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Properties

Label 34.34.3249216995...2297.1
Degree $34$
Signature $[34, 0]$
Discriminant $137^{33}$
Root discriminant $118.54$
Ramified prime $137$
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![1, 77, 1328, 1806, -61867, -116296, 1118049, 1347387, -9042178, -7334043, 37938129, 22664245, -91025976, -41728477, 134553261, 47931532, -128721615, -35828109, 82324817, 17958849, -35945371, -6136823, 10839684, 1435905, -2262360, -228291, 323844, 24095, -31023, -1607, 1891, 61, -66, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^34 - x^33 - 66*x^32 + 61*x^31 + 1891*x^30 - 1607*x^29 - 31023*x^28 + 24095*x^27 + 323844*x^26 - 228291*x^25 - 2262360*x^24 + 1435905*x^23 + 10839684*x^22 - 6136823*x^21 - 35945371*x^20 + 17958849*x^19 + 82324817*x^18 - 35828109*x^17 - 128721615*x^16 + 47931532*x^15 + 134553261*x^14 - 41728477*x^13 - 91025976*x^12 + 22664245*x^11 + 37938129*x^10 - 7334043*x^9 - 9042178*x^8 + 1347387*x^7 + 1118049*x^6 - 116296*x^5 - 61867*x^4 + 1806*x^3 + 1328*x^2 + 77*x + 1)
 
gp: K = bnfinit(x^34 - x^33 - 66*x^32 + 61*x^31 + 1891*x^30 - 1607*x^29 - 31023*x^28 + 24095*x^27 + 323844*x^26 - 228291*x^25 - 2262360*x^24 + 1435905*x^23 + 10839684*x^22 - 6136823*x^21 - 35945371*x^20 + 17958849*x^19 + 82324817*x^18 - 35828109*x^17 - 128721615*x^16 + 47931532*x^15 + 134553261*x^14 - 41728477*x^13 - 91025976*x^12 + 22664245*x^11 + 37938129*x^10 - 7334043*x^9 - 9042178*x^8 + 1347387*x^7 + 1118049*x^6 - 116296*x^5 - 61867*x^4 + 1806*x^3 + 1328*x^2 + 77*x + 1, 1)
 

Normalized defining polynomial

\( x^{34} - x^{33} - 66 x^{32} + 61 x^{31} + 1891 x^{30} - 1607 x^{29} - 31023 x^{28} + 24095 x^{27} + 323844 x^{26} - 228291 x^{25} - 2262360 x^{24} + 1435905 x^{23} + 10839684 x^{22} - 6136823 x^{21} - 35945371 x^{20} + 17958849 x^{19} + 82324817 x^{18} - 35828109 x^{17} - 128721615 x^{16} + 47931532 x^{15} + 134553261 x^{14} - 41728477 x^{13} - 91025976 x^{12} + 22664245 x^{11} + 37938129 x^{10} - 7334043 x^{9} - 9042178 x^{8} + 1347387 x^{7} + 1118049 x^{6} - 116296 x^{5} - 61867 x^{4} + 1806 x^{3} + 1328 x^{2} + 77 x + 1 \)

