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Properties

Label 36.36.1176590270...8125.1
Degree $36$
Signature $[36, 0]$
Discriminant $3^{88}\cdot 5^{27}\cdot 7^{18}$
Root discriminant $129.74$
Ramified primes $3, 5, 7$
Class number Not computed
Class group Not computed
Galois group $C_{36}$ (as 36T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13538501, -474397875, 4783445352, -20282160951, 30223063710, 44758761705, -203664646266, 125024057829, 373616574660, -552047771387, -214390074411, 823286390904, -161390547741, -646861351536, 335224535589, 290388199548, -242670922227, -69735280059, 100934354919, 3870494730, -26805831327, 2841546570, 4697849808, -985627989, -541366140, 165335742, 39162357, -16861368, -1512459, 1085229, 5643, -42939, 2088, 951, -81, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 9*x^35 - 81*x^34 + 951*x^33 + 2088*x^32 - 42939*x^31 + 5643*x^30 + 1085229*x^29 - 1512459*x^28 - 16861368*x^27 + 39162357*x^26 + 165335742*x^25 - 541366140*x^24 - 985627989*x^23 + 4697849808*x^22 + 2841546570*x^21 - 26805831327*x^20 + 3870494730*x^19 + 100934354919*x^18 - 69735280059*x^17 - 242670922227*x^16 + 290388199548*x^15 + 335224535589*x^14 - 646861351536*x^13 - 161390547741*x^12 + 823286390904*x^11 - 214390074411*x^10 - 552047771387*x^9 + 373616574660*x^8 + 125024057829*x^7 - 203664646266*x^6 + 44758761705*x^5 + 30223063710*x^4 - 20282160951*x^3 + 4783445352*x^2 - 474397875*x + 13538501)
 
gp: K = bnfinit(x^36 - 9*x^35 - 81*x^34 + 951*x^33 + 2088*x^32 - 42939*x^31 + 5643*x^30 + 1085229*x^29 - 1512459*x^28 - 16861368*x^27 + 39162357*x^26 + 165335742*x^25 - 541366140*x^24 - 985627989*x^23 + 4697849808*x^22 + 2841546570*x^21 - 26805831327*x^20 + 3870494730*x^19 + 100934354919*x^18 - 69735280059*x^17 - 242670922227*x^16 + 290388199548*x^15 + 335224535589*x^14 - 646861351536*x^13 - 161390547741*x^12 + 823286390904*x^11 - 214390074411*x^10 - 552047771387*x^9 + 373616574660*x^8 + 125024057829*x^7 - 203664646266*x^6 + 44758761705*x^5 + 30223063710*x^4 - 20282160951*x^3 + 4783445352*x^2 - 474397875*x + 13538501, 1)
 

Normalized defining polynomial

\( x^{36} - 9 x^{35} - 81 x^{34} + 951 x^{33} + 2088 x^{32} - 42939 x^{31} + 5643 x^{30} + 1085229 x^{29} - 1512459 x^{28} - 16861368 x^{27} + 39162357 x^{26} + 165335742 x^{25} - 541366140 x^{24} - 985627989 x^{23} + 4697849808 x^{22} + 2841546570 x^{21} - 26805831327 x^{20} + 3870494730 x^{19} + 100934354919 x^{18} - 69735280059 x^{17} - 242670922227 x^{16} + 290388199548 x^{15} + 335224535589 x^{14} - 646861351536 x^{13} - 161390547741 x^{12} + 823286390904 x^{11} - 214390074411 x^{10} - 552047771387 x^{9} + 373616574660 x^{8} + 125024057829 x^{7} - 203664646266 x^{6} + 44758761705 x^{5} + 30223063710 x^{4} - 20282160951 x^{3} + 4783445352 x^{2} - 474397875 x + 13538501 \)

