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Properties

Label 34.34.1809456815...3125.1
Degree $34$
Signature $[34, 0]$
Discriminant $5^{17}\cdot 137^{32}$
Root discriminant $229.36$
Ramified primes $5, 137$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{34}$ (as 34T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25802981, -862099703, -11421005318, -27416557293, 60848439526, 205690825367, -149394480868, -617008992418, 251420356724, 1031937684646, -344727618637, -1070683452806, 370896183799, 717036544368, -283122055990, -312514932409, 145098023403, 87535918282, -49173609150, -15074113129, 11012856773, 1363075041, -1623065077, -6829935, 154470461, -12475837, -9059463, 1365891, 292815, -67889, -3513, 1634, -41, -15, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^34 - 15*x^33 - 41*x^32 + 1634*x^31 - 3513*x^30 - 67889*x^29 + 292815*x^28 + 1365891*x^27 - 9059463*x^26 - 12475837*x^25 + 154470461*x^24 - 6829935*x^23 - 1623065077*x^22 + 1363075041*x^21 + 11012856773*x^20 - 15074113129*x^19 - 49173609150*x^18 + 87535918282*x^17 + 145098023403*x^16 - 312514932409*x^15 - 283122055990*x^14 + 717036544368*x^13 + 370896183799*x^12 - 1070683452806*x^11 - 344727618637*x^10 + 1031937684646*x^9 + 251420356724*x^8 - 617008992418*x^7 - 149394480868*x^6 + 205690825367*x^5 + 60848439526*x^4 - 27416557293*x^3 - 11421005318*x^2 - 862099703*x + 25802981)
 
gp: K = bnfinit(x^34 - 15*x^33 - 41*x^32 + 1634*x^31 - 3513*x^30 - 67889*x^29 + 292815*x^28 + 1365891*x^27 - 9059463*x^26 - 12475837*x^25 + 154470461*x^24 - 6829935*x^23 - 1623065077*x^22 + 1363075041*x^21 + 11012856773*x^20 - 15074113129*x^19 - 49173609150*x^18 + 87535918282*x^17 + 145098023403*x^16 - 312514932409*x^15 - 283122055990*x^14 + 717036544368*x^13 + 370896183799*x^12 - 1070683452806*x^11 - 344727618637*x^10 + 1031937684646*x^9 + 251420356724*x^8 - 617008992418*x^7 - 149394480868*x^6 + 205690825367*x^5 + 60848439526*x^4 - 27416557293*x^3 - 11421005318*x^2 - 862099703*x + 25802981, 1)
 

Normalized defining polynomial

\( x^{34} - 15 x^{33} - 41 x^{32} + 1634 x^{31} - 3513 x^{30} - 67889 x^{29} + 292815 x^{28} + 1365891 x^{27} - 9059463 x^{26} - 12475837 x^{25} + 154470461 x^{24} - 6829935 x^{23} - 1623065077 x^{22} + 1363075041 x^{21} + 11012856773 x^{20} - 15074113129 x^{19} - 49173609150 x^{18} + 87535918282 x^{17} + 145098023403 x^{16} - 312514932409 x^{15} - 283122055990 x^{14} + 717036544368 x^{13} + 370896183799 x^{12} - 1070683452806 x^{11} - 344727618637 x^{10} + 1031937684646 x^{9} + 251420356724 x^{8} - 617008992418 x^{7} - 149394480868 x^{6} + 205690825367 x^{5} + 60848439526 x^{4} - 27416557293 x^{3} - 11421005318 x^{2} - 862099703 x + 25802981 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $34$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[34, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(180945681559339099999375313948611491462714365067909795752956331915039825439453125=5^{17}\cdot 137^{32}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $229.36$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $5, 137$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(685=5\cdot 137\)
