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Properties

Label 34.34.1964632229...3125.1
Degree $34$
Signature $[34, 0]$
Discriminant $5^{17}\cdot 103^{32}$
Root discriminant $175.36$
Ramified primes $5, 103$
Class number Not computed
Class group Not computed
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![354479, 4815433, 10164521, -103423227, -454863738, 217702907, 3558673713, 3013305089, -9723162977, -14175769302, 12172233277, 27308330133, -6257310733, -29464147828, -1432253924, 19723433725, 3841748798, -8604733168, -2357419205, 2530727241, 783518411, -518298014, -160733104, 75969733, 21149768, -8082262, -1770304, 617655, 87991, -32289, -2025, 1030, -9, -15, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^34 - 15*x^33 - 9*x^32 + 1030*x^31 - 2025*x^30 - 32289*x^29 + 87991*x^28 + 617655*x^27 - 1770304*x^26 - 8082262*x^25 + 21149768*x^24 + 75969733*x^23 - 160733104*x^22 - 518298014*x^21 + 783518411*x^20 + 2530727241*x^19 - 2357419205*x^18 - 8604733168*x^17 + 3841748798*x^16 + 19723433725*x^15 - 1432253924*x^14 - 29464147828*x^13 - 6257310733*x^12 + 27308330133*x^11 + 12172233277*x^10 - 14175769302*x^9 - 9723162977*x^8 + 3013305089*x^7 + 3558673713*x^6 + 217702907*x^5 - 454863738*x^4 - 103423227*x^3 + 10164521*x^2 + 4815433*x + 354479)
 
gp: K = bnfinit(x^34 - 15*x^33 - 9*x^32 + 1030*x^31 - 2025*x^30 - 32289*x^29 + 87991*x^28 + 617655*x^27 - 1770304*x^26 - 8082262*x^25 + 21149768*x^24 + 75969733*x^23 - 160733104*x^22 - 518298014*x^21 + 783518411*x^20 + 2530727241*x^19 - 2357419205*x^18 - 8604733168*x^17 + 3841748798*x^16 + 19723433725*x^15 - 1432253924*x^14 - 29464147828*x^13 - 6257310733*x^12 + 27308330133*x^11 + 12172233277*x^10 - 14175769302*x^9 - 9723162977*x^8 + 3013305089*x^7 + 3558673713*x^6 + 217702907*x^5 - 454863738*x^4 - 103423227*x^3 + 10164521*x^2 + 4815433*x + 354479, 1)
 

Normalized defining polynomial

\( x^{34} - 15 x^{33} - 9 x^{32} + 1030 x^{31} - 2025 x^{30} - 32289 x^{29} + 87991 x^{28} + 617655 x^{27} - 1770304 x^{26} - 8082262 x^{25} + 21149768 x^{24} + 75969733 x^{23} - 160733104 x^{22} - 518298014 x^{21} + 783518411 x^{20} + 2530727241 x^{19} - 2357419205 x^{18} - 8604733168 x^{17} + 3841748798 x^{16} + 19723433725 x^{15} - 1432253924 x^{14} - 29464147828 x^{13} - 6257310733 x^{12} + 27308330133 x^{11} + 12172233277 x^{10} - 14175769302 x^{9} - 9723162977 x^{8} + 3013305089 x^{7} + 3558673713 x^{6} + 217702907 x^{5} - 454863738 x^{4} - 103423227 x^{3} + 10164521 x^{2} + 4815433 x + 354479 \)

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:  \(19646322293740163065077262137177497346623569818755784434079377930450439453125=5^{17}\cdot 103^{32}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $175.36$
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:  $5, 103$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(515=5\cdot 103\)