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

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[36, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(11765902701633698332267748143401268612491790640019253062374889850616455078125=3^{88}\cdot 5^{27}\cdot 7^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $129.74$
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:  $3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(945=3^{3}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{945}(1,·)$, $\chi_{945}(643,·)$, $\chi_{945}(517,·)$, $\chi_{945}(904,·)$, $\chi_{945}(13,·)$, $\chi_{945}(526,·)$, $\chi_{945}(274,·)$, $\chi_{945}(538,·)$, $\chi_{945}(412,·)$, $\chi_{945}(799,·)$, $\chi_{945}(421,·)$, $\chi_{945}(169,·)$, $\chi_{945}(433,·)$, $\chi_{945}(307,·)$, $\chi_{945}(694,·)$, $\chi_{945}(316,·)$, $\chi_{945}(832,·)$, $\chi_{945}(328,·)$, $\chi_{945}(841,·)$, $\chi_{945}(202,·)$, $\chi_{945}(631,·)$, $\chi_{945}(589,·)$, $\chi_{945}(64,·)$, $\chi_{945}(211,·)$, $\chi_{945}(853,·)$, $\chi_{945}(727,·)$, $\chi_{945}(223,·)$, $\chi_{945}(736,·)$, $\chi_{945}(97,·)$, $\chi_{945}(484,·)$, $\chi_{945}(106,·)$, $\chi_{945}(748,·)$, $\chi_{945}(622,·)$, $\chi_{945}(118,·)$, $\chi_{945}(937,·)$, $\chi_{945}(379,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $\frac{1}{269} a^{33} + \frac{24}{269} a^{32} - \frac{113}{269} a^{31} - \frac{101}{269} a^{30} - \frac{66}{269} a^{29} - \frac{72}{269} a^{28} + \frac{2}{269} a^{27} + \frac{51}{269} a^{26} - \frac{32}{269} a^{25} + \frac{33}{269} a^{24} - \frac{19}{269} a^{23} - \frac{106}{269} a^{22} + \frac{33}{269} a^{21} + \frac{75}{269} a^{20} + \frac{39}{269} a^{19} - \frac{94}{269} a^{18} - \frac{121}{269} a^{17} - \frac{105}{269} a^{16} - \frac{2}{269} a^{15} + \frac{104}{269} a^{14} + \frac{65}{269} a^{13} - \frac{11}{269} a^{12} - \frac{27}{269} a^{11} - \frac{91}{269} a^{10} - \frac{27}{269} a^{9} - \frac{28}{269} a^{8} + \frac{119}{269} a^{7} + \frac{69}{269} a^{6} + \frac{83}{269} a^{5} - \frac{70}{269} a^{4} + \frac{101}{269} a^{3} + \frac{110}{269} a^{2} + \frac{4}{269} a$, $\frac{1}{115939} a^{34} + \frac{80}{115939} a^{33} + \frac{32704}{115939} a^{32} - \frac{26604}{115939} a^{31} - \frac{10026}{115939} a^{30} - \frac{48422}{115939} a^{29} - \frac{33082}{115939} a^{28} + \frac{45893}{115939} a^{27} - \frac{36181}{115939} a^{26} - \frac{16554}{115939} a^{25} + \frac{19045}{115939} a^{24} - \frac{18117}{115939} a^{23} - \frac{12359}{115939} a^{22} - \frac{29819}{115939} a^{21} - \frac{26427}{115939} a^{20} + \frac{5856}{115939} a^{19} + \frac{23936}{115939} a^{18} - \frac{51535}{115939} a^{17} - \frac{56454}{115939} a^{16} + \frac{39266}{115939} a^{15} + \frac{23374}{115939} a^{14} - \frac{14394}{115939} a^{13} - \frac{10058}{115939} a^{12} - \frac{42760}{115939} a^{11} + \frac{46794}{115939} a^{10} + \frac{38810}{115939} a^{9} + \frac{51813}{115939} a^{8} - \frac{1606}{115939} a^{7} + \frac{3140}{115939} a^{6} + \frac{43314}{115939} a^{5} + \frac{32765}{115939} a^{4} - \frac{57718}{115939} a^{3} + \frac{16386}{115939} a^{2} + \frac{33311}{115939} a + \frac{194}{431}$, $\frac{1}{131981568018369042344862030605246912398733426180635489705745413285106717453444151245211751437888017791192042664545787517500129179766799340834672611224184888848411} a^{35} - \frac{310050781894589817191857911899757524270672851530593548543193550111414939168630005579691838849739802983149568363274524923619619084820147088965957485797952895}{131981568018369042344862030605246912398733426180635489705745413285106717453444151245211751437888017791192042664545787517500129179766799340834672611224184888848411} a^{34} - \frac{36537604138796126178362541874802809181634823529393935508662273936493960294140362213314501832784206286755885731271548924050593011693849911617113286719764357762}{131981568018369042344862030605246912398733426180635489705745413285106717453444151245211751437888017791192042664545787517500129179766799340834672611224184888848411} a^{33} - \frac{687315429141785698660841386249885262995607591862025078077823285668746015281