Dirichlet character group:    $\lbrace$$\chi_{685}(256,·)$, $\chi_{685}(1,·)$, $\chi_{685}(259,·)$, $\chi_{685}(389,·)$, $\chi_{685}(396,·)$, $\chi_{685}(526,·)$, $\chi_{685}(16,·)$, $\chi_{685}(534,·)$, $\chi_{685}(324,·)$, $\chi_{685}(671,·)$, $\chi_{685}(544,·)$, $\chi_{685}(34,·)$, $\chi_{685}(549,·)$, $\chi_{685}(681,·)$, $\chi_{685}(171,·)$, $\chi_{685}(564,·)$, $\chi_{685}(56,·)$, $\chi_{685}(59,·)$, $\chi_{685}(586,·)$, $\chi_{685}(449,·)$, $\chi_{685}(196,·)$, $\chi_{685}(74,·)$, $\chi_{685}(461,·)$, $\chi_{685}(334,·)$, $\chi_{685}(209,·)$, $\chi_{685}(211,·)$, $\chi_{685}(471,·)$, $\chi_{685}(346,·)$, $\chi_{685}(604,·)$, $\chi_{685}(484,·)$, $\chi_{685}(621,·)$, $\chi_{685}(499,·)$, $\chi_{685}(119,·)$, $\chi_{685}(636,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{41} a^{25} - \frac{12}{41} a^{24} + \frac{5}{41} a^{23} + \frac{16}{41} a^{22} + \frac{11}{41} a^{21} + \frac{10}{41} a^{20} + \frac{2}{41} a^{19} - \frac{14}{41} a^{18} + \frac{10}{41} a^{17} - \frac{3}{41} a^{16} + \frac{12}{41} a^{15} + \frac{3}{41} a^{14} + \frac{1}{41} a^{13} - \frac{5}{41} a^{12} - \frac{7}{41} a^{11} - \frac{8}{41} a^{10} - \frac{5}{41} a^{9} + \frac{9}{41} a^{8} + \frac{9}{41} a^{7} + \frac{8}{41} a^{6} + \frac{14}{41} a^{5} - \frac{11}{41} a^{3} - \frac{16}{41} a^{2} + \frac{11}{41} a$, $\frac{1}{41} a^{26} - \frac{16}{41} a^{24} - \frac{6}{41} a^{23} - \frac{2}{41} a^{22} + \frac{19}{41} a^{21} - \frac{1}{41} a^{20} + \frac{10}{41} a^{19} + \frac{6}{41} a^{18} - \frac{6}{41} a^{17} + \frac{17}{41} a^{16} - \frac{17}{41} a^{15} - \frac{4}{41} a^{14} + \frac{7}{41} a^{13} + \frac{15}{41} a^{12} - \frac{10}{41} a^{11} - \frac{19}{41} a^{10} - \frac{10}{41} a^{9} - \frac{6}{41} a^{8} - \frac{7}{41} a^{7} - \frac{13}{41} a^{6} + \frac{4}{41} a^{5} - \frac{11}{41} a^{4} + \frac{16}{41} a^{3} - \frac{17}{41} a^{2} + \frac{9}{41} a$, $\frac{1}{41} a^{27} + \frac{7}{41} a^{24} - \frac{4}{41} a^{23} - \frac{12}{41} a^{22} + \frac{11}{41} a^{21} + \frac{6}{41} a^{20} - \frac{3}{41} a^{19} + \frac{16}{41} a^{18} + \frac{13}{41} a^{17} + \frac{17}{41} a^{16} - \frac{17}{41} a^{15} + \frac{14}{41} a^{14} - \frac{10}{41} a^{13} - \frac{8}{41} a^{12} - \frac{8}{41} a^{11} - \frac{15}{41} a^{10} - \frac{4}{41} a^{9} + \frac{14}{41} a^{8} + \frac{8}{41} a^{7} + \frac{9}{41} a^{6} + \frac{8}{41} a^{5} + \frac{16}{41} a^{4} + \frac{12}{41} a^{3} - \frac{1}{41} a^{2} + \frac{12}{41} a$, $\frac{1}{1517} a^{28} - \frac{14}{1517} a^{27} + \frac{7}{1517} a^{26} + \frac{17}{1517} a^{25} + \frac{568}{1517} a^{24} - \frac{522}{1517} a^{23} - \frac{372}{1517} a^{22} - \frac{397}{1517} a^{21} - \frac{445}{1517} a^{20} - \frac{344}{1517} a^{19} - \frac{22}{1517} a^{18} + \frac{467}{1517} a^{17} - \frac{2}{1517} a^{16} - \frac{239}{1517} a^{15} - \frac{204}{1517} a^{14} - \frac{752}{1517} a^{13} - \frac{5}{1517} a^{12} - \frac{371}{1517} a^{11} + \frac{362}{1517} a^{10} - \frac{337}{1517} a^{9} - \frac{427}{1517} a^{8} + \frac{717}{1517} a^{7} - \frac{580}{1517} a^{6} - \frac{420}{1517} a^{5} + \frac{695}{1517} a^{4} - \frac{331}{1517} a^{3} + \frac{75}{1517} a^{2} + \frac{415}{1517} a + \frac{1}{37}$, $\frac{1}{1517} a^{29} - \frac{4}{1517} a^{27} + \frac{4}{1517} a^{26} - \frac{8}{1517} a^{25} + \frac{548}{1517} a^{24} + \frac{312}{1517} a^{23} + \frac{611}{1517} a^{22} + \frac{139}{1517} a^{21} + \frac{160}{1517} a^{20} - \frac{546}{1517} a^{19} + \frac{196}{1517} a^{18} - \frac{50}{1517} a^{17} + \frac{399}{1517} a^{16} + \frac{594}{1517} a^{15} + \frac{18}{1517} a^{14} - \frac{321}{1517} a^{13} + \frac{484}{1517} a^{12} + \frac{496}{1517} a^{11} - \frac{42}{1517} a^{10} - \frac{705}{1517} a^{9} - \frac{229}{1517} a^{8} - \frac{162}{1517} a^{7} + \frac{192}{1517} a^{6} - \frac{375}{1517} a^{5} - \frac{73}{1517} a^{4} + \frac{288}{1517} a^{3} - \frac{496}{1517} a^{2} - \frac{365}{1517} a + \frac{14}{37}$, $\frac{1}{192659} a^{30} + \frac{33}{192659} a^{29} + \frac{25}{192659} a^{28} - \frac{1792}{192659} a^{27} - \frac{1042}{192659} a^{26} + \frac{34}{5207} a^{25} - \frac{29105}{192659} a^{24} + \frac{34175}{192659} a^{23} + \frac{68418}{192659} a^{22} - \frac{92902}{192659} a^{21} - \frac{6506}{192659} a^{20} + \frac{36064}{192659} a^{19} + \frac{76894}{192659} a^{18} + \frac{81778}{192659} a^{17} + \frac{13111}{192659} a^{16} + \frac{23678}{192659} a^{15} + \frac{10970}{192659} a^{14} + \frac{4935}{192659} a^{13} + \frac{61093}{192659} a^{12} + \frac{48709}{192659} a^{11} - \frac{38546}{192659} a^{10} + \frac{20975}{192659} a^{9} - \frac{57028}{192659} a^{8} - \frac{70016}{192659} a^{7} + \frac{11600}{192659} a^{6} + \frac{92477}{192659} a^{5} - \frac{76538}{192659} a^{4} + \frac{73927}{192659} a^{3} + \frac{75093}{192659} a^{2} - \frac{17011}{192659} a - \frac{1618}{4699}$, $\frac{1}{7899019} a^{31} + \frac{11}{7899019} a^{30} + \frac{1077}{7899019} a^{29} + \frac{1214}{7899019} a^{28} - \frac{56106}{7899019} a^{27} - \frac{14299}{7899019} a^{26} + \frac{45835}{7899019} a^{25} - \frac{2886468}{7899019} a^{24} - \frac{1900346}{7899019} a^{23} - \frac{340290}{7899019} a^{22} + \frac{510925}{7899019} a^{21} + \frac{2565272}{7899019} a^{20} + \frac{2775224}{7899019} a^{19} - \frac{1997494}{7899019} a^{18} - \frac{3184021}{7899019} a^{17} + \frac{935640}{7899019} a^{16} - \frac{101641}{7899019} a^{15} - \frac{427032}{7899019} a^{14} - \frac{848847}{7899019} a^{13} - \frac{2839657}{7899019} a^{12} + \frac{2338541}{7899019} a^{11} - \frac{29466}{213487} a^{10} - \frac{3360357}{7899019} a^{9} - \frac{336860}{7899019} a^{8} + \frac{660539}{7899019} a^{7} - \frac{3937290}{7899019} a^{6} + \frac{3063837}{7899019} a^{5} + \frac{2731980}{7899019} a^{4} + \frac{786388}{7899019} a^{3} + \frac{8812}{192659} a^{2} + \frac{2788722}{7899019} a + \frac{82840}{192659}$, $\frac{1}{18814829235266292672075562159} a^{32} - \frac{1144880763499126065638}{18814829235266292672075562159} a^{31} + \frac{42669817878293693063910}{18814829235266292672075562159} a^{30} - \frac{384044137686260426875303}{18814829235266292672075562159} a^{29} + \frac{5303758042718435755977126}{18814829235266292672075562159} a^{28} - \frac{99253403450435920880959622}{18814829235266292672075562159} a^{27} - \frac{199510695568419093840733059}{18814829235266292672075562159} a^{26} - \frac{152730812518207124482424318}{18814829235266292672075562159} a^{25} - \frac{4522620702220769248141320392}{18814829235266292672075562159} a^{24} + \frac{398491009375948798097803976}{18814829235266292672075562159} a^{23} - \frac{695287636407998208999542135}{18814829235266292672075562159} a^{22} + \frac{4100071426894902904102971439}{18814829235266292672075562159} a^{21} + \frac{2504489270594805395818643746}{18814829235266292672075562159} a^{20} - \frac{166608732736124831151153925}{508508898250440342488528707} a^{19} + \frac{9261354702739096834204460407}{18814829235266292672075562159} a^{18} + \frac{1252573095938982223026284559}{18814829235266292672075562159} a^{17} + \frac{1752771520788937001307785482}{18814829235266292672075562159} a^{16} + \frac{2983345172708179974583102400}{18814829235266292672075562159} a^{15} + \frac{189074435240109197503136446}{18814829235266292672075562159} a^{14} - \frac{61873370672510548765894948}{458898274030885187123794199} a^{13} - \frac{9163667222705687658365455542}{18814829235266292672075562159} a^{12} - \frac{1411010137084742