Global number field 34.0.4074534520224700001793556316626715832180494310341751036114303141235744313442304.1

• Label: 34.0.40745345202...2304.1
• Degree: $34$
• Signature: $[0, 17]$
• Discriminant: $-\,2^{34}\cdot 137^{32}$
• Root discriminant: $205.15$
• Ramified primes: $2, 137$
• Class number: Not computed
• Class group: Not computed
• Galois group: $C_{34}$ (as 34T1)

Normalized defining polynomial

\( x^{34} + 129 x^{32} + 7138 x^{30} + 224913 x^{28} + 4510720 x^{26} + 60883886 x^{24} + 569615128 x^{22} + 3744031929 x^{20} + 17312897409 x^{18} + 55753623955 x^{16} + 122133706853 x^{14} + 174558029891 x^{12} + 152022207918 x^{10} + 72865356108 x^{8} + 17601512191 x^{6} + 2160930954 x^{4} + 127784577 x^{2} + 2825761 \)

Invariants

Degree:		$34$
Signature:		$[0, 17]$
Discriminant:		\(-4074534520224700001793556316626715832180494310341751036114303141235744313442304=-\,2^{34}\cdot 137^{32}\)
Root discriminant:		$205.15$
Ramified primes:		$2, 137$
This field is Galois and abelian over $\Q$.
Conductor:		\(548=2^{2}\cdot 137\)
Dirichlet character group:		$\lbrace$$\chi_{548}(1,·)$, $\chi_{548}(259,·)$, $\chi_{548}(389,·)$, $\chi_{548}(193,·)$, $\chi_{548}(393,·)$, $\chi_{548}(397,·)$, $\chi_{548}(275,·)$, $\chi_{548}(533,·)$, $\chi_{548}(407,·)$, $\chi_{548}(153,·)$, $\chi_{548}(133,·)$, $\chi_{548}(427,·)$, $\chi_{548}(175,·)$, $\chi_{548}(115,·)$, $\chi_{548}(187,·)$, $\chi_{548}(445,·)$, $\chi_{548}(449,·)$, $\chi_{548}(197,·)$, $\chi_{548}(211,·)$, $\chi_{548}(73,·)$, $\chi_{548}(333,·)$, $\chi_{548}(461,·)$, $\chi_{548}(209,·)$, $\chi_{548}(467,·)$, $\chi_{548}(471,·)$, $\chi_{548}(347,·)$, $\chi_{548}(225,·)$, $\chi_{548}(59,·)$, $\chi_{548}(485,·)$, $\chi_{548}(171,·)$, $\chi_{548}(483,·)$, $\chi_{548}(499,·)$, $\chi_{548}(119,·)$, $\chi_{548}(123,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{41} a^{21} + \frac{9}{41} a^{19} + \frac{17}{41} a^{17} + \frac{9}{41} a^{15} - \frac{18}{41} a^{13} + \frac{4}{41} a^{11} - \frac{2}{41} a^{9} + \frac{15}{41} a^{7} + \frac{6}{41} a^{5} + \frac{4}{41} a^{3} - \frac{4}{41} a$, $\frac{1}{1517} a^{22} - \frac{729}{1517} a^{20} + \frac{6}{41} a^{18} - \frac{606}{1517} a^{16} + \frac{720}{1517} a^{14} - \frac{201}{1517} a^{12} + \frac{736}{1517} a^{10} - \frac{272}{1517} a^{8} + \frac{621}{1517} a^{6} - \frac{119}{1517} a^{4} + \frac{734}{1517} a^{2} + \frac{10}{37}$, $\frac{1}{1517} a^{23} + \frac{11}{1517} a^{21} - \frac{19}{41} a^{19} - \frac{162}{1517} a^{17} - \frac{5}{37} a^{15} + \frac{132}{1517} a^{13} + \frac{662}{1517} a^{11} - \frac{235}{1517} a^{9} - \frac{415}{1517} a^{7} - \frac{230}{1517} a^{5} + \frac{660}{1517} a^{3} + \frac{484}{1517} a$, $\frac{1}{1517} a^{24} - \frac{269}{1517} a^{20} + \frac{430}{1517} a^{18} + \frac{393}{1517} a^{16} - \frac{203}{1517} a^{14} - \frac{161}{1517} a^{12} - \frac{746}{1517} a^{10} - \frac{457}{1517} a^{8} + \frac{524}{1517} a^{6} + \frac{452}{1517} a^{4} - \frac{5}{1517} a^{2} + \frac{1}{37}$, $\frac{1}{1517} a^{25} - \frac{10}{1517} a^{21} - \frac{273}{1517} a^{19} + \frac{245}{1517} a^{17} + \frac{611}{1517} a^{15} - \frac{272}{1517} a^{13} + \frac{290}{1517} a^{11} + \frac{542}{1517} a^{9} - \frac{142}{1517} a^{7} + \frac{489}{1517} a^{5} - \frac{486}{1517} a^{3} + \frac{522}{1517} a$, $\frac{1}{1517} a^{26} + \frac{22}{1517} a^{20} - \frac{569}{1517} a^{18} + \frac{619}{1517} a^{16} - \frac{657}{1517} a^{14} - \frac{203}{1517} a^{12} + \frac{317}{1517} a^{10} + \frac{172}{1517} a^{8} + \frac{631}{1517} a^{6} - \frac{159}{1517} a^{4} + \frac{277}{1517} a^{2} - \frac{11}{37}$, $\frac{1}{1517} a^{27} - \frac{15}{1517} a^{21} + \frac{15}{37} a^{19} - \frac{10}{1517} a^{17} + \frac{527}{1517} a^{15} + \frac{463}{1517} a^{13} + \frac{169}{1517} a^{11} + \frac{6}{37} a^{9} + \frac{76}{1517} a^{7} - \frac{381}{1517} a^{5} + \frac{129}{1517} a^{3} - \frac{303}{1517} a$, $\frac{1}{1517} a^{28} + \frac{299}{1517} a^{20} + \frac{286}{1517} a^{18} + \frac{539}{1517} a^{16} + \frac{644}{1517} a^{14} + \frac{188}{1517} a^{12} + \frac{667}{1517} a^{10} + \frac{547}{1517} a^{8} - \frac{168}{1517} a^{6} - \frac{139}{1517} a^{4} + \frac{88}{1517} a^{2} + \frac{2}{37}$, $\frac{1}{1517} a^{29} + \frac{3}{1517} a^{21} + \frac{16}{37} a^{19} + \frac{58}{1517} a^{17} - \frac{503}{1517} a^{15} - \frac{552}{1517} a^{13} - \frac{517}{1517} a^{11} - \frac{378}{1517} a^{9} - \frac{57}{1517} a^{7} - \frac{398}{1517} a^{5} + \frac{421}{1517} a^{3} - \frac{251}{1517} a$, $\frac{1}{7899019} a^{30} - \frac{876}{7899019} a^{28} - \frac{214}{7899019} a^{26} + \frac{791}{7899019} a^{24} + \frac{427}{7899019} a^{22} + \frac{1255548}{7899019} a^{20} + \frac{84026}{213487} a^{18} + \frac{2960641}{7899019} a^{16} + \frac{1035848}{7899019} a^{14} + \frac{2432670}{7899019} a^{12} + \frac{2117123}{7899019} a^{10} - \frac{3033217}{7899019} a^{8} + \frac{2980194}{7899019} a^{6} + \frac{2615910}{7899019} a^{4} + \frac{9655}{213487} a^{2} - \frac{747}{4699}$, $\frac{1}{7899019} a^{31} - \frac{876}{7899019} a^{29} - \frac{214}{7899019} a^{27} + \frac{791}{7899019} a^{25} + \frac{427}{7899019} a^{23} - \frac{93065}{7899019} a^{21} - \frac{30528}{213487} a^{19} + \frac{3731277}{7899019} a^{17} - \frac{3202650}{7899019} a^{15} + \frac{3010647}{7899019} a^{13} - \frac{3277329}{7899019} a^{11} - \frac{335991}{7899019} a^{9} - \frac{1450963}{7899019} a^{7} + \frac{2423251}{7899019} a^{5} + \frac{77346}{213487} a^{3} - \frac{91714}{192659} a$, $\frac{1}{55684500633153294848851025854302819771832104008321649251} a^{32} - \frac{2927803565157543530401001803095097229043510385748}{55684500633153294848851025854302819771832104008321649251} a^{30} - \frac{845764547446496280799097208762174990037142959891037}{55684500633153294848851025854302819771832104008321649251} a^{28} + \frac{4707118211775679565333825124917624778575376438757298}{55684500633153294848851025854302819771832104008321649251} a^{26} - \frac{12851322087117530781564161753027075049504852528785487}{55684500633153294848851025854302819771832104008321649251} a^{24} - \frac{155797541883094264077492009882965237780235609200783}{1504986503598737698617595293359535669508975784008693223} a^{22} + \frac{27525987922499966976565832686843642567358348164530448824}{55684500633153294848851025854302819771832104008321649251} a^{20} - \frac{14727557495908655665194287510624932826272402678228253925}{55684500633153294848851025854302819771832104008321649251} a^{18} - \frac{11202976731109457355620299762618868154519796799570800317}{55684500633153294848851025854302819771832104008321649251} a^{16} + \frac{6699078047630430588740889134719318199825350440674666401}{55684500633153294848851025854302819771832104008321649251} a^{14} - \frac{1148289600392906325122611026207774567787558130668793337}{55684500633153294848851025854302819771832104008321649251} a^{12} + \frac{21797511922437671919414281226759047941328570953217256402}{55684500633153294848851025854302819771832104008321649251} a^{10} + \frac{7018019087938216151931141193759361900211233136889112219}{55684500633153294848851025854302819771832104008321649251} a^{8} + \frac{7240577585423668979467964273419726463365134898243630520}{55684500633153294848851025854302819771832104008321649251} a^{6} - \frac{2790894032250271458396919859066338193279959620006880322}{55684500633153294848851025854302819771832104008321649251} a^{4} - \frac{18587641030048413311229350488923722623664331636912920364}{55684500633153294848851025854302819771832104008321649251} a^{2} + \frac{13826452354720891878291316318058016018920596295353407}{33125818342149491284265928527247364528157111248257971}$, $\frac{1}{2283064525959285088802892060026415610645116264341187619291} a^{33} + \frac{138063123619986066958068581202852400230659027380832}{2283064525959285088802892060026415610645116264341187619291} a^{31} - \frac{564837671473457966956775123149153354328339234533764353}{2283064525959285088802892060026415610645116264341187619291} a^{29} + \frac{121363011364763502883493558104038584176733256186825590}{2283064525959285088802892060026415610645116264341187619291} a^{27} + \frac{649277319706113149974911117438653859895623190457515838}{2283064525959285088802892060026415610645116264341187619291} a^{25} - \frac{13409897338686493268983419769083962076677812689919888}{61704446647548245643321407027740962449868007144356422143} a^{23} - \frac{4756380395949839686674924253987242246745584592046510642}{2283064525959285088802892060026415610645116264341187619291} a^{21} + \frac{50187689635519583934142086177868772577995156328940271216}{2283064525959285088802892060026415610645116264341187619291} a^{19} + \frac{921109462091472860366743730546331727798959270225725105244}{2283064525959285088802892060026415610645116264341187619291} a^{17} - \frac{457179062833800872243022367673639507343271856855192549207}{2283064525959285088802892060026415610645116264341187619291} a^{15} + \frac{752403768814404575212540820104538974405748335006670392318}{2283064525959285088802892060026415610645116264341187619291} a^{13} + \frac{871888553720314160600438481753297253696227828434252236523}{2283064525959285088802892060026415610645116264341187619291} a^{11} + \frac{264130801041624122247932756515797041124568662364112040824}{2283064525959285088802892060026415610645116264341187619291} a^{9} + \frac{206554946686937625788662307165750663599602775042776134289}{2283064525959285088802892060026415610645116264341187619291} a^{7} - \frac{5592863124571504487078471269976735189295612034057231294}{2283064525959285088802892060026415610645116264341187619291} a^{5} - \frac{595946947109307864497880848558462836277744931712323904467}{2283064525959285088802892060026415610645116264341187619291} a^{3} + \frac{554907333613146525849891035019982727020021025482157473}{1358158552028129142654903069617141945654441561178576811} a$

