Global number field 34.0.442395848806444333196713449710663325979115564010458272524910178228286521344.1

• Label: 34.0.44239584880...1344.1
• Degree: $34$
• Signature: $[0, 17]$
• Discriminant: $-\,2^{34}\cdot 103^{32}$
• Root discriminant: $156.84$
• Ramified primes: $2, 103$
• Class number: Not computed
• Class group: Not computed
• Galois group: $C_{34}$ (as 34T1)

Normalized defining polynomial

\( x^{34} + 97 x^{32} + 4040 x^{30} + 96737 x^{28} + 1499007 x^{26} + 15996211 x^{24} + 121823703 x^{22} + 675554879 x^{20} + 2752450407 x^{18} + 8237696660 x^{16} + 17941689717 x^{14} + 27884282812 x^{12} + 29930432911 x^{10} + 21101216892 x^{8} + 9069664015 x^{6} + 2134903778 x^{4} + 234085541 x^{2} + 8048569 \)

Invariants

Degree:		$34$
Signature:		$[0, 17]$
Discriminant:		\(-442395848806444333196713449710663325979115564010458272524910178228286521344=-\,2^{34}\cdot 103^{32}\)
Root discriminant:		$156.84$
Ramified primes:		$2, 103$
This field is Galois and abelian over $\Q$.
Conductor:		\(412=2^{2}\cdot 103\)
Dirichlet character group:		$\lbrace$$\chi_{412}(1,·)$, $\chi_{412}(133,·)$, $\chi_{412}(385,·)$, $\chi_{412}(9,·)$, $\chi_{412}(267,·)$, $\chi_{412}(13,·)$, $\chi_{412}(23,·)$, $\chi_{412}(409,·)$, $\chi_{412}(111,·)$, $\chi_{412}(285,·)$, $\chi_{412}(287,·)$, $\chi_{412}(219,·)$, $\chi_{412}(167,·)$, $\chi_{412}(169,·)$, $\chi_{412}(299,·)$, $\chi_{412}(175,·)$, $\chi_{412}(179,·)$, $\chi_{412}(137,·)$, $\chi_{412}(61,·)$, $\chi_{412}(215,·)$, $\chi_{412}(117,·)$, $\chi_{412}(323,·)$, $\chi_{412}(203,·)$, $\chi_{412}(207,·)$, $\chi_{412}(81,·)$, $\chi_{412}(339,·)$, $\chi_{412}(343,·)$, $\chi_{412}(79,·)$, $\chi_{412}(93,·)$, $\chi_{412}(229,·)$, $\chi_{412}(317,·)$, $\chi_{412}(373,·)$, $\chi_{412}(375,·)$, $\chi_{412}(381,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $\frac{1}{149} a^{26} + \frac{50}{149} a^{24} - \frac{51}{149} a^{22} - \frac{40}{149} a^{20} - \frac{8}{149} a^{18} - \frac{1}{149} a^{16} - \frac{39}{149} a^{14} + \frac{58}{149} a^{12} - \frac{13}{149} a^{10} + \frac{20}{149} a^{8} - \frac{54}{149} a^{6} + \frac{49}{149} a^{4} + \frac{44}{149} a^{2} + \frac{42}{149}$, $\frac{1}{149} a^{27} + \frac{50}{149} a^{25} - \frac{51}{149} a^{23} - \frac{40}{149} a^{21} - \frac{8}{149} a^{19} - \frac{1}{149} a^{17} - \frac{39}{149} a^{15} + \frac{58}{149} a^{13} - \frac{13}{149} a^{11} + \frac{20}{149} a^{9} - \frac{54}{149} a^{7} + \frac{49}{149} a^{5} + \frac{44}{149} a^{3} + \frac{42}{149} a$, $\frac{1}{149} a^{28} - \frac{18}{149} a^{24} - \frac{23}{149} a^{22} + \frac{55}{149} a^{20} - \frac{48}{149} a^{18} + \frac{11}{149} a^{16} + \frac{71}{149} a^{14} + \frac{67}{149} a^{12} + \frac{74}{149} a^{10} - \frac{11}{149} a^{8} + \frac{67}{149} a^{6} - \frac{22}{149} a^{4} - \frac{72}{149} a^{2} - \frac{14}{149}$, $\frac{1}{149} a^{29} - \frac{18}{149} a^{25} - \frac{23}{149} a^{23} + \frac{55}{149} a^{21} - \frac{48}{149} a^{19} + \frac{11}{149} a^{17} + \frac{71}{149} a^{15} + \frac{67}{149} a^{13} + \frac{74}{149} a^{11} - \frac{11}{149} a^{9} + \frac{67}{149} a^{7} - \frac{22}{149} a^{5} - \frac{72}{149} a^{3} - \frac{14}{149} a$, $\frac{1}{7612261} a^{30} - \frac{19974}{7612261} a^{28} + \frac{1006}{7612261} a^{26} - \frac{754918}{7612261} a^{24} - \frac{2875684}{7612261} a^{22} - \frac{54262}{7612261} a^{20} + \frac{2778503}{7612261} a^{18} + \frac{3085792}{7612261} a^{16} + \frac{2129301}{7612261} a^{14} + \frac{1364766}{7612261} a^{12} + \frac{2298267}{7612261} a^{10} + \frac{3435119}{7612261} a^{8} + \frac{1434444}{7612261} a^{6} - \frac{2516096}{7612261} a^{4} - \frac{2563819}{7612261} a^{2} - \frac{2519233}{7612261}$, $\frac{1}{7612261} a^{31} - \frac{19974}{7612261} a^{29} + \frac{1006}{7612261} a^{27} - \frac{754918}{7612261} a^{25} - \frac{2875684}{7612261} a^{23} - \frac{54262}{7612261} a^{21} + \frac{2778503}{7612261} a^{19} + \frac{3085792}{7612261} a^{17} + \frac{2129301}{7612261} a^{15} + \frac{1364766}{7612261} a^{13} + \frac{2298267}{7612261} a^{11} + \frac{3435119}{7612261} a^{9} + \frac{1434444}{7612261} a^{7} - \frac{2516096}{7612261} a^{5} - \frac{2563819}{7612261} a^{3} - \frac{2519233}{7612261} a$, $\frac{1}{41459914193042653511794766424621173802262050853} a^{32} + \frac{2672555155916283345437810930852973789649}{41459914193042653511794766424621173802262050853} a^{30} + \frac{86622555525057432426725855438335789352256966}{41459914193042653511794766424621173802262050853} a^{28} + \frac{95699460414135503811643629377313122542504222}{41459914193042653511794766424621173802262050853} a^{26} - \frac{6141006957694377450466797609348189401322245293}{41459914193042653511794766424621173802262050853} a^{24} - \frac{293619159496225352547901669335087771949504}{882125833894524542804143966481301570260894699} a^{22} - \frac{16135848828423262998620410436903923708920981730}{41459914193042653511794766424621173802262050853} a^{20} - \frac{6541059315966023407502360003692108970809346296}{41459914193042653511794766424621173802262050853} a^{18} - \frac{4118009070296874051046473154910057955601961070}{41459914193042653511794766424621173802262050853} a^{16} + \frac{2020198268674419258113223362076146620328759450}{41459914193042653511794766424621173802262050853} a^{14} + \frac{17876752021536342044456299033328494808868024358}{41459914193042653511794766424621173802262050853} a^{12} + \frac{9855810794759762830014987691555253581002477801}{41459914193042653511794766424621173802262050853} a^{10} + \frac{8921293199882189823538324208727272216365111295}{41459914193042653511794766424621173802262050853} a^{8} - \frac{18029685186105652119160328419764768304279008811}{41459914193042653511794766424621173802262050853} a^{6} - \frac{15504887792856039256361419327820918623379078992}{41459914193042653511794766424621173802262050853} a^{4} - \frac{1862256263349437143092680506922247436914236670}{41459914193042653511794766424621173802262050853} a^{2} - \frac{4129061530144514594700920428339869105416621725}{41459914193042653511794766424621173802262050853}$, $\frac{1}{117621776565662008012961752346650270077017438269961} a^{33} - \frac{923226519089863137119140360020668465948761}{117621776565662008012961752346650270077017438269961} a^{31} + \frac{221428030306980687859325705030942385441781453419}{117621776565662008012961752346650270077017438269961} a^{29} + \frac{20868092690094203379441672493564186960718952128}{117621776565662008012961752346650270077017438269961} a^{27} - \frac{12694263342656253317945463591769419156691307611280}{117621776565662008012961752346650270077017438269961} a^{25} - \frac{28773496256564256262535165655121901967094165476520}{117621776565662008012961752346650270077017438269961} a^{23} + \frac{9078766433348042511195761970760577965154352292675}{117621776565662008012961752346650270077017438269961} a^{21} - \frac{38111692150530998076138594758044286828823195846032}{117621776565662008012961752346650270077017438269961} a^{19} - \frac{51522946031679861469322139431287989699744923170544}{117621776565662008012961752346650270077017438269961} a^{17} - \frac{15984618150416383396731827468645446467424261602456}{117621776565662008012961752346650270077017438269961} a^{15} + \frac{22268968884956391166801757021821377913647007588374}{117621776565662008012961752346650270077017438269961} a^{13} + \frac{47188025913586692863597881992433199193356054243337}{117621776565662008012961752346650270077017438269961} a^{11} - \frac{23045420339090550614021427691541743758557626660326}{117621776565662008012961752346650270077017438269961} a^{9} - \frac{41249817068129879622604240078511401817005677478530}{117621776565662008012961752346650270077017438269961} a^{7} + \frac{9362887991053246528871546442642991354775751020772}{117621776565662008012961752346650270077017438269961} a^{5} + \frac{27404024648949653279809820468716490300183832436937}{117621776565662008012961752346650270077017438269961} a^{3} + \frac{4877793782164889162328752154901496451999318089719}{117621776565662008012961752346650270077017438269961} a$

