Global number field 36.0.80731161945559438248836517604483794680492496928434000672170721.1

• Label: 36.0.80731161945...0721.1
• Degree: $36$
• Signature: $[0, 18]$
• Discriminant: $3^{18}\cdot 37^{34}$
• Root discriminant: $52.44$
• Ramified primes: $3, 37$
• Class number: $12996$ (GRH)
• Class group: $[19, 684]$ (GRH)
• Galois group: $C_2\times C_{18}$ (as 36T2)

Normalized defining polynomial

\( x^{36} - x^{35} + 18 x^{34} - 15 x^{33} + 185 x^{32} - 137 x^{31} + 1281 x^{30} - 831 x^{29} + 6616 x^{28} - 3799 x^{27} + 26339 x^{26} - 13196 x^{25} + 83006 x^{24} - 36260 x^{23} + 208286 x^{22} - 77735 x^{21} + 418163 x^{20} - 132518 x^{19} + 666068 x^{18} - 173318 x^{17} + 834766 x^{16} - 177139 x^{15} + 804267 x^{14} - 129870 x^{13} + 582511 x^{12} - 73129 x^{11} + 302060 x^{10} - 23507 x^{9} + 107217 x^{8} - 7128 x^{7} + 23244 x^{6} - 78 x^{5} + 2775 x^{4} - 165 x^{3} + 126 x^{2} + 9 x + 1 \)

