Global number field 34.34.52741270066792486921251681213328814961789907155674264592984618377892546447546437542608896.1

• Label: 34.34.5274127006...8896.1
• Degree: $34$
• Signature: $[34, 0]$
• Discriminant: $2^{34}\cdot 239^{33}$
• Root discriminant: $406.89$
• Ramified primes: $2, 239$
• Class number: Not computed
• Class group: Not computed
• Galois group: $C_{34}$ (as 34T1)

Normalized defining polynomial

\( x^{34} - 239 x^{32} + 25334 x^{30} - 1571903 x^{28} + 63464060 x^{26} - 1753657720 x^{24} + 33979252356 x^{22} - 465613265023 x^{20} + 4498814472085 x^{18} - 30263057086263 x^{16} + 138661899949444 x^{14} - 419815968533922 x^{12} + 810250249909457 x^{10} - 961723188339514 x^{8} + 674590960979032 x^{6} - 266007049819311 x^{4} + 53533295641866 x^{2} - 4207542058871 \)

Invariants

Degree:		$34$
Signature:		$[34, 0]$
Discriminant:		\(52741270066792486921251681213328814961789907155674264592984618377892546447546437542608896=2^{34}\cdot 239^{33}\)
Root discriminant:		$406.89$
Ramified primes:		$2, 239$
This field is Galois and abelian over $\Q$.
Conductor:		\(956=2^{2}\cdot 239\)
Dirichlet character group:		$\lbrace$$\chi_{956}(1,·)$, $\chi_{956}(261,·)$, $\chi_{956}(641,·)$, $\chi_{956}(23,·)$, $\chi_{956}(529,·)$, $\chi_{956}(403,·)$, $\chi_{956}(405,·)$, $\chi_{956}(407,·)$, $\chi_{956}(665,·)$, $\chi_{956}(411,·)$, $\chi_{956}(545,·)$, $\chi_{956}(291,·)$, $\chi_{956}(549,·)$, $\chi_{956}(551,·)$, $\chi_{956}(553,·)$, $\chi_{956}(427,·)$, $\chi_{956}(689,·)$, $\chi_{956}(695,·)$, $\chi_{956}(315,·)$, $\chi_{956}(245,·)$, $\chi_{956}(267,·)$, $\chi_{956}(199,·)$, $\chi_{956}(203,·)$, $\chi_{956}(845,·)$, $\chi_{956}(849,·)$, $\chi_{956}(855,·)$, $\chi_{956}(933,·)$, $\chi_{956}(955,·)$, $\chi_{956}(101,·)$, $\chi_{956}(107,·)$, $\chi_{956}(711,·)$, $\chi_{956}(111,·)$, $\chi_{956}(753,·)$, $\chi_{956}(757,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $\frac{1}{139} a^{26} + \frac{53}{139} a^{24} + \frac{12}{139} a^{22} - \frac{51}{139} a^{20} - \frac{51}{139} a^{18} - \frac{8}{139} a^{16} - \frac{68}{139} a^{14} - \frac{3}{139} a^{12} - \frac{17}{139} a^{10} + \frac{29}{139} a^{8} + \frac{14}{139} a^{6} + \frac{24}{139} a^{4} + \frac{33}{139} a^{2} + \frac{32}{139}$, $\frac{1}{139} a^{27} + \frac{53}{139} a^{25} + \frac{12}{139} a^{23} - \frac{51}{139} a^{21} - \frac{51}{139} a^{19} - \frac{8}{139} a^{17} - \frac{68}{139} a^{15} - \frac{3}{139} a^{13} - \frac{17}{139} a^{11} + \frac{29}{139} a^{9} + \frac{14}{139} a^{7} + \frac{24}{139} a^{5} + \frac{33}{139} a^{3} + \frac{32}{139} a$, $\frac{1}{139} a^{28} - \frac{17}{139} a^{24} + \frac{8}{139} a^{22} + \frac{11}{139} a^{20} + \frac{54}{139} a^{18} - \frac{61}{139} a^{16} - \frac{13}{139} a^{14} + \frac{3}{139} a^{12} - \frac{43}{139} a^{10} + \frac{6}{139} a^{8} - \frac{23}{139} a^{6} + \frac{12}{139} a^{4} - \frac{49}{139} a^{2} - \frac{28}{139}$, $\frac{1}{139} a^{29} - \frac{17}{139} a^{25} + \frac{8}{139} a^{23} + \frac{11}{139} a^{21} + \frac{54}{139} a^{19} - \frac{61}{139} a^{17} - \frac{13}{139} a^{15} + \frac{3}{139} a^{13} - \frac{43}{139} a^{11} + \frac{6}{139} a^{9} - \frac{23}{139} a^{7} + \frac{12}{139} a^{5} - \frac{49}{139} a^{3} - \frac{28}{139} a$, $\frac{1}{105779} a^{30} + \frac{258}{105779} a^{28} - \frac{372}{105779} a^{26} - \frac{48491}{105779} a^{24} - \frac{52225}{105779} a^{22} - \frac{32379}{105779} a^{20} - \frac{30574}{105779} a^{18} - \frac{11799}{105779} a^{16} - \frac{2007}{105779} a^{14} + \frac{24731}{105779} a^{12} - \frac{19509}{105779} a^{10} - \frac{8353}{105779} a^{8} + \frac{15518}{105779} a^{6} + \frac{13709}{105779} a^{4} + \frac{42752}{105779} a^{2} + \frac{19363}{105779}$, $\frac{1}{29300783} a^{31} + \frac{17761}{29300783} a^{29} - \frac{58208}{29300783} a^{27} + \frac{13090174}{29300783} a^{25} + \frac{1932463}{29300783} a^{23} + \frac{1417326}{29300783} a^{21} - \frac{9781267}{29300783} a^{19} - \frac{6223081}{29300783} a^{17} + \frac{14492760}{29300783} a^{15} + \frac{7126383}{29300783} a^{13} + \frac{13010333}{29300783} a^{11} - \frac{2109474}{29300783} a^{9} - \frac{9024401}{29300783} a^{7} + \frac{1374377}{29300783} a^{5} - \frac{3569715}{29300783} a^{3} + \frac{2227024}{29300783} a$, $\frac{1}{105801193806468905548558434400043609639922484778837050690132414700212613503546511526376518565334570107721} a^{32} + \frac{417612112431198040617611256015652375174198304411360378517094244545878502347954599410547934121369599}{105801193806468905548558434400043609639922484778837050690132414700212613503546511526376518565334570107721} a^{30} - \frac{350244417484504283087469271966476497721757210678744526457595153315087851028221347279274763948843297023}{105801193806468905548558434400043609639922484778837050690132414700212613503546511526376518565334570107721} a^{28} - \frac{236727892181518844632886225994651108215836303957142353014911931957349620674739527566484766611653147334}{105801193806468905548558434400043609639922484778837050690132414700212613503546511526376518565334570107721} a^{26} + \frac{96574681567381984816625564383267019805164557253604992559608635619384213904561584245022481131001953259}{462013946753139325539556482096260304104464999034222928777870806551146783858281709722168203342072358549} a^{24} - \frac{37334283392331555519019249747382941199984928928430156836664628420667536276155999133098243239229517357745}{105801193806468905548558434400043609639922484778837050690132414700212613503546511526376518565334570107721} a^{22} + \frac{9271433075018275509952157407277827318871825605376654461954328192399663425290329259878310951008355023150}{105801193806468905548558434400043609639922484778837050690132414700212613503546511526376518565334570107721} a^{20} + \frac{6105629547567361419221999403313516669376075031610467633944763979477646162711354760302251115493849304799}{105801193806468905548558434400043609639922484778837050690132414700212613503546511526376518565334570107721} a^{18} - \frac{27359007930222542852475903209639684088555754390591818177709678954241075823406019524899391163095139438339}{105801193806468905548558434400043609639922484778837050690132414700212613503546511526376518565334570107721} a^{16} + \frac{23695745958791816003534859086561896655625917879814467197788036825962594654042133137590896260808446156936}{105801193806468905548558434400043609639922484778837050690132414700212613503546511526376518565334570107721} a^{14} - \frac{50243550485649235059803674067949045372278946778694178281888337921116796931130332678428423221296049140333}{105801193806468905548558434400043609639922484778837050690132414700212613503546511526376518565334570107721} a^{12} + \frac{32591737310690585461014784386941740181882366999028188616624719081403102835284979440113560729873332919849}{105801193806468905548558434400043609639922484778837050690132414700212613503546511526376518565334570107721} a^{10} + \frac{9217924970755337471405215356350268344991760755896195229110935538009261115537212429438592407946474448051}{105801193806468905548558434400043609639922484778837050690132414700212613503546511526376518565334570107721} a^{8} - \frac{24676058486478295549047346123053320596305973960903448074278691862500636507377071333458474665226865684526}{105801193806468905548558434400043609639922484778837050690132414700212613503546511526376518565334570107721} a^{6} - \frac{5150550605660103035824032231733734365199710010735479185409018175020518619133132743654287878467644949407}{105801193806468905548558434400043609639922484778837050690132414700212613503546511526376518565334570107721} a^{4} + \frac{28042459233202544668772250674761519856575423679169683874995059641556976337185523249438188226925472677880}{105801193806468905548558434400043609639922484778837050690132414700212613503546511526376518565334570107721} a^{2} - \frac{85849202399755553356641485912870433186407466609248835311237589457740756601069020823894175017304567889}{381953768254400381041727200000157435523185865627570580108781280506182720229409788903886348611316137573}$, $\frac{1}{50678771833298605757759490077620889017522870209062947280573426641401841868198779021134352392795259081598359} a^{33} + \frac{269566616648193704948803504004322575720142012607406911529695682442308875773626979135641380901316832}{50678771833298605757759490077620889017522870209062947280573426641401841868198779021134352392795259081598359} a^{31} + \frac{34317143