278545339166}{18814829235266292672075562159} a^{11} + \frac{7627915137350914117277817906}{18814829235266292672075562159} a^{10} + \frac{1157582234394892208912519543}{18814829235266292672075562159} a^{9} - \frac{5549228438627085546475232857}{18814829235266292672075562159} a^{8} - \frac{199224176989817593028514140}{458898274030885187123794199} a^{7} + \frac{8358990212520775282573768057}{18814829235266292672075562159} a^{6} + \frac{26153521040483037195677844}{458898274030885187123794199} a^{5} - \frac{3025476022298289695683044879}{18814829235266292672075562159} a^{4} + \frac{309018891044367209617959063}{18814829235266292672075562159} a^{3} + \frac{5797524326661619632040777039}{18814829235266292672075562159} a^{2} - \frac{3894568902046974400131099661}{18814829235266292672075562159} a + \frac{207700619268513071755459384}{458898274030885187123794199}$, $\frac{1}{2359523158047737748461694770315297379493854565530021560052649200140948521448744319193515594779108409283536723070443508753681} a^{33} - \frac{46283119718226429277215743399076856635451808542856477560108721712872160800244028338319918694241}{2359523158047737748461694770315297379493854565530021560052649200140948521448744319193515594779108409283536723070443508753681} a^{32} - \frac{62403283764417589507739860716829983552492489280124630680329943962104485568729492608551494514779920600047523537800634}{2359523158047737748461694770315297379493854565530021560052649200140948521448744319193515594779108409283536723070443508753681} a^{31} + \frac{3584201597150925646433608054501172539405647786908276584192248164521952322737757092125574562188681780957911712263481711}{2359523158047737748461694770315297379493854565530021560052649200140948521448744319193515594779108409283536723070443508753681} a^{30} - \frac{702649550900354661587967802433315752504023140754365055168243644020181863036341790462369792510431624675038434492010204698}{2359523158047737748461694770315297379493854565530021560052649200140948521448744319193515594779108409283536723070443508753681} a^{29} - \frac{743755593787975716946225904624491244642790749102489424048589885472318264629561898089515674842342184050178336688661916157}{2359523158047737748461694770315297379493854565530021560052649200140948521448744319193515594779108409283536723070443508753681} a^{28} - \frac{6392064570188040379208395561526306178135715591350536767293858798711112276997879851463737594857110610477648916279404139061}{2359523158047737748461694770315297379493854565530021560052649200140948521448744319193515594779108409283536723070443508753681} a^{27} - \frac{14452587350923254986723594769066456539825342099870168150359346163795352768124855086339264988851288703191813179312311467276}{2359523158047737748461694770315297379493854565530021560052649200140948521448744319193515594779108409283536723070443508753681} a^{26} + \frac{9390335481184745373990904525537638162378150661756804112197881498240920945770027492418781928386269441560038530314456988372}{2359523158047737748461694770315297379493854565530021560052649200140948521448744319193515594779108409283536723070443508753681} a^{25} + \frac{922260562383598286812331136163257224486733289055884271801290594930352166954866994389140891463381732007109328221911494898388}{2359523158047737748461694770315297379493854565530021560052649200140948521448744319193515594779108409283536723070443508753681} a^{24} + \frac{89562089684063636655256076471916829909554750989866245481230779944380612862570252842060576433155548246716252033167563710061