487531923879751924695508846063480250051860758776773989189243901481115897995303687768}{131981568018369042344862030605246912398733426180635489705745413285106717453444151245211751437888017791192042664545787517500129179766799340834672611224184888848411} a^{32} - \frac{12927734192318516552466707897059631714984723757347988302482690444731473141522724332005227360132137704961268196289759105380924378541859238026588392061342217618914}{131981568018369042344862030605246912398733426180635489705745413285106717453444151245211751437888017791192042664545787517500129179766799340834672611224184888848411} a^{31} - \frac{19948638769601849131473379670850672068022819350207381833980796379826036822955006842021254520917399865566814697167036220623543595471861620524883107878380731034261}{131981568018369042344862030605246912398733426180635489705745413285106717453444151245211751437888017791192042664545787517500129179766799340834672611224184888848411} a^{30} - \frac{19566932764756869404978009088402242225065253617280768355273922975805500672922479866072368604992489653742336571962356952070367595869286624810902669841383610704314}{131981568018369042344862030605246912398733426180635489705745413285106717453444151245211751437888017791192042664545787517500129179766799340834672611224184888848411} a^{29} - \frac{197261149499810534812421988567950810418094181619248257480914872722320244845277460084918213064958646432104044471007169057233797239799435696246078920249644509758}{490637799324791978977182269907981086984139130783031560244406740836827945923584205372534391962408988071345883511322630176580405872739031006820344279643809995719} a^{28} + \frac{31833489147443456829339627260931061254153414067088853325477437866432196175537537090815321799865625794334531821228717360467877014041141455798855740511448624102510}{131981568018369042344862030605246912398733426180635489705745413285106717453444151245211751437888017791192042664545787517500129179766799340834672611224184888848411} a^{27} - \frac{45262107377148884121317524815173005890849204469428656592494406463444102705908172526001071750764167872191429493727042861254569536339953993641979835368965353467068}{131981568018369042344862030605246912398733426180635489705745413285106717453444151245211751437888017791192042664545787517500129179766799340834672611224184888848411} a^{26} + \frac{59737149214786306139173483214569058953314128649795080496158401834909240957194194716583863922464885009429246582639596419265905954973411401756748805488066005496145}{131981568018369042344862030605246912398733426180635489705745413285106717453444151245211751437888017791192042664545787517500129179766799340834672611224184888848411} a^{25} - \frac{20686156830804441462315613677945415950826205202051251894130384672019858796148126690653634846936712166465830186250366832981408119810616454149600266479885736742834}{131981568018369042344862030605246912398733426180635489705745413285106717453444151245211751437888017791192042664545787517500129179766799340834672611224184888848411} a^{24} + \frac{52858667108985440723246037874120664851035865382134658949224307463098745319677126840919996430073350782902767494610192310670978863956105931458736902038931188755210}{131981568018369042344862030605246912398733426180635489705745413285106717453444151245211751437888017791192042664545787517500129179766799340834672611224184888848411} a^{23} + \frac{11372024152044652505297510134486742509256678405792564628682226839267346401561481454712487495315266942510376559589801354411741542046480177970802879349215097727305}{131981568018369042344862030605246912398733426180635489705745413285106717453444151245211751437888017791192042664545787517500129179766799340834672611224184888848411} a^{22} - \frac{30215781065158247280280352671242389781805781112878388812710180154703380136508902826114264482828355744954493012975630685580377806326175585647775433688436377145565}{131981568018369042344862030605246912398733426180635489705745413285106717453444151245211751437888017791192042664545787517500129179766799340834672611224184888848411} a^{21} + \frac{15359713979353778