Dirichlet character group:    $\lbrace$$\chi_{515}(1,·)$, $\chi_{515}(9,·)$, $\chi_{515}(14,·)$, $\chi_{515}(409,·)$, $\chi_{515}(34,·)$, $\chi_{515}(164,·)$, $\chi_{515}(421,·)$, $\chi_{515}(169,·)$, $\chi_{515}(426,·)$, $\chi_{515}(299,·)$, $\chi_{515}(306,·)$, $\chi_{515}(179,·)$, $\chi_{515}(184,·)$, $\chi_{515}(61,·)$, $\chi_{515}(446,·)$, $\chi_{515}(64,·)$, $\chi_{515}(66,·)$, $\chi_{515}(196,·)$, $\chi_{515}(76,·)$, $\chi_{515}(79,·)$, $\chi_{515}(81,·)$, $\chi_{515}(339,·)$, $\chi_{515}(214,·)$, $\chi_{515}(219,·)$, $\chi_{515}(476,·)$, $\chi_{515}(484,·)$, $\chi_{515}(229,·)$, $\chi_{515}(104,·)$, $\chi_{515}(491,·)$, $\chi_{515}(236,·)$, $\chi_{515}(111,·)$, $\chi_{515}(116,·)$, $\chi_{515}(381,·)$, $\chi_{515}(126,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $\frac{1}{149} a^{27} - \frac{10}{149} a^{26} + \frac{11}{149} a^{25} - \frac{48}{149} a^{24} - \frac{13}{149} a^{23} - \frac{43}{149} a^{22} + \frac{19}{149} a^{21} - \frac{10}{149} a^{20} - \frac{36}{149} a^{19} + \frac{6}{149} a^{18} - \frac{33}{149} a^{17} + \frac{43}{149} a^{16} + \frac{24}{149} a^{15} - \frac{2}{149} a^{14} - \frac{59}{149} a^{13} + \frac{12}{149} a^{12} + \frac{65}{149} a^{11} - \frac{19}{149} a^{10} - \frac{63}{149} a^{9} + \frac{43}{149} a^{8} - \frac{25}{149} a^{7} - \frac{35}{149} a^{6} + \frac{30}{149} a^{5} + \frac{68}{149} a^{4} + \frac{25}{149} a^{3} + \frac{43}{149} a^{2} + \frac{33}{149} a + \frac{43}{149}$, $\frac{1}{149} a^{28} + \frac{60}{149} a^{26} + \frac{62}{149} a^{25} - \frac{46}{149} a^{24} - \frac{24}{149} a^{23} + \frac{36}{149} a^{22} + \frac{31}{149} a^{21} + \frac{13}{149} a^{20} - \frac{56}{149} a^{19} + \frac{27}{149} a^{18} + \frac{11}{149} a^{17} + \frac{7}{149} a^{16} - \frac{60}{149} a^{15} + \frac{70}{149} a^{14} + \frac{18}{149} a^{13} + \frac{36}{149} a^{12} + \frac{35}{149} a^{11} + \frac{45}{149} a^{10} + \frac{9}{149} a^{9} - \frac{42}{149} a^{8} + \frac{13}{149} a^{7} - \frac{22}{149} a^{6} + \frac{70}{149} a^{5} - \frac{40}{149} a^{4} - \frac{5}{149} a^{3} + \frac{16}{149} a^{2} - \frac{74}{149} a - \frac{17}{149}$, $\frac{1}{149} a^{29} + \frac{66}{149} a^{26} + \frac{39}{149} a^{25} + \frac{25}{149} a^{24} + \frac{71}{149} a^{23} - \frac{71}{149} a^{22} + \frac{65}{149} a^{21} - \frac{52}{149} a^{20} - \frac{48}{149} a^{19} - \frac{51}{149} a^{18} + \frac{50}{149} a^{17} + \frac{42}{149} a^{16} - \frac{29}{149} a^{15} - \frac{11}{149} a^{14} + \frac{60}{149} a^{12} + \frac{19}{149} a^{11} - \frac{43}{149} a^{10} + \frac{13}{149} a^{9} - \frac{34}{149} a^{8} - \frac{12}{149} a^{7} - \frac{65}{149} a^{6} - \frac{52}{149} a^{5} - \frac{62}{149} a^{4} + \frac{6}{149} a^{3} + \frac{28}{149} a^{2} - \frac{60}{149} a - \frac{47}{149}$, $\frac{1}{7003} a^{30} - \frac{2}{7003} a^{29} - \frac{15}{7003} a^{28} - \frac{15}{7003} a^{27} - \frac{2716}{7003} a^{26} - \frac{86}{7003} a^{25} - \frac{1659}{7003} a^{24} + \frac{1051}{7003} a^{23} + \frac{21}{7003} a^{22} - \frac{696}{7003} a^{21} + \frac{820}{7003} a^{20} + \frac{2311}{7003} a^{19} + \frac{2837}{7003} a^{18} + \frac{2450}{7003} a^{17} - \frac{2956}{7003} a^{16} - \frac{550}{7003} a^{15} + \frac{2412}{7003} a^{14} + \frac{56}{149} a^{13} - \frac{421}{7003} a^{12} - \frac{2295}{7003} a^{11} - \frac{2315}{7003} a^{10} + \frac{438}{7003} a^{9} - \frac{3244}{7003} a^{8} - \frac{446}{7003} a^{7} + \frac{710}{7003} a^{6} + \frac{585}{7003} a^{5} - \frac{755}{7003} a^{4} - \frac{1934}{7003} a^{3} + \frac{3462}{7003} a^{2} - \frac{8}{47} a - \frac{2389}{7003}$, $\frac{1}{7003} a^{31} - \frac{19}{7003} a^{29} + \frac{2}{7003} a^{28} - \frac{20}{7003} a^{27} - \frac{1946}{7003} a^{26} + \frac{3057}{7003} a^{25} - \frac{2220}{7003} a^{24} + \frac{572}{7003} a^{23} + \frac{2871}{7003} a^{22} - \frac{3345}{7003} a^{21} - \frac{1689}{7003} a^{20} - \frac{2270}{7003} a^{19} - \frac{2263}{7003} a^{18} - \frac{3461}{7003} a^{17} - \frac{963}{7003} a^{16} + \frac{889}{7003} a^{15} - \frac{1709}{7003} a^{14} - \frac{1079}{7003} a^{13} + \frac{3255}{7003} a^{12} - \frac{3145}{7003} a^{11} + \frac{2153}{7003} a^{10} + \frac{1392}{7003} a^{9} + \frac{3265}{7003} a^{8} + \frac{2309}{7003} a^{7} - \frac{3400}{7003} a^{6} + \frac{1449}{7003} a^{5} - \frac{2034}{7003} a^{4} - \frac{2521}{7003} a^{3} - \frac{2352}{7003} a^{2} - \frac{2329}{7003} a - \frac{407}{7003}$, $\frac{1}{99752558263349} a^{32} + \frac{3482444903}{99752558263349} a^{31} + \frac{4131118368}{99752558263349} a^{30} + \frac{156688382699}{99752558263349} a^{29} + \frac{171906021145}{99752558263349} a^{28} - \frac{296506233685}{99752558263349} a^{27} + \frac{43317168770277}{99752558263349} a^{26} + \frac{9332642689673}{99752558263349} a^{25} - \frac{48912597076201}{99752558263349} a^{24} - \frac{45965646092843}{99752558263349} a^{23} + \frac{46552241264622}{99752558263349} a^{22} - \frac{32004709835377}{99752558263349} a^{21} + \frac{22394846172269}{99752558263349} a^{20} + \frac{18214689591554}{99752558263349} a^{19} - \frac{19649031484127}{99752558263349} a^{18} - \frac{36453429712136}{99752558263349} a^{17} + \frac{36580545919413}{99752558263349} a^{16} - \frac{31563792257714}{99752558263349} a^{15} + \frac{36710924498523}{99752558263349} a^{14} + \frac{13046058338560}{99752558263349} a^{13} - \frac{19495296301852}{99752558263349} a^{12} - \frac{8026600298740}{99752558263349} a^{11} - \frac{324829708258}{669480256801} a^{10} - \frac{41157851320991}{99752558263349} a^{9} - \frac{30834468607737}{99752558263349} a^{8} + \frac{25831858404695}{99752558263349} a^{7} - \frac{3842836499801}{99752558263349} a^{6} + \frac{19094798403961}{99752558263349} a^{5} - \frac{36933021228644}{99752558263349} a^{4} - \frac{12695571997271}{99752558263349} a^{3} - \frac{47100155516758}{99752558263349} a^{2} - \frac{38783543343331}{99752558263349} a - \frac{12337488289498}{99752558263349}$, $\frac{1}{2202059049728140978773165329488662295107103247312592992840286271068305771928873798493272177320471831053801} a^{33} + \frac{1360359973190841520860587477107352341317145678756282722751753899209746570115913731551205169}{2202059049728140978773165329488662295107103247312592992840286271068305771928873798493272177320471831053801} a^{32} - \frac{7232763618331749643781449113558253029817943197733759771040746098915265628938832944133667367173842598}{2202059049728140978773165329488662295107103247312592992840286271068305771928873798493272177320471831053801} a^{31} + \frac{90625506665378846248509940919465263779119620428956014574529725465565484185659294524085962186651666473}{2202059049728140978773165329488662295107103247312592992840286271068305771928873798493272177320471831053801} a^{30} + \frac{2045352030522120637398212246730977525840009674596094021178885036999334491692187284562450616980682843214}{2202059049728140978773165329488662295107103247312592992840286271068305771928873798493272177320471831053801} a^{29} - \frac{117002941198446551818870398181235077197895381396183907455536092181164616452023038323816704181262506063}{46852320206981722952620538925290687129938366964097723251920984490815016424018591457303663347244081511783} a^{28} + \frac{440769554995516981769088756984798514987960882242617588857244157710861631