Invariants

Degree:		$36$
Signature:		$[0, 18]$
Discriminant:		\(80731161945559438248836517604483794680492496928434000672170721=3^{18}\cdot 37^{34}\)
Root discriminant:		$52.44$
Ramified primes:		$3, 37$
This field is Galois and abelian over $\Q$.
Conductor:		\(111=3\cdot 37\)
Dirichlet character group:		$\lbrace$$\chi_{111}(1,·)$, $\chi_{111}(4,·)$, $\chi_{111}(7,·)$, $\chi_{111}(10,·)$, $\chi_{111}(11,·)$, $\chi_{111}(16,·)$, $\chi_{111}(25,·)$, $\chi_{111}(26,·)$, $\chi_{111}(28,·)$, $\chi_{111}(34,·)$, $\chi_{111}(38,·)$, $\chi_{111}(40,·)$, $\chi_{111}(41,·)$, $\chi_{111}(44,·)$, $\chi_{111}(46,·)$, $\chi_{111}(47,·)$, $\chi_{111}(49,·)$, $\chi_{111}(53,·)$, $\chi_{111}(58,·)$, $\chi_{111}(62,·)$, $\chi_{111}(64,·)$, $\chi_{111}(65,·)$, $\chi_{111}(67,·)$, $\chi_{111}(70,·)$, $\chi_{111}(71,·)$, $\chi_{111}(73,·)$, $\chi_{111}(77,·)$, $\chi_{111}(83,·)$, $\chi_{111}(85,·)$, $\chi_{111}(86,·)$, $\chi_{111}(95,·)$, $\chi_{111}(100,·)$, $\chi_{111}(101,·)$, $\chi_{111}(104,·)$, $\chi_{111}(107,·)$, $\chi_{111}(110,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $\frac{1}{73} a^{33} - \frac{6}{73} a^{32} - \frac{7}{73} a^{31} - \frac{23}{73} a^{30} + \frac{3}{73} a^{29} + \frac{1}{73} a^{28} - \frac{19}{73} a^{27} - \frac{12}{73} a^{26} - \frac{25}{73} a^{25} - \frac{15}{73} a^{24} - \frac{1}{73} a^{23} + \frac{25}{73} a^{22} - \frac{18}{73} a^{21} - \frac{15}{73} a^{20} + \frac{6}{73} a^{19} + \frac{28}{73} a^{17} + \frac{8}{73} a^{16} - \frac{31}{73} a^{15} - \frac{14}{73} a^{14} - \frac{30}{73} a^{13} + \frac{32}{73} a^{12} + \frac{22}{73} a^{11} - \frac{27}{73} a^{10} + \frac{27}{73} a^{9} + \frac{23}{73} a^{8} - \frac{11}{73} a^{7} + \frac{33}{73} a^{6} + \frac{17}{73} a^{5} - \frac{34}{73} a^{4} + \frac{23}{73} a^{3} + \frac{8}{73} a^{2} - \frac{10}{73} a - \frac{9}{73}$, $\frac{1}{73} a^{34} + \frac{30}{73} a^{32} + \frac{8}{73} a^{31} + \frac{11}{73} a^{30} + \frac{19}{73} a^{29} - \frac{13}{73} a^{28} + \frac{20}{73} a^{27} - \frac{24}{73} a^{26} - \frac{19}{73} a^{25} - \frac{18}{73} a^{24} + \frac{19}{73} a^{23} - \frac{14}{73} a^{22} + \frac{23}{73} a^{21} - \frac{11}{73} a^{20} + \frac{36}{73} a^{19} + \frac{28}{73} a^{18} + \frac{30}{73} a^{17} + \frac{17}{73} a^{16} + \frac{19}{73} a^{15} + \frac{32}{73} a^{14} - \frac{2}{73} a^{13} - \frac{5}{73} a^{12} + \frac{32}{73} a^{11} + \frac{11}{73} a^{10} - \frac{34}{73} a^{9} - \frac{19}{73} a^{8} - \frac{33}{73} a^{7} - \frac{4}{73} a^{6} - \frac{5}{73} a^{5} - \frac{35}{73} a^{4} - \frac{35}{73} a^{2} + \frac{4}{73} a + \frac{19}{73}$, $\frac{1}{365727950135022534390908589813592896926517587901066835721264058193} a^{35} + \frac{1950516105845598691982369338104695891133902050074740674662109205}{365727950135022534390908589813592896926517587901066835721264058193} a^{34} + \frac{253581610741919985230813716809373485462908442829787779860168936}{365727950135022534390908589813592896926517587901066835721264058193} a^{33} - \frac{64829502577752596333899968231393035469031506954221300144246189544}{365727950135022534390908589813592896926517587901066835721264058193} a^{32} - \frac{171426671930428359662171926960367245031760919856537319692678265813}{365727950135022534390908589813592896926517587901066835721264058193} a^{31} + \frac{167259138011320175210398583166312336497107077372367930476994643193}{365727950135022534390908589813592896926517587901066835721264058193} a^{30} - \frac{125930194659218157536168437367198991365660747305603345211056167281}{365727950135022534390908589813592896926517587901066835721264058193} a^{29} + \frac{159648004432607467770462332776310761408142595578047781320532346123}{365727950135022534390908589813592896926517587901066835721264058193} a^{28} - \frac{45901791025683591345493937166712956812180620755409277034890139991}{365727950135022534390908589813592896926517587901066835721264058193} a^{27} + \frac{76587425406251914295978433060341707671296399192275963108746180435}{365727950135022534390908589813