Global number field 34.0.2652335238355663972863781109929452800183143879582476922663961790823.1

• Label: 34.0.26523352383...0823.1
• Degree: $34$
• Signature: $[0, 17]$
• Discriminant: $-\,103^{33}$
• Root discriminant: $89.87$
• Ramified prime: $103$
• Class number: $5105$ (GRH)
• Class group: $[5105]$ (GRH)
• Galois group: $C_{34}$ (as 34T1)

Normalized defining polynomial

\( x^{34} - x^{33} + 2 x^{32} + 64 x^{31} - 57 x^{30} + 107 x^{29} + 1603 x^{28} - 1261 x^{27} + 2206 x^{26} + 20286 x^{25} - 13891 x^{24} + 22349 x^{23} + 140429 x^{22} - 82189 x^{21} + 120996 x^{20} + 538574 x^{19} - 264119 x^{18} + 327327 x^{17} + 1143572 x^{16} - 499882 x^{15} + 128724 x^{14} + 1425880 x^{13} - 737516 x^{12} - 1133952 x^{11} + 1294431 x^{10} - 1061173 x^{9} - 1305757 x^{8} + 907693 x^{7} - 16051 x^{6} - 713165 x^{5} + 590348 x^{4} + 474529 x^{3} + 129602 x^{2} - 93135 x + 56857 \)