915310119007173882658883049658}{2202059049728140978773165329488662295107103247312592992840286271068305771928873798493272177320471831053801} a^{27} - \frac{46409249488628837411098825466038448438642727898946631433032690110533898921621248859525739529944657603363}{2202059049728140978773165329488662295107103247312592992840286271068305771928873798493272177320471831053801} a^{26} - \frac{172199486464770231881144425536806899417312196283436969916006488464624803235274125854454400361312013695502}{2202059049728140978773165329488662295107103247312592992840286271068305771928873798493272177320471831053801} a^{25} - \frac{466710130886894091898101272427196403536801136544666690867076646582700642027389984043204554497251400650188}{2202059049728140978773165329488662295107103247312592992840286271068305771928873798493272177320471831053801} a^{24} - \frac{368287283047641051016969670673348010888450686706958147315505307701470399732995519496816284532334547221390}{2202059049728140978773165329488662295107103247312592992840286271068305771928873798493272177320471831053801} a^{23} - \frac{1063735076537151895343559662813390234969146156786998494039435172734613643312338182730163116583040030214780}{2202059049728140978773165329488662295107103247312592992840286271068305771928873798493272177320471831053801} a^{22} - \frac{566470420999678645572744364536622389788761849152922606000095770076741162969226644903185200758876810894042}{2202059049728140978773165329488662295107103247312592992840286271068305771928873798493272177320471831053801} a^{21} + \frac{1017420028945647559952129665603199526223261736421010194278427941553224544811221477319207869318094655144699}{2202059049728140978773165329488662295107103247312592992840286271068305771928873798493272177320471831053801} a^{20} - \frac{694882825625445663075098684930928694673118705168818482927289078027153981824557005179335398090650874663511}{2202059049728140978773165329488662295107103247312592992840286271068305771928873798493272177320471831053801} a^{19} + \frac{608875294062479210303644244747971908735695167551428439050369505399436672165909738080114684048487010761753}{2202059049728140978773165329488662295107103247312592992840286271068305771928873798493272177320471831053801} a^{18} + \frac{52913966443103246551077056175715448635561683815158747316601594566039937400279681845095898790592278782860}{2202059049728140978773165329488662295107103247312592992840286271068305771928873798493272177320471831053801} a^{17} - \frac{308728701983009515572233537766924785309315659900178677486577840274439935839720296358683411181734959579013}{2202059049728140978773165329488662295107103247312592992840286271068305771928873798493272177320471831053801} a^{16} - \frac{664421102885479099732597875164053631188294181664414395039495794071153346684795257524451058248117218169075}{2202059049728140978773165329488662295107103247312592992840286271068305771928873798493272177320471831053801} a^{15} - \frac{1045998882184434676111058008351053153137373765796164328853967209389762635664417689320081992392561854190291}{2202059049728140978773165329488662295107103247312592992840286271068305771928873798493272177320471831053801} a^{14} - \frac{507861739547514027765031310207414506742626486452156696126539662136679698680137274039673999427411560852817}{2202059049728140978773165329488662295107103247312592992840286271068