113847107658704537314039123850383966168015112308881425390700565568379774788165815609153461287117472223704614249259656308965064869476759} a^{27} + \frac{264500665407544943043548235231621778496936544180642539712587781249545018135617320769856094358033371490115837999151765625925418822997638159260217983160767985386600}{1105391053730327536508814873113847107658704537314039123850383966168015112308881425390700565568379774788165815609153461287117472223704614249259656308965064869476759} a^{26} - \frac{351400168687884195658604149217071383788610189996575509206732950678698734038903455872346701554659924232314390033529461580495354295051550095394712772702727655016864}{1105391053730327536508814873113847107658704537314039123850383966168015112308881425390700565568379774788165815609153461287117472223704614249259656308965064869476759} a^{25} - \frac{413408242180459494414774670395161264157055918895849317388841782877359780408279371277277329687382636157212699989362568858798675065891347569006855824040226433750471}{1105391053730327536508814873113847107658704537314039123850383966168015112308881425390700565568379774788165815609153461287117472223704614249259656308965064869476759} a^{24} - \frac{431173509545516002012433207991727067999677235974033296102403488769994738361896387858817185697912537799959649175995777806412468616948144875235862363908143966752915}{1105391053730327536508814873113847107658704537314039123850383966168015112308881425390700565568379774788165815609153461287117472223704614249259656308965064869476759} a^{23} + \frac{51578036755020824809110084759168090264163754792526714440787502924411636139398700361008326716601083467866564864353338126545383467382231959943206341296596099958995}{1105391053730327536508814873113847107658704537314039123850383966168015112308881425390700565568379774788165815609153461287117472223704614249259656308965064869476759} a^{22} - \frac{113615313734611983619332200378948671091979040364170370985748825392108724361171849294660286104372509063830045410376381710405537478073587025521758876773661516009384}{1105391053730327536508814873113847107658704537314039123850383966168015112308881425390700565568379774788165815609153461287117472223704614249259656308965064869476759} a^{21} - \frac{100903359544214686090294213380948903669033911621987891353215309053634856400129415281360117133406118811861178855660799801292335101333785072281874647489400211063616}{1105391053730327536508814873113847107658704537314039123850383966168015112308881425390700565568379774788165815609153461287117472223704614249259656308965064869476759} a^{20} - \frac{3943060549022968574092485176340841928550847005949638770159837687069546461903992563501196397655525544183461650055229457883823681972340733731283825388302767271788}{9782221714427677314237299762069443430608004754991496671242335983787744356715764826466376686445838714939520492116402312275375860386766497781058905389071370526343} a^{19} + \frac{473139579030440121439163493500064968990598766175118524370444731825369376499389230281307140531233925345613342142694689120537801408227505793545651390020577945399620}{1105391053730327536508814873113847107658704537314039123850383966168015112308881425390700565568379774788165815609153461287117472223704614249259656308965064869476759} a^{18} - \frac{519862189367669574767283625325730861382526514319103025747891102582389703053249423472806904726550841971153814800146544773802928500564729096510895098624106061228344}{1105391053730327536508814873113847107658704537314039123850383966168015112308881425390700565568379774788165815609153461287117472223704614249259656308965064869476759} a^{17} + \frac{526601567358413236870775914828541969981366212364748452314749593706870814632854052722111273002532052116290548562819881408263880527707309647973975438744656787498751}{1105391053730327536508814873113847107658704537314039123850383966168015112308881425390700565568379774788165815609153461287117472223704614249259656308965064869476759} a^{16} - \frac{131066904928537104823833431385707730929437323585060887615436515993331239825935926598664409916055385369169680660487386466247838134970729072361314922485114392889253}{1105391053730327536508814873113847107658704537314039123850383966168015112308881425390700565568379774788165815609153461287117472223704614249259656308965064869476759} a^{15} - \frac{368927602583112916945236417375395708108481012436443783176453242580067938512829974024251043067343243202778942297997372286602040213538527160658322357361025640143546}{1105391053730327536508814873113847107658704537314039123850383966168015112308881425390700565568379774788165815609153461287117472223704614249259656308965064869476759} a^{14} - \frac{298364406499695908104565639680075835100541531514372025955017686359432893979088390631066197584698590792085412819107151016389144953959754139869775104967427386423438}{1105391053730327536508814873113847107658704537314