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:  \(32492169951483601485711825975325195410331229243708069666036144501372297=137^{33}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $118.54$
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:  $137$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(137\)
Dirichlet character group:    $\lbrace$$\chi_{137}(1,·)$, $\chi_{137}(4,·)$, $\chi_{137}(133,·)$, $\chi_{137}(136,·)$, $\chi_{137}(14,·)$, $\chi_{137}(15,·)$, $\chi_{137}(16,·)$, $\chi_{137}(18,·)$, $\chi_{137}(22,·)$, $\chi_{137}(34,·)$, $\chi_{137}(38,·)$, $\chi_{137}(49,·)$, $\chi_{137}(50,·)$, $\chi_{137}(56,·)$, $\chi_{137}(59,·)$, $\chi_{137}(60,·)$, $\chi_{137}(63,·)$, $\chi_{137}(64,·)$, $\chi_{137}(65,·)$, $\chi_{137}(72,·)$, $\chi_{137}(73,·)$, $\chi_{137}(74,·)$, $\chi_{137}(77,·)$, $\chi_{137}(78,·)$, $\chi_{137}(81,·)$, $\chi_{137}(87,·)$, $\chi_{137}(88,·)$, $\chi_{137}(99,·)$, $\chi_{137}(103,·)$, $\chi_{137}(115,·)$, $\chi_{137}(119,·)$, $\chi_{137}(121,·)$, $\chi_{137}(122,·)$, $\chi_{137}(123,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{37} a^{23} + \frac{3}{37} a^{22} + \frac{16}{37} a^{21} + \frac{8}{37} a^{20} + \frac{16}{37} a^{19} - \frac{9}{37} a^{18} + \frac{3}{37} a^{17} - \frac{11}{37} a^{16} - \frac{8}{37} a^{15} - \frac{8}{37} a^{14} + \frac{15}{37} a^{13} + \frac{8}{37} a^{12} + \frac{11}{37} a^{11} + \frac{15}{37} a^{10} + \frac{1}{37} a^{9} + \frac{5}{37} a^{8} + \frac{16}{37} a^{7} + \frac{16}{37} a^{5} + \frac{16}{37} a^{4} - \frac{9}{37} a^{3} - \frac{11}{37} a^{2} + \frac{13}{37} a + \frac{1}{37}$, $\frac{1}{37} a^{24} + \frac{7}{37} a^{22} - \frac{3}{37} a^{21} - \frac{8}{37} a^{20} + \frac{17}{37} a^{19} - \frac{7}{37} a^{18} + \frac{17}{37} a^{17} - \frac{12}{37} a^{16} + \frac{16}{37} a^{15} + \frac{2}{37} a^{14} - \frac{13}{37} a^{12} - \frac{18}{37} a^{11} - \frac{7}{37} a^{10} + \frac{2}{37} a^{9} + \frac{1}{37} a^{8} - \frac{11}{37} a^{7} + \frac{16}{37} a^{6} + \frac{5}{37} a^{5} + \frac{17}{37} a^{4} + \frac{16}{37} a^{3} + \frac{9}{37} a^{2} - \frac{1}{37} a - \frac{3}{37}$, $\frac{1}{37} a^{25} + \frac{13}{37} a^{22} - \frac{9}{37} a^{21} - \frac{2}{37} a^{20} - \frac{8}{37} a^{19} + \frac{6}{37} a^{18} + \frac{4}{37} a^{17} - \frac{18}{37} a^{16} - \frac{16}{37} a^{15} - \frac{18}{37} a^{14} - \frac{7}{37} a^{13} - \frac{10}{37} a^{11} + \frac{8}{37} a^{10} - \frac{6}{37} a^{9} - \frac{9}{37} a^{8} + \frac{15}{37} a^{7} + \frac{5}{37} a^{6} + \frac{16}{37} a^{5} + \frac{15}{37} a^{4} - \frac{2}{37} a^{3} + \frac{2}{37} a^{2} + \frac{17}{37} a - \frac{7}{37}$, $\frac{1}{37} a^{26} - \frac{11}{37} a^{22} + \frac{12}{37} a^{21} - \frac{1}{37} a^{20} - \frac{17}{37} a^{19} + \frac{10}{37} a^{18} + \frac{17}{37} a^{17} + \frac{16}{37} a^{16} + \frac{12}{37} a^{15} - \frac{14}{37} a^{14} - \frac{10}{37} a^{13} - \frac{3}{37} a^{12} + \frac{13}{37} a^{11} - \frac{16}{37} a^{10} + \frac{15}{37} a^{9} - \frac{13}{37} a^{8} - \frac{18}{37} a^{7} + \frac{16}{37} a^{6} - \frac{8}{37} a^{5} + \frac{12}{37} a^{4} + \frac{8}{37} a^{3} + \frac{12}{37} a^{2} + \frac{9}{37} a - \frac{13}{37}$, $\frac{1}{37} a^{27} + \frac{8}{37} a^{22} - \frac{10}{37} a^{21} - \frac{3}{37} a^{20} + \frac{1}{37} a^{19} - \frac{8}{37} a^{18} + \frac{12}{37} a^{17} + \frac{2}{37} a^{16} + \frac{9}{37} a^{15} + \frac{13}{37} a^{14} + \frac{14}{37} a^{13} - \frac{10}{37} a^{12} - \frac{6}{37} a^{11} - \frac{5}{37} a^{10} - \frac{2}{37} a^{9} + \frac{7}{37} a^{7} - \frac{8}{37} a^{6} + \frac{3}{37} a^{5} - \frac{1}{37} a^{4} - \frac{13}{37} a^{3} - \frac{1}{37} a^{2} - \frac{18}{37} a + \frac{11}{37}$, $\frac{1}{37} a^{28} + \frac{3}{37} a^{22} + \frac{17}{37} a^{21} + \frac{11}{37} a^{20} + \frac{12}{37} a^{19} + \frac{10}{37} a^{18} + \frac{15}{37} a^{17} - \frac{14}{37} a^{16} + \frac{3}{37} a^{15} + \frac{4}{37} a^{14} + \frac{18}{37} a^{13} + \frac{4}{37} a^{12} + \frac{18}{37} a^{11} - \frac{11}{37} a^{10} - \frac{8}{37} a^{9} + \frac{4}{37} a^{8} + \frac{12}{37} a^{7} + \frac{3}{37} a^{6} - \frac{18}{37} a^{5} + \frac{7}{37} a^{4} - \frac{3}{37} a^{3} - \frac{4}{37} a^{2} + \frac{18}{37} a - \frac{8}{37}$, $\frac{1}{37} a^{29} + \frac{8}{37} a^{22} - \frac{12}{37} a^{20} - \frac{1}{37} a^{19} + \frac{5}{37} a^{18} + \frac{14}{37} a^{17} - \frac{1}{37} a^{16} - \frac{9}{37} a^{15} + \frac{5}{37} a^{14} - \frac{4}{37} a^{13} - \frac{6}{37} a^{12} - \frac{7}{37} a^{11} - \frac{16}{37} a^{10} + \frac{1}{37} a^{9} - \frac{3}{37} a^{8} - \frac{8}{37} a^{7} - \frac{18}{37} a^{6} - \frac{4}{37} a^{5} - \frac{14}{37} a^{4} - \frac{14}{37} a^{3} + \frac{14}{37} a^{2} - \frac{10}{37} a - \frac{3}{37}$, $\frac{1}{37} a^{30} + \frac{13}{37} a^{22} + \frac{8}{37} a^{21} + \frac{9}{37} a^{20} - \frac{12}{37} a^{19} + \frac{12}{37} a^{18} + \frac{12}{37} a^{17} + \frac{5}{37} a^{16} - \frac{5}{37} a^{15} - \frac{14}{37} a^{14} - \frac{15}{37} a^{13} + \frac{3}{37} a^{12} + \frac{7}{37} a^{11} - \frac{8}{37} a^{10} - \frac{11}{37} a^{9} - \frac{11}{37} a^{8} + \frac{2}{37} a^{7} - \frac{4}{37} a^{6} + \frac{6}{37} a^{5} + \frac{6}{37} a^{4} + \frac{12}{37} a^{3} + \frac{4}{37} a^{2} + \frac{4}{37} a - \frac{8}{37}$, $\frac{1}{59459} a^{31} - \frac{70}{59459} a^{30} - \frac{390}{59459} a^{29} + \frac{121}{59459} a^{28} - \frac{422}{59459} a^{27} + \frac{463}{59459} a^{26} + \frac{699}{59459} a^{25} + \frac{178}{59459} a^{24} + \frac{336}{59459} a^{23} + \frac{15158}{59459} a^{22} - \frac{26154}{59459} a^{21} - \frac{26071}{59459} a^{20} - \frac{610}{1607} a^{19} - \frac{20864}{59459} a^{18} + \frac{2232}{59459} a^{17} + \frac{21414}{59459} a^{16} + \frac{13103}{59459} a^{15} + \frac{23448}{59459} a^{14} + \frac{2937}{59459} a^{13} - \frac{6377}{59459} a^{12} + \frac{20750}{59459} a^{11} - \frac{9778}{59459} a^{10} - \frac{14862}{59459} a^{9} + \frac{1381}{59459} a^{8} + \frac{3949}{59459} a^{7} + \frac{1005}{59459} a^{6} - \frac{28868}{59459} a^{5} + \frac{12574}{59459} a^{4} + \frac{27830}{59459} a^{3} - \frac{24512}{59459} a^{2} - \frac{2766}{59459} a - \frac{15817}{59459}$, $\frac{1}{684194713} a^{32} - \frac{5472}{684194713} a^{31} - \frac{903029}{684194713} a^{30} - \frac{4104154}{684194713} a^{29} + \frac{6190149}{684194713} a^{28} - \frac{5592586}{684194713} a^{27} - \frac{1369099}{684194713} a^{26} + \frac{7020006}{684194713} a^{25} + \frac{7599269}{684194713} a^{24} + \frac{5429}{499777} a^{23} - \frac{174839266}{684194713} a^{22} + \frac{204533462}{684194713} a^{21} + \frac{119542720}{684194713} a^{20} + \frac{19436742}{684194713} a^{19} - \frac{291256493}{684194713} a^{18} - \frac{129353278}{684194713} a^{17} + \frac{246142690}{684194713} a^{16} + \frac{168632618}{684194713} a^{15} - \frac{182517658}{684194713} a^{14} + \frac{164409244}{684194713} a^{13} + \frac{149287847}{684194713} a^{12} - \frac{308081356}{684194713} a^{11} + \frac{261320572}{684194713} a^{10} + \frac{330290109}{684194713} a^{9} + \frac{272513720}{684194713} a^{8} - \frac{223985442}{684194713} a^{7} - \frac{162914952}{684194713} a^{6} - \frac{1639373}{684194713} a^{5} + \frac{29856892}{684194713} a^{4} + \frac{136415013}{684194713} a^{3} + \frac{11032741}{684194713} a^{2} - \frac{49461554}{684194713} a - \frac{105336392}{684194713}$, $\frac{1}{568895401910821043946802347535088660794754333172608863447039} a^{33} - \frac{245575892804673345549971097902419659227777729114260}{568895401910821043946802347535088660794754333172608863447039} a^{32} + \frac{4682131793057490220457726478541516858815353805476451681}{568895401910821043946802347535088660794754333172608863447039} a^{31} - \frac{2457722046177397971832768756453751568010255814222974221357}{568895401910821043946802347535088660794754333172608863447039} a^{30} + \frac{6393091983066365565837977418223555542854810248750062882943}{568895401910821043946802347535088660794754333172608863447039} a^{29} - \frac{1607372636787310537994634033206699609011707388184721574217}{568895401910821043946802347535088660794754333172608863447039} a^{28} + \frac{4510008457881128255281042636774720577835357260517315030055}{568895401910821043946802347535088660794754333172608863447039} a^{27} + \frac{5329775190620889837305133571130149103926580405663823061552}{568895401910821043946802347535088660794754333172608863447039