247863026012389219594215955886377639308310773801174098987396635476664131866661630877413352999501}{50678771833298605757759490077620889017522870209062947280573426641401841868198779021134352392795259081598359} a^{29} + \frac{150717562181101535206411803696664682798445474154133772989033350110102186060252914359096059852412290617395}{50678771833298605757759490077620889017522870209062947280573426641401841868198779021134352392795259081598359} a^{27} + \frac{97116053956104640979126796059779551223749762438272387327847788886210263587842086897158208530315073835642}{221304680494753736933447554924108685666038734537392782884600116337999309468116938956918569400852659744971} a^{25} - \frac{7733316091100829042284731339906083142497371713634318006867210485770455363360798211230357563014167311652031}{50678771833298605757759490077620889017522870209062947280573426641401841868198779021134352392795259081598359} a^{23} - \frac{7790080343643005114774062425446801484969393255251983946235544642612409795257300653355170776620105782831461}{50678771833298605757759490077620889017522870209062947280573426641401841868198779021134352392795259081598359} a^{21} - \frac{22306182034114323065676762268603084981323543554006723939518807410581922176940817894830103952764164890368036}{50678771833298605757759490077620889017522870209062947280573426641401841868198779021134352392795259081598359} a^{19} - \frac{21458274696856247421648447057300693659523025571325808169045658491487560987886769966100171514911629656794686}{50678771833298605757759490077620889017522870209062947280573426641401841868198779021134352392795259081598359} a^{17} + \frac{18925695036972958160834466263550481355795659565817611154834314423786345001624618812326290092753712856464844}{50678771833298605757759490077620889017522870209062947280573426641401841868198779021134352392795259081598359} a^{15} + \frac{18470231878339541819295263764106708512746842018166797463290394638908826521423946627658635890497361156823508}{50678771833298605757759490077620889017522870209062947280573426641401841868198779021134352392795259081598359} a^{13} - \frac{10149067217551555947499166023289741962060905095952027865661391836877045609305139733844617710758319956906756}{50678771833298605757759490077620889017522870209062947280573426641401841868198779021134352392795259081598359} a^{11} + \frac{19485233002131088416790558015840669120282051366714478808929921097610953617713362486245123278707267457883412}{50678771833298605757759490077620889017522870209062947280573426641401841868198779021134352392795259081598359} a^{9} + \frac{20220835485689543257019101358211163435693544470934790701741272040515759502016057620768293612203872164927018}{50678771833298605757759490077620889017522870209062947280573426641401841868198779021134352392795259081598359} a^{7} + \frac{1221598140593093067449761064023068134119700743522398037556842527097909256295240865687661607681750117993015}{50678771833298605757759490077620889017522870209062947280573426641401841868198779021134352392795259081598359} a^{5} + \frac{1204269563401474748942140442300109500839705033669575046780496984738661702781357712668762690244052748503174}{50678771833298605757759490077620889017522870209062947280573426641401841868198779021134352392795259081598359} a^{3} + \frac{6225221906427501362327694562492444493685333260633482414774890807846019856377729847610003361023790470391516}{50678771833298605757759490077620889017522870209062947280573426641401841868198779021134352392795259081598359} a$

Class group and class number

Not computed

Unit group

Rank:		$33$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		Not computed
Regulator:		Not computed

Galois group

$C_{34}$ (as 34T1):

A cyclic group of order 34

The 34 conjugacy class representatives for $C_{34}$

Character table for $C_{34}$ is not computed

Intermediate fields

\(\Q(\sqrt{239}) \), 17.17.113335617496346216833223278514633468161.1

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	R	$34$	$17^{2}$	$17^{2}$	$34$	$34$	$17^{2}$	$17^{2}$	$17^{2}$	$17^{2}$	$34$	$34$	$34$	$17^{2}$	$17^{2}$	$34$	$17^{2}$

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
2	Data not computed
239	Data not computed