Invariants

Degree:		$34$
Signature:		$[0, 17]$
Discriminant:		\(-2652335238355663972863781109929452800183143879582476922663961790823=-\,103^{33}\)
Root discriminant:		$89.87$
Ramified primes:		$103$
This field is Galois and abelian over $\Q$.
Conductor:		\(103\)
Dirichlet character group:		$\lbrace$$\chi_{103}(1,·)$, $\chi_{103}(3,·)$, $\chi_{103}(8,·)$, $\chi_{103}(9,·)$, $\chi_{103}(10,·)$, $\chi_{103}(13,·)$, $\chi_{103}(14,·)$, $\chi_{103}(22,·)$, $\chi_{103}(23,·)$, $\chi_{103}(24,·)$, $\chi_{103}(27,·)$, $\chi_{103}(30,·)$, $\chi_{103}(31,·)$, $\chi_{103}(34,·)$, $\chi_{103}(37,·)$, $\chi_{103}(39,·)$, $\chi_{103}(42,·)$, $\chi_{103}(61,·)$, $\chi_{103}(64,·)$, $\chi_{103}(66,·)$, $\chi_{103}(69,·)$, $\chi_{103}(72,·)$, $\chi_{103}(73,·)$, $\chi_{103}(76,·)$, $\chi_{103}(79,·)$, $\chi_{103}(80,·)$, $\chi_{103}(81,·)$, $\chi_{103}(89,·)$, $\chi_{103}(90,·)$, $\chi_{103}(93,·)$, $\chi_{103}(94,·)$, $\chi_{103}(95,·)$, $\chi_{103}(100,·)$, $\chi_{103}(102,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $\frac{1}{149} a^{30} + \frac{34}{149} a^{29} + \frac{52}{149} a^{28} - \frac{30}{149} a^{27} - \frac{69}{149} a^{26} + \frac{41}{149} a^{25} - \frac{40}{149} a^{23} - \frac{33}{149} a^{22} - \frac{43}{149} a^{21} - \frac{58}{149} a^{20} - \frac{64}{149} a^{19} - \frac{64}{149} a^{18} - \frac{21}{149} a^{16} - \frac{36}{149} a^{15} + \frac{26}{149} a^{14} + \frac{46}{149} a^{13} + \frac{49}{149} a^{12} - \frac{13}{149} a^{11} + \frac{62}{149} a^{10} + \frac{21}{149} a^{9} + \frac{56}{149} a^{8} - \frac{19}{149} a^{7} + \frac{2}{149} a^{6} + \frac{26}{149} a^{5} + \frac{34}{149} a^{4} + \frac{61}{149} a^{3} - \frac{2}{149} a^{2} + \frac{69}{149} a + \frac{61}{149}$, $\frac{1}{149} a^{31} - \frac{61}{149} a^{29} - \frac{10}{149} a^{28} + \frac{57}{149} a^{27} + \frac{3}{149} a^{26} - \frac{53}{149} a^{25} - \frac{40}{149} a^{24} - \frac{14}{149} a^{23} + \frac{36}{149} a^{22} + \frac{63}{149} a^{21} - \frac{29}{149} a^{20} + \frac{26}{149} a^{19} - \frac{59}{149} a^{18} - \frac{21}{149} a^{17} - \frac{67}{149} a^{16} + \frac{58}{149} a^{15} + \frac{56}{149} a^{14} - \frac{25}{149} a^{13} - \frac{40}{149} a^{12} + \frac{57}{149} a^{11} - \frac{1}{149} a^{10} - \frac{62}{149} a^{9} + \frac{14}{149} a^{8} + \frac{52}{149} a^{7} - \frac{42}{149} a^{6} + \frac{44}{149} a^{5} - \frac{52}{149} a^{4} + \frac{10}{149} a^{3} - \frac{12}{149} a^{2} - \frac{50}{149} a + \frac{12}{149}$, $\frac{1}{92231} a^{32} - \frac{309}{92231} a^{31} + \frac{136}{92231} a^{30} + \frac{10935}{92231} a^{29} + \frac{44681}{92231} a^{28} - \frac{41698}{92231} a^{27} + \frac{6883}{92231} a^{26} + \frac{36185}{92231} a^{25} + \frac{44679}{92231} a^{24} - \frac{33914}{92231} a^{23} - \frac{35889}{92231} a^{22} + \frac{23885}{92231} a^{21} + \frac{37940}{92231} a^{20} - \frac{586}{92231} a^{19} - \frac{41929}{92231} a^{18} + \frac{3293}{92231} a^{17} + \frac{28246}{92231} a^{16} + \frac{44178}{92231} a^{15} + \frac{1501}{92231} a^{14} - \frac{28400}{92231} a^{13} - \frac{35593}{92231} a^{12} - \frac{4828}{92231} a^{11} - \frac{39242}{92231} a^{10} + \frac{38805}{92231} a^{9} + \frac{30151}{92231} a^{8} + \frac{32744}{92231} a^{7} + \frac{9989}{92231} a^{6} - \frac{1374}{92231} a^{5} - \frac{33248}{92231} a^{4} + \frac{14875}{92231} a^{3} - \frac{29665}{92231} a^{2} - \frac{141}{619} a + \frac{9799}{92231}$, $\frac{1}{6752650066678406903506955757204780092587011487009140509586392009691898835763639653622203262014937290251963} a^{33} + \frac{33994232572275801577207751496766017462846425487364708741863437531840922450626734513766081714548560583}{6752650066678406903506955757204780092587011487009140509586392009691898835763639653622203262014937290251963} a^{32} + \frac{488840384612849590640424128915080919605844087599393164351197859952259790849228837896787083312293056320}{6752650066678406903506955757204780092587011487009140509586392009691898835763639653622203262014937290251963} a^{31} + \frac{6037054662546263378673811621887881812582148095062289817654847478691447668167377444148097653911307455656}{6752650066678406903506955757204780092587011487009140509586392009691898835763639653622203262014937290251963} a^{30} + \frac{1902166207645474860802993146912374978251450840811506054009196674056514981287536960090520684187827966664307}{6752650066678406903506955757204780092587011487009140509586392009691898835763639653622203262014937290251963} a^{29} + \frac{453895460271242661934050179605123160180156999295354853067514139500749320392448327700190009702294126223598}{6752650066678406903506955757204780092587011487009140509586392009691898835763639653622203262014937290251963} a^{28} - \frac{2262171654488820719483070010048526123705299978573149498474770416416612164040415872942489547184540962361638}{6752650066678406903506955757204780092587011487009140509586392009691898835763639653622203262014937290251963} a^{27} - \frac{39694669681549749719261567246526512605280572043683244854387454297616558609421761727953349620108208944803}{6752650066678406903506955757204780092587011487009140509586392009691898835763639653622203262014937290251963} a^{26} + \frac{3121395594335938669926497027262365358073292832979321174121342189126952401016229543113744072949129299971134}{6752650066678406903506955757204780092587011487009140509586392009691898835763639653622203262014937290251963} a^{25} + \frac{2744424510378572181217284157740932564948435413082805059237803411256654476566733591163500470613352388575558}{6752650066678406903506955757204780092587011487009140509586392009691898835763639653622203262014937290251963} a^{24} - \frac{473728667543398111312504574791933883325484896105743769763899532417354506994153990331802312096686812493902}{6752650066678406903506955757204780092587011487009140509586392009691898835763639653622203262014937290251963} a^{23} + \frac{2034826951250197634631632695733003385397044645551560393609160039