743335501541024874711291256771429451913647438482436862450601545123841529630001596414232515798175471283313875365818503682319463473574003900254499}{131981568018369042344862030605246912398733426180635489705745413285106717453444151245211751437888017791192042664545787517500129179766799340834672611224184888848411} a^{20} + \frac{42951912635121002006170570659041146806809609994978143354962402605546876536877854577626840122753137807561309985587834064928701616127704299317374421644375119991487}{131981568018369042344862030605246912398733426180635489705745413285106717453444151245211751437888017791192042664545787517500129179766799340834672611224184888848411} a^{19} - \frac{11017747416133081971414856269571750468700543135850603484391872207691455828317034592105650116212029828513493800467979962582922968897321665740125976393125650793266}{131981568018369042344862030605246912398733426180635489705745413285106717453444151245211751437888017791192042664545787517500129179766799340834672611224184888848411} a^{18} + \frac{37156733885723822574749395447102668665176759245012710111916819955068760901518178412292551657469422429092485632702196017943819909173070431863770430048501773557913}{131981568018369042344862030605246912398733426180635489705745413285106717453444151245211751437888017791192042664545787517500129179766799340834672611224184888848411} a^{17} - \frac{19385427069930635568958842817570850112575714662503998044083483049333839311968427375703913325491066546965234695749765705028765402354575460258084951374026627245517}{131981568018369042344862030605246912398733426180635489705745413285106717453444151245211751437888017791192042664545787517500129179766799340834672611224184888848411} a^{16} - \frac{61810726245994243518113637903930080269971515390408894452784422429703672274462594726725460064214011092000931872311258085114567206972389773239314161227790127743259}{131981568018369042344862030605246912398733426180635489705745413285106717453444151245211751437888017791192042664545787517500129179766799340834672611224184888848411} a^{15} - \frac{32042515443023001637217530478231462899851498985095248746728042622200895263595868974112628429201834525053673109797390314762236592809290193640633439507315403454607}{131981568018369042344862030605246912398733426180635489705745413285106717453444151245211751437888017791192042664545787517500129179766799340834672611224184888848411} a^{14} - \frac{7728840707361777521463685430865757264165313446735920296123336490602970685242095002342609650556845612900945110644393332220326667662330131268587764801717534380576}{131981568018369042344862030605246912398733426180635489705745413285106717453444151245211751437888017791192042664545787517500129179766799340834672611224184888848411} a^{13} + \frac{16133590205753513741240671951350070955472238661506700572381012298347200311798037910647700770918783574104205932618851576291857773933742465290351861615338416457203}{131981568018369042344862030605246912398733426180635489705745413285106717453444151245211751437888017791192042664545787517500129179766799340834672611224184888848411} a^{12} + \frac{30231889695152301714191234422805281515871812680077057451730691957463588068521776988332151520584515404257418383784877347666988008376006696408692071898668924052261}{131981568018369042344862030605246912398733426180635489705745413285106717453444151245211751437888017791192042664545787517500129179766799340834672611224184888848411} a^{11} + \frac{48994192980874194355200123015143906939618678439292568029414325304423838862016488657074845514586773914819451621436844071921025236037577111105237224747899452294803}{131981568018369042344862030605246912398733426180635489705745413285106717453444151245211751437888017791192042664545787517500129179766799340834672611224184888848411} a^{10} - \frac{17265366538503007824902198555317910188998428857752868565787131697314578314014944622366163354891976731530152662086808022417661721021579379668640137290055070043919}{131981568018369042344862030605246912398733426180635489705745413285106717453444151245211751437888017791192042664545787517500129179766