305771928873798493272177320471831053801} a^{13} - \frac{234188116762855565532226636970192055307865289508648392103294037192994612499684703046286769489047117205009}{2202059049728140978773165329488662295107103247312592992840286271068305771928873798493272177320471831053801} a^{12} - \frac{89917547820419785718367516765678827239004658073927691541978813734709653453856109181756685682089180051430}{2202059049728140978773165329488662295107103247312592992840286271068305771928873798493272177320471831053801} a^{11} - \frac{269578969981538008742046837358078520147389175610718183791593702872068686441797913302910726866702248238809}{2202059049728140978773165329488662295107103247312592992840286271068305771928873798493272177320471831053801} a^{10} + \frac{672200509447612002329809040407850447732274719996331165827368591341683631218204848301382365141654421823312}{2202059049728140978773165329488662295107103247312592992840286271068305771928873798493272177320471831053801} a^{9} + \frac{834562585394396950030951474666233801931229023672297777188729374473030367892881274648605665844534230717812}{2202059049728140978773165329488662295107103247312592992840286271068305771928873798493272177320471831053801} a^{8} - \frac{487598283667353194555473942435368425193745323116417121796795360657656789621140392966204476195969428373690}{2202059049728140978773165329488662295107103247312592992840286271068305771928873798493272177320471831053801} a^{7} - \frac{382919781931991642249643545380725953663628586659345243250288614154104640640968205165208266678174474765553}{2202059049728140978773165329488662295107103247312592992840286271068305771928873798493272177320471831053801} a^{6} - \frac{627703339654220357704927704600324229541430470471190313566085746426728004642333005931783619933589505749440}{2202059049728140978773165329488662295107103247312592992840286271068305771928873798493272177320471831053801} a^{5} - \frac{1088141789184054352782692311580196242627282315314854906619137653984684842913697138442301351047139856570012}{2202059049728140978773165329488662295107103247312592992840286271068305771928873798493272177320471831053801} a^{4} - \frac{541604578085469799849298217144601772136767990190148621424141516962238048661486998675106338213188732537918}{2202059049728140978773165329488662295107103247312592992840286271068305771928873798493272177320471831053801} a^{3} - \frac{118387820685300230694574698233252978775022647077325693537598668971636508413099818933356186506034701108113}{2202059049728140978773165329488662295107103247312592992840286271068305771928873798493272177320471831053801} a^{2} + \frac{684006800044467004788703563958424143718947616886980732221964217025254672165858499460192122674665130681535}{2202059049728140978773165329488662295107103247312592992840286271068305771928873798493272177320471831053801} a + \frac{351437177066126413153824756829207336618885351708069720315182786956839343971596938361388653569861337161044}{2202059049728140978773165329488662295107103247312592992840286271068305771928873798493272177320471831053801}$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
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

\(\Q(\sqrt{5}) \), 17.17.160470643909878751793805444097921.1

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

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
103Data not computed