}{2359523158047737748461694770315297379493854565530021560052649200140948521448744319193515594779108409283536723070443508753681} a^{23} + \frac{492994329918185149874283314881049798166726489299772854426171758915649266647169166217536110058955285955755661096744357298356}{2359523158047737748461694770315297379493854565530021560052649200140948521448744319193515594779108409283536723070443508753681} a^{22} - \frac{569945160248614586971583538715643885762959028172521945864433262828869476254746806066022882989704961203712392286296787495742}{2359523158047737748461694770315297379493854565530021560052649200140948521448744319193515594779108409283536723070443508753681} a^{21} + \frac{72575669111631044154354073557710434301583080230097211278220551711331651151162848809360700867085973450821637917414096436732}{2359523158047737748461694770315297379493854565530021560052649200140948521448744319193515594779108409283536723070443508753681} a^{20} - \frac{814732046852137786637184417567710104458858659900217691966071508237704590394633057508125437228442260178188702974687668130017}{2359523158047737748461694770315297379493854565530021560052649200140948521448744319193515594779108409283536723070443508753681} a^{19} - \frac{761574079363349224331518215636214025958968421150840418305240785903942211445512311230265274824563907645843576859243192288955}{2359523158047737748461694770315297379493854565530021560052649200140948521448744319193515594779108409283536723070443508753681} a^{18} + \frac{28520629315156842613508808688089895929737942320805429691161602195765646365512108050264883766095632129369476029393267826423}{63770896163452371580045804603116145391725799068378961082504032436241851931047143761986907967002929980636127650552527263613} a^{17} - \frac{710177405290932620917299632457025380525521780132189047553367630249487003604079447807378607892746113711962248551478407028488}{2359523158047737748461694770315297379493854565530021560052649200140948521448744319193515594779108409283536723070443508753681} a^{16} + \frac{322951182625548964059135440999281282392715458163854077467331126930083172116629324618218429776334363486685240538109851385350}{2359523158047737748461694770315297379493854565530021560052649200140948521448744319193515594779108409283536723070443508753681} a^{15} - \frac{271912914494383993725114832884624531305287668504552522531479271016604091429634281525636784804725159730492998021179211448469}{2359523158047737748461694770315297379493854565530021560052649200140948521448744319193515594779108409283536723070443508753681} a^{14} - \frac{854956692637380189949609275088701029990616319696815013797093213085958723965077894840913088384947508256246513505985585755055}{2359523158047737748461694770315297379493854565530021560052649200140948521448744319193515594779108409283536723070443508753681} a^{13} - \frac{259293531718901192409953267801443444109371637692022680615856495436030463392531864534090981747936808596269507837995397271953}{2359523158047737748461694770315297379493854565530021560052649200140948521448744319193515594779108409283536723070443508753681} a^{12} + \frac{404794220467243653271290726140655180038405882735488791451734490504291095021284940997814829547071834170921432643998996716389}{2359523158047737748461694770315297379493854565530021560052649200140948521448744319193515594779108409283536723070443508753681} a^{11} + \frac{454697903253688933315406216680288332870862226385354576734740493119327601158351748887792314954744900194242838164593742755404}{2359523158047737748