592896926517587901066835721264058193} a^{26} - \frac{60868644311315486407762965704995128761089545621461198353276435441}{365727950135022534390908589813592896926517587901066835721264058193} a^{25} - \frac{11311474690528038469653724942476985522239834185643115200943373733}{365727950135022534390908589813592896926517587901066835721264058193} a^{24} - \frac{155287263259564423639849315663016422982780181290304081681914942654}{365727950135022534390908589813592896926517587901066835721264058193} a^{23} + \frac{71420828342690238556594425765290849081259924609286594882171392991}{365727950135022534390908589813592896926517587901066835721264058193} a^{22} + \frac{126795280253375455903532827045566037192486930791567620869715314366}{365727950135022534390908589813592896926517587901066835721264058193} a^{21} - \frac{170263197343136887072257872347830332967784980065685434714163068916}{365727950135022534390908589813592896926517587901066835721264058193} a^{20} + \frac{38750430200732286887134608451486558950070427648346568473412092256}{365727950135022534390908589813592896926517587901066835721264058193} a^{19} + \frac{27434364450839131400884822942040960719188780831672475612790472340}{365727950135022534390908589813592896926517587901066835721264058193} a^{18} - \frac{115538956163578731820495750034415692015138637068201828193844902309}{365727950135022534390908589813592896926517587901066835721264058193} a^{17} + \frac{21772243557807424717202136862906079521230864492390634622966285446}{365727950135022534390908589813592896926517587901066835721264058193} a^{16} + \frac{128043828562029553068061055765444902564633588700766076742065158817}{365727950135022534390908589813592896926517587901066835721264058193} a^{15} + \frac{105452076280122806475167967986561496805477714342914344768731747704}{365727950135022534390908589813592896926517587901066835721264058193} a^{14} - \frac{127978273150494699885419828016130698374737616292520359879491593949}{365727950135022534390908589813592896926517587901066835721264058193} a^{13} + \frac{56368882361189773381703948375084124089397300872671347794455582784}{365727950135022534390908589813592896926517587901066835721264058193} a^{12} - \frac{142981749077426597632427662054800806145864324679766787518031917577}{365727950135022534390908589813592896926517587901066835721264058193} a^{11} - \frac{132073894054437580159302600730369241114260831797267873532847936570}{365727950135022534390908589813592896926517587901066835721264058193} a^{10} - \frac{31866929027198246826228983572688307698094353817658484334382381888}{365727950135022534390908589813592896926517587901066835721264058193} a^{9} + \frac{108461861466797502270050134013768132372002883477314458664407962837}{365727950135022534390908589813592896926517587901066835721264058193} a^{8} - \frac{7660053924441216511529945047414771725863643283466958195510097507}{365727950135022534390908589813592896926517587901066835721264058193} a^{7} + \frac{88291840788151329482617327533939042407199044501605834678007913153}{365727950135022534390908589813592896926517587901066835721264058193} a^{6} - \frac{163909424897598882429955677180278357648572526362635398335670180543}{365727950135022534390908589813592896926517587901066835721264058193} a^{5} - \frac{141058274403677446485745764155509429880853755749095887524255606441}{365727950135022534390908589813592896926517587901066835721264058193} a^{4} + \frac{38448035992474492236835156598294347626911260334088540362079321077}{365727950135022534390908589813592896926517587901066835721264058193} a^{3} - \frac{172688077016890610843428113561917382504165886437569587029636200349}{365727950135022534390908589813592896926517587901066835721264058193} a^{2} - \frac{148052034755178136356607951409769550931675265852953978830629741537}{365727950135022534390908589813592896926517587901066835721264058193} a + \frac{141728930178830187211731044814389788867332397448176597678434429495}{365727950135022534390908589813592896926517587901066835721264058193}$