} a^{26} - \frac{6410744707247966824008022264235337072000680346898174159050}{568895401910821043946802347535088660794754333172608863447039} a^{25} - \frac{2349280323536325201500979991199417365602726911131981686368}{568895401910821043946802347535088660794754333172608863447039} a^{24} + \frac{3070503157088406560557008034860654648820292889577128785709}{568895401910821043946802347535088660794754333172608863447039} a^{23} + \frac{278794181719503861228017628790942344132828888869309620282956}{568895401910821043946802347535088660794754333172608863447039} a^{22} - \frac{189817671745646564288761185151861884246451278346839161920928}{568895401910821043946802347535088660794754333172608863447039} a^{21} + \frac{270001094171664851773762472355663718358074558513931335448200}{568895401910821043946802347535088660794754333172608863447039} a^{20} + \frac{234950344227564608831077951950166881189067923891953202771384}{568895401910821043946802347535088660794754333172608863447039} a^{19} - \frac{273485958652776559589428794819213599896559440007609897541650}{568895401910821043946802347535088660794754333172608863447039} a^{18} + \frac{98734486468445713022026750599269955646891281151666647298949}{568895401910821043946802347535088660794754333172608863447039} a^{17} + \frac{100828605482647849981690286585218612452024174725262073025129}{568895401910821043946802347535088660794754333172608863447039} a^{16} + \frac{33935042555079951459622248195682930463114915343469052598673}{568895401910821043946802347535088660794754333172608863447039} a^{15} + \frac{4559525880707912673119062298255573522932301096503460532038}{13875497607581001071873227988660699043774495931039240571879} a^{14} + \frac{193258088530835969457363971311237487655944832686172683551483}{568895401910821043946802347535088660794754333172608863447039} a^{13} + \frac{191154695144751441339710465722249213078208416848878757381694}{568895401910821043946802347535088660794754333172608863447039} a^{12} + \frac{222306564778058121493926175688649910518861557198813397891460}{568895401910821043946802347535088660794754333172608863447039} a^{11} + \frac{4363782130557780773604424544850696288539071476776069069274}{15375551402995163349913576960407801643101468464124563876947} a^{10} + \frac{128106582017588951374960642952817219460629248179549810520936}{568895401910821043946802347535088660794754333172608863447039} a^{9} + \frac{69518105838709229786771411437351351437715472770785481006927}{568895401910821043946802347535088660794754333172608863447039} a^{8} + \frac{170498011439833663916785381863143010759044362871558999043473}{568895401910821043946802347535088660794754333172608863447039} a^{7} + \frac{30477898004417354417802163298980952260694208870122946149066}{568895401910821043946802347535088660794754333172608863447039} a^{6} - \frac{4636899991316096742617910882870802511978056567308450316113}{15375551402995163349913576960407801643101468464124563876947} a^{5} + \frac{282795230272027074988750850213402802866015851383256281916532}{568895401910821043946802347535088660794754333172608863447039} a^{4} - \frac{109116413978394963619692461413987567366615414752149151890713}{568895401910821043946802347535088660794754333172608863447039} a^{3} - \frac{33097858004998666753840755806862947860301508711143455131159}{568895401910821043946802347535088660794754333172608863447039} a^{2} - \frac{203956949945124666425178053930294990178995274986974238477434}{568895401910821043946802347535088660794754333172608863447039} a - \frac{107478533662378531994906900377446515805066933737389615286742}{568895401910821043946802347535088660794754333172608863447039}$

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:  \( 5641198150779374000000000 \) (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

\(\Q(\sqrt{137}) \), 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 $17^{2}$ $34$ $34$ $17^{2}$ $17^{2}$ $34$ $17^{2}$ $17^{2}$ $34$ $34$ $34$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{34}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{17}$ $34$ $34$ $34$ $17^{2}$

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
137Data not computed