461694770315297379493854565530021560052649200140948521448744319193515594779108409283536723070443508753681} a^{10} + \frac{893512888692075930272804698065438455328058661460401514348362431092800136159677337850053975457937034675677969767175823728050}{2359523158047737748461694770315297379493854565530021560052649200140948521448744319193515594779108409283536723070443508753681} a^{9} - \frac{406519501178813525126820025697087882354155297923796671245548090679914993294649434534122688759833570770151469398994724050971}{2359523158047737748461694770315297379493854565530021560052649200140948521448744319193515594779108409283536723070443508753681} a^{8} + \frac{478267686777679703832757917135868538883361634514723055547146240468207212242830563868637308796270630757523016990851726656018}{2359523158047737748461694770315297379493854565530021560052649200140948521448744319193515594779108409283536723070443508753681} a^{7} - \frac{1070690067630496349294105133936545163744024513355595263208232647083630889861165571977288132425054085154693681407975242227404}{2359523158047737748461694770315297379493854565530021560052649200140948521448744319193515594779108409283536723070443508753681} a^{6} + \frac{376517253099024758925122062181223071167934457199930619933977390053617896816187730717334140534123745714012988564942409763270}{2359523158047737748461694770315297379493854565530021560052649200140948521448744319193515594779108409283536723070443508753681} a^{5} + \frac{4861202900461025106652589051806961982851999823969393328855037144162408857380009024305449283401303940088571488095319801587}{10126708832822908791681093434829602487098088264077345751298923605755143868878730983663157059137804331688998811461130938857} a^{4} - \frac{956354292211112942304844733769566654535785396384306047132411043941037535276137479740395793453566494188658046911091405117543}{2359523158047737748461694770315297379493854565530021560052649200140948521448744319193515594779108409283536723070443508753681} a^{3} - \frac{344537549236595687166124760214841116706282516945918637498835273660280063004339524112265451468572195075880502804883785894256}{2359523158047737748461694770315297379493854565530021560052649200140948521448744319193515594779108409283536723070443508753681} a^{2} + \frac{548840383443253558776073088345270219781761497357845039705659360209331687828753280369828840901342716657244091176847671057295}{2359523158047737748461694770315297379493854565530021560052649200140948521448744319193515594779108409283536723070443508753681} a + \frac{13989154037156123348392033553837003921301824218394543751280547595168710804951869087356618449657974356804089003391542768999}{57549345318237506060041335861348716573020843061707842928113395125388988328018154126671112067783131933744798123669353872041}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $33$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 382425771471362240000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 34
The 34 conjugacy class representatives for $C_{34}$
Character table for $C_{34}$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 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 $34$ $34$ R $34$ $17^{2}$ $34$ $34$ $17^{2}$ $34$ $17^{2}$ $17^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{17}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{34}$ $34$ $34$ $34$ $17^{2}$

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

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
5Data not computed
137Data not computed