Class group and class number

Not computed

Unit group

Rank:		$16$
Torsion generator:		\( \frac{351698461005093867635640633865}{7547225783741208839494601563648153} a^{33} + \frac{45326710339686136569410314554907}{7547225783741208839494601563648153} a^{31} + \frac{2504960414991987091671692577952602}{7547225783741208839494601563648153} a^{29} + \frac{78799645153584267727630767906492674}{7547225783741208839494601563648153} a^{27} + \frac{1576916361018072393180171343393728243}{7547225783741208839494601563648153} a^{25} + \frac{573587294854233007647852861728139117}{203979075236248887553908150368869} a^{23} + \frac{197775325622847612116000209946161919533}{7547225783741208839494601563648153} a^{21} + \frac{1292939488059702145284780375046478190670}{7547225783741208839494601563648153} a^{19} + \frac{5933144374668455342937634781796678936369}{7547225783741208839494601563648153} a^{17} + \frac{18893724514256224266272620880715269457612}{7547225783741208839494601563648153} a^{15} + \frac{40678553594685382123375828846253539994524}{7547225783741208839494601563648153} a^{13} + \frac{56493114304815954335656765711883435654690}{7547225783741208839494601563648153} a^{11} + \frac{46664308347284656904248507411921064622365}{7547225783741208839494601563648153} a^{9} + \frac{20009680956490569963383110155101667513908}{7547225783741208839494601563648153} a^{7} + \frac{3782518471865652693931225125628796682247}{7547225783741208839494601563648153} a^{5} + \frac{305023500716804528993523753895742867880}{7547225783741208839494601563648153} a^{3} + \frac{4937835720308113405912038947929541}{4489723845176209898569066962313} a \) (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

\(\Q(\sqrt{-1}) \), 17.17.15400296222263289476715621650663041.1

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	$34$	$17^{2}$	$34$	$34$	$17^{2}$	$17^{2}$	$34$	$34$	$17^{2}$	$34$	${\href{/LocalNumberField/37.1.0.1}{1} }^{34}$	${\href{/LocalNumberField/41.1.0.1}{1} }^{34}$	$34$	$34$	$17^{2}$	$34$

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
137	Data not computed