750924004907392062505076833416071049228483}{6752650066678406903506955757204780092587011487009140509586392009691898835763639653622203262014937290251963} a^{22} - \frac{2815958766479045276756979289875996490355893537998939327856984807202328257305213101584891982194328551398489}{6752650066678406903506955757204780092587011487009140509586392009691898835763639653622203262014937290251963} a^{21} - \frac{542442177050689272190974873511843978184644162937942429732937185284991384267208123674523533617595301196403}{6752650066678406903506955757204780092587011487009140509586392009691898835763639653622203262014937290251963} a^{20} - \frac{1404664080400743109710918522637220647040083647883128994583568385140147753280565344166547394972935219428734}{6752650066678406903506955757204780092587011487009140509586392009691898835763639653622203262014937290251963} a^{19} - \frac{2141713834558049352411027988109738591809376411786399498570318548911041488880773314213549226197875032530446}{6752650066678406903506955757204780092587011487009140509586392009691898835763639653622203262014937290251963} a^{18} + \frac{1047422150490468630782046062391579144191441330348174475024514044486382584943665761687368056710309655137220}{6752650066678406903506955757204780092587011487009140509586392009691898835763639653622203262014937290251963} a^{17} + \frac{2172406483709431272492547054373933465110624239430751487953539476303054042045023222084095932699494861574146}{6752650066678406903506955757204780092587011487009140509586392009691898835763639653622203262014937290251963} a^{16} - \frac{3360208457157252904495870161485430499913665229813129336907461612176720632841070678946264154579212041775846}{6752650066678406903506955757204780092587011487009140509586392009691898835763639653622203262014937290251963} a^{15} - \frac{2473324163078103512380494542715263542241680863116321274409507728655789568009208141479022611981176787000898}{6752650066678406903506955757204780092587011487009140509586392009691898835763639653622203262014937290251963} a^{14} - \frac{143313672889977595551528760732741617366977950710917207326266022958465168967287701872977166406538168206790}{6752650066678406903506955757204780092587011487009140509586392009691898835763639653622203262014937290251963} a^{13} - \frac{895041301165365385530195000644460831525034757799301805567361934841275227068897271713498134447877255048388}{6752650066678406903506955757204780092587011487009140509586392009691898835763639653622203262014937290251963} a^{12} - \frac{1803536684108917067599759871988113421771376685761140253167828102669266925958712968277352095355098539035739}{6752650066678406903506955757204780092587011487009140509586392009691898835763639653622203262014937290251963} a^{11} + \frac{3343056686482373405498142167394080064912864824028873915359263890081346894948083452386399510931495499171557}{6752650066678406903506955757204780092587011487009140509586392009691898835763639653622203262014937290251963} a^{10} + \frac{1729791508147249333936102273768248275074951803465329960420201658878414384439836320917175198395308051805323}{6752650066678406903506955757204780092587011487009140509586392009691898835763639653622203262014937290251963} a^{9} + \frac{1970087247177807499669489608560894891896129281385866614435190159332807189145849404873033484103208696258454}{6752650066678406903506955757204780092587011487009140509586392009691898835763639653622203262014937290251963} a^{8} + \frac{1626602346706880180448360096993652144026163140824313230050115691598540197410927148648082249480229808316173}{6752650066678406903506955757204780092587011487009140509586392009691898835763639653622203262014937290251963} a^{7} + \frac{2604838560248154587399950613744963110237865163290762440946252668661959954364143764815086959291455290546385}{6752650066678406903506955757204780092587011487009140509586392009691898835763639653622203262014937290251963} a^{6} - \frac{2878402221795916507245913784768838468914868260218190321055534469170682623293975740185455736109379139319917}{6752650066678406903506955757204780092587011487009140509586392009691898835763639653622203262014937290251963} a^{5} - \frac{339159819026687893202623241113574328762754185643989837139738376645007961546939474162013819206965390127023}{6752650066678406903506955757204780092587011487009140509586392009691898835763639653622203262014937290251963} a^{4} - \frac{449775789685154168062235794001290575651787635125002443057976638034751405931701575791763049649932154321181}{6752650066678406903506955757204780092587011487009140509586392009691898835763639653622203262014937290251963} a^{3} - \frac{204904182549550543073962007791012110965962601151745785816362891168129837610268785108339501650861129858760}{6752650066678406903506955757204780092587011487009140509586392009691898835763639653622203262014937290251963} a^{2} - \frac{629333261795607754991084766757849840029512049093655154714069494108237886257758722213230850849622458169762}{6752650066678406903506955757204780092587011487009140509586392009691898835763639653622203262014937290251963} a - \frac{6797541791940109243948010122815816578468045287717586790078629118155758136350498880177367525108008205}{118765500583541286095062274780673973171060933341701822283736250764055416848649060865367558295635318259}$

Class group and class number

$C_{5105}$, which has order $5105$ (assuming GRH)

Unit group

Rank:		$16$
Torsion generator:		\( -1 \) (order $2$)
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)
Regulator:		\( 2016418740785133.8 \) (assuming GRH)

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

\(\Q(\sqrt{-103}) \), 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	$17^{2}$	$34$	$34$	$17^{2}$	$34$	$17^{2}$	$17^{2}$	$17^{2}$	$17^{2}$	$17^{2}$	$34$	$34$	$17^{2}$	$34$	${\href{/LocalNumberField/47.2.0.1}{2} }^{17}$	$34$	$17^{2}$

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
103	Data not computed