Class group and class number

Not computed

Unit group

Rank:		$16$
Torsion generator:		\( \frac{6584604691613314584410232644810856}{175691180390737874904529991916776925161} a^{33} + \frac{618612329626957855436301613128915806}{175691180390737874904529991916776925161} a^{31} + \frac{24713817074185250208824167189890342855}{175691180390737874904529991916776925161} a^{29} + \frac{561540738274339995777521393051402837985}{175691180390737874904529991916776925161} a^{27} + \frac{8156131082120196637394360453692187207549}{175691180390737874904529991916776925161} a^{25} + \frac{80425757876536952948256112821231646257473}{175691180390737874904529991916776925161} a^{23} + \frac{556543964208175475927708689737503525208867}{175691180390737874904529991916776925161} a^{21} + \frac{2748129970372826328650487127736848762555126}{175691180390737874904529991916776925161} a^{19} + \frac{9725901127645103558088469325986608258544041}{175691180390737874904529991916776925161} a^{17} + \frac{24507991166461980786549605223032688635076258}{175691180390737874904529991916776925161} a^{15} + \frac{43170973671177914805721133534382400780860678}{175691180390737874904529991916776925161} a^{13} + \frac{51452192474828329626048270856314689082454663}{175691180390737874904529991916776925161} a^{11} + \frac{39415022973935949991658147747138208085265001}{175691180390737874904529991916776925161} a^{9} + \frac{17989903463441247259896493217492020586460844}{175691180390737874904529991916776925161} a^{7} + \frac{4396569637969126262942417169215696981937497}{175691180390737874904529991916776925161} a^{5} + \frac{490204284213911053895895068871093321232306}{175691180390737874904529991916776925161} a^{3} + \frac{16239318835584517704845353142103350429369}{175691180390737874904529991916776925161} a \) (order $4$)
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

\(\Q(\sqrt{-1}) \), 17.17.160470643909878751793805444097921.1

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	$34$	$17^{2}$	$34$	$34$	$17^{2}$	$17^{2}$	$34$	$34$	$17^{2}$	$34$	$17^{2}$	$17^{2}$	$34$	${\href{/LocalNumberField/47.2.0.1}{2} }^{17}$	$17^{2}$	$34$

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
103	Data not computed