Class group and class number

$C_{19}\times C_{684}$, which has order $12996$ (assuming GRH)

Unit group

Rank:		$17$
Torsion generator:		\( -\frac{34122480476536495885823411117725648848297922887380250870498727234}{365727950135022534390908589813592896926517587901066835721264058193} a^{35} + \frac{35920099990306144158226830773237501966815584652708354942618883933}{365727950135022534390908589813592896926517587901066835721264058193} a^{34} - \frac{615083437181100973102290398099363536394447387659071528670198770753}{365727950135022534390908589813592896926517587901066835721264058193} a^{33} + \frac{543294620918538617109207244987576078895238727979936775551656239507}{365727950135022534390908589813592896926517587901066835721264058193} a^{32} - \frac{6323219576533506173543594384637229613051597192385634606383939877972}{365727950135022534390908589813592896926517587901066835721264058193} a^{31} + \frac{4994028550074475773614939668982845334911542526376809072066813446520}{365727950135022534390908589813592896926517587901066835721264058193} a^{30} - \frac{43789545700798284266678688992786530335633250234760390102543103751356}{365727950135022534390908589813592896926517587901066835721264058193} a^{29} + \frac{30538101137546885171065725683196131944400618465349807317471070295845}{365727950135022534390908589813592896926517587901066835721264058193} a^{28} - \frac{226094852310675109725082640471747046842141478058709699943311770029233}{365727950135022534390908589813592896926517587901066835721264058193} a^{27} + \frac{140802803241538726678484986478660388481558873898343767651597561524073}{365727950135022534390908589813592896926517587901066835721264058193} a^{26} - \frac{899662743420468921090965311067708352400951449363031696002455350422472}{365727950135022534390908589813592896926517587901066835721264058193} a^{25} + \frac{494370034684118631348844025413733664758589959107270464644516063424382}{365727950135022534390908589813592896926517587901066835721264058193} a^{24} - \frac{2832702632307964074097956305645971571255004367493074655008528218257566}{365727950135022534390908589813592896926517587901066835721264058193} a^{23} + \frac{1375361385329746730339724095795553192320654228521346424514192109698348}{365727950135022534390908589813592896926517587901066835721264058193} a^{22} - \frac{7099305470203584634525615669076328860278060023228367082535085912640170}{365727950135022534390908589813592896926517587901066835721264058193} a^{21} + \frac{2996903950309203456086012668935968640961389811412156144802040975546348}{365727950135022534390908589813592896926517587901066835721264058193} a^{20} - \frac{14226830177776055252324451680478952567919670003585093236728241215416985}{365727950135022534390908589813592896926517587901066835721264058193} a^{19} + \frac{5210809126661414118826730544576553890988514785410355002755432457682249}{365727950135022534390908589813592896926517587901066835721264058193} a^{18} - \frac{22605602178149331114252766356866281505343532953417190044279124244844161}{365727950135022534390908589813592896926517587901066835721264058193} a^{17} + \frac{7007791415035400592789121544589243795962853911924902271068832990226289}{365727950135022534390908589813592896926517587901066835721264058193} a^{16} - \frac{28230029416107238778025746906923996814056773011687080990664209928227583}{365727950135022534390908589813592896926517587901066835721264058193} a^{15} + \frac{7416059244792492650399321473801859407320669821663505240054337179613445}{365727950135022534390908589813592896926517587901066835721264058193} a^{14} - \frac{27065934546758807409030469320616305346669578910218733914306996055971366}{365727950135022534390908589813592896926517587901066835721264058193} a^{13} + \frac{5753779239451731686726543155643944040830672515091752316957551478826250}{365727950135022534390908589813592896926517587901066835721264058193} a^{12} - \frac{19456343624872687763264086627413575996271004192247238151574420547824420}{365727950135022534390908589813592896926517587901066835721264058193} a^{11} + \frac{3462966144439364019763530187242271972835021163410861262040525403851913}{365727950135022534390908589813592896926517587901066835721264058193} a^{10} - \frac{9980823467549781215685173095720091111087833315498145689198934890360111}{365727950135022534390908589813592896926517587901066835721264058193} a^{9} + \frac{1307617305524707519457541155870299299414462296446468970040965316641557}{365727950135022534390908589813592896926517587901066835721264058193} a^{8} - \frac{3475551741590602543175147416232313481626086833511671804190450452709766}{365727950135022534390908589813592896926517587901066835721264058193} a^{7} + \frac{431341116566667804301373140731883003832073698261304135400115610321480}{365727950135022534390908589813592896926517587901066835721264058193} a^{6} - \frac{731868384586680047520962298759092079999212586283239884341466674884120}{365727950135022534390908589813592896926517587901066835721264058193} a^{5} + \frac{43979378537352129542691076866218379786253337005261913936392627312728}{365727950135022534390908589813592896926517587901066835721264058193} a^{4} - \frac{80982907038501973778097211728899009297524862646644087808361795677936}{365727950135022534390908589813592896926517587901066835721264058193} a^{3} + \frac{12761208633604858136842663688957528159502826724118708182623353365328}{365727950135022534390908589813592896926517587901066835721264058193} a^{2} - \frac{3288990076144576669983553907357450657151358179358877400431188778463}{365727950135022534390908589813592896926517587901066835721264058193} a + \frac{138091802744566568529774057644096308278359388757109784234311498641}{365727950135022534390908589813592896926517587901066835721264058193} \) (order $6$)
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)
Regulator:		\( 3171872728760.222 \) (assuming GRH)

Galois group

$C_2\times C_{18}$ (as 36T2):

An abelian group of order 36

The 36 conjugacy class representatives for $C_2\times C_{18}$

Character table for $C_2\times C_{18}$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{37}) \), \(\Q(\sqrt{-111}) \), 3.3.1369.1, \(\Q(\sqrt{-3}, \sqrt{37})\), 6.0.50602347.1, 6.6.69343957.1, 6.0.1872286839.1, 9.9.3512479453921.1, 12.0.3505458007492611921.1, 18.0.242839247007536485508643885603.1, \(\Q(\zeta_{37})^+\), 18.0.8985052139278849963819823767311.1

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	$18^{2}$	R	$18^{2}$	${\href{/LocalNumberField/7.9.0.1}{9} }^{4}$	${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$	$18^{2}$	$18^{2}$	$18^{2}$	${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$	${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$	${\href{/LocalNumberField/31.2.0.1}{2} }^{18}$	R	$18^{2}$	${\href{/LocalNumberField/43.2.0.1}{2} }^{18}$	${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$	$18^{2}$	$18^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
3	Data not computed
37	Data not computed