039123850383966168015112308881425390700565568379774788165815609153461287117472223704614249259656308965064869476759} a^{13} - \frac{253649525713802558098860027571523206818032792894030470666628568191929947365430603035747107408077229715423889629641265572300150353252244688011115145416310803011768}{1105391053730327536508814873113847107658704537314039123850383966168015112308881425390700565568379774788165815609153461287117472223704614249259656308965064869476759} a^{12} - \frac{434058189308503717389969926118148656490660835090246293763921321213101365422824858514402500018474354818168898436959772548654102708776094170764087059704801485445137}{1105391053730327536508814873113847107658704537314039123850383966168015112308881425390700565568379774788165815609153461287117472223704614249259656308965064869476759} a^{11} + \frac{58833240080060572368115406190554991979055980963515383913968825135832300783590161005965461445694312415046957747473825093751797165109051588012263949094845263421539}{1105391053730327536508814873113847107658704537314039123850383966168015112308881425390700565568379774788165815609153461287117472223704614249259656308965064869476759} a^{10} + \frac{13335894332897896453686753443025306496097705649933401660696396608924187631466329136943252786328935364390189660441842792950290114191733676488238170783894875302060}{1105391053730327536508814873113847107658704537314039123850383966168015112308881425390700565568379774788165815609153461287117472223704614249259656308965064869476759} a^{9} + \frac{395727068940271744463031348184919236992528318552265074091028675396172330098576795637770202064857885419877427974236578914573766603357137410456144470510122439495215}{1105391053730327536508814873113847107658704537314039123850383966168015112308881425390700565568379774788165815609153461287117472223704614249259656308965064869476759} a^{8} + \frac{448197895630285760655387172028554336181402523624364001107705973189021099718186063007165911213989753236357994846821131004797921511804075929783714578936541014853310}{1105391053730327536508814873113847107658704537314039123850383966168015112308881425390700565568379774788165815609153461287117472223704614249259656308965064869476759} a^{7} + \frac{170315333402887064981846972864608283629383795381118450793592841866187162988200042062312233623935588926320755324104336964044716770725920657314611650212051657283418}{1105391053730327536508814873113847107658704537314039123850383966168015112308881425390700565568379774788165815609153461287117472223704614249259656308965064869476759} a^{6} + \frac{3259118325191853788182773979436852075406769305920433224061841717006553125114151329299453984941074056459167215708742181521051989645912735068901494988959188272783}{10731951977964345014648688088483952501540820750621739066508582195805971964163897333890296753091065774642386559312169527059392934210724410186986954455971503587153} a^{5} + \frac{84399255387271783067059099917791544401092422168308352977949772203630132922910260245659089705566648729415664249533144576850166841369199772596796268546825277931555}{1105391053730327536508814873113847107658704537314039123850383966168015112308881425390700565568379774788165815609153461287117472223704614249259656308965064869476759} a^{4} + \frac{447202008401466916369117288879168170559423519569192080118536058373529862033492688950247077935227632147646838368951577510748498018086604411242802560595806436514035}{1105391053730327536508814873113847107658704537314039123850383966168015112308881425390700565568379774788165815609153461287117472223704614249259656308965064869476759} a^{3} + \frac{383471119736910909492398582853091475906756865235834990446233808637187661506105985927424934961102792470488794410845140400439823664538785858325016835957112964723718}{1105391053730327536508814873113847107658704537314039123850383966168015112308881425390700565568379774788165815609153461287117472223704614249259656308965064869476759} a^{2} - \frac{4930126196950562036739506540904371868607751689477611009181352018028524664227605979906468058926070380055882697387227804046610782158345310349274671513553269370379}{18121164815251271090308440542849952584568926841213756128694819117508444464080023367060665009317701226035505173920548545690450364323026463102617316540410899499619} a - \frac{123132585296167417004564118387407403156572408332952074662453739531703957737411347324124456017961738794243590831180609327759783448674136674770023184259173360734}{297068275659856903119810500702458239091293882642848461126144575696859745312787268312469918185536085672713199572468008945745087939721745296764218303941162286879}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $31$
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:  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:  \( 108860537045226350000000000000 \) (assuming GRH)
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