799340834672611224184888848411} a^{9} + \frac{29415008058125365096446984763386042266183829940785506884111595766156464997366518536715169826986593944762256322870632773234214516603151052084324091933582861556808}{131981568018369042344862030605246912398733426180635489705745413285106717453444151245211751437888017791192042664545787517500129179766799340834672611224184888848411} a^{8} + \frac{27678499751508164957949043528088164447150677997090846530802295770177169526404995117715213975969590924762839540381228751088797883287898124926577560102954433629521}{131981568018369042344862030605246912398733426180635489705745413285106717453444151245211751437888017791192042664545787517500129179766799340834672611224184888848411} a^{7} + \frac{38858667888185880485659953161963887974625140229658838911526341371632840223818507622786423685437303205554386791998164523442521723438300325805962785775429127543882}{131981568018369042344862030605246912398733426180635489705745413285106717453444151245211751437888017791192042664545787517500129179766799340834672611224184888848411} a^{6} - \frac{893590748967732069996592784981041671269128466251146433354021692890100334567883683042626539827668495661024676668637284931676374899332609172537195552102671267494}{131981568018369042344862030605246912398733426180635489705745413285106717453444151245211751437888017791192042664545787517500129179766799340834672611224184888848411} a^{5} - \frac{35485532343019722808278709003387212324076265222402050723323723433125639213193393345242579589972383528728032895577284880943369023572415219967677283940758730990090}{131981568018369042344862030605246912398733426180635489705745413285106717453444151245211751437888017791192042664545787517500129179766799340834672611224184888848411} a^{4} - \frac{33293907118045121137445623702798570220678616792191610062617457434546862289151787398144445501033737586430306732136688435393622134133830998510821915560858065546481}{131981568018369042344862030605246912398733426180635489705745413285106717453444151245211751437888017791192042664545787517500129179766799340834672611224184888848411} a^{3} - \frac{52795161827302925724225188141215940784887362711018973100785899789297512686555724009687547317699121992345367149369839339882057358118192600675818976847571733517831}{131981568018369042344862030605246912398733426180635489705745413285106717453444151245211751437888017791192042664545787517500129179766799340834672611224184888848411} a^{2} + \frac{1223823946293020265539642294978250845773275704161964113151037824690310200364453413585134693404776424587909857346070640887165530405150974120817778970082783763333}{131981568018369042344862030605246912398733426180635489705745413285106717453444151245211751437888017791192042664545787517500129179766799340834672611224184888848411} a + \frac{49798666530504945166725128097382576242345917075945741338526480139230232887452600224115537246252569017736220394323449728171700231238381115501962978109788967413}{490637799324791978977182269907981086984139130783031560244406740836827945923584205372534391962408988071345883511322630176580405872739031006820344279643809995719}$

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:  $35$
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

$C_{36}$ (as 36T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 36
The 36 conjugacy class representatives for $C_{36}$
Character table for $C_{36}$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), 4.4.6125.1, 6.6.820125.1, \(\Q(\zeta_{27})^+\), 12.12.9891413435408203125.1, 18.18.1923380668327365689220703125.1

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 $36$ R R R ${\href{/LocalNumberField/11.9.0.1}{9} }^{4}$ $36$ ${\href{/LocalNumberField/17.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{12}$ $36$ $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$ $18^{2}$ $36$ $36$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{9}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{4}$

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
3Data not computed
5Data not computed
7Data not computed