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Properties

Label 34.0.22067477015...2464.1
Degree $34$
Signature $[0, 17]$
Discriminant $-\,2^{34}\cdot 239^{32}$
Root discriminant $346.36$
Ramified primes $2, 239$
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![383161, 0, 1846678070, 0, 373964654925, 0, 3800340397568, 0, 11370254800894, 0, 16357214524365, 0, 13458854494998, 0, 6901883479220, 0, 2318506170329, 0, 525386382057, 0, 81579185137, 0, 8726161392, 0, 640400568, 0, 31753524, 0, 1030237, 0, 20590, 0, 225, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^34 + 225*x^32 + 20590*x^30 + 1030237*x^28 + 31753524*x^26 + 640400568*x^24 + 8726161392*x^22 + 81579185137*x^20 + 525386382057*x^18 + 2318506170329*x^16 + 6901883479220*x^14 + 13458854494998*x^12 + 16357214524365*x^10 + 11370254800894*x^8 + 3800340397568*x^6 + 373964654925*x^4 + 1846678070*x^2 + 383161)
 
gp: K = bnfinit(x^34 + 225*x^32 + 20590*x^30 + 1030237*x^28 + 31753524*x^26 + 640400568*x^24 + 8726161392*x^22 + 81579185137*x^20 + 525386382057*x^18 + 2318506170329*x^16 + 6901883479220*x^14 + 13458854494998*x^12 + 16357214524365*x^10 + 11370254800894*x^8 + 3800340397568*x^6 + 373964654925*x^4 + 1846678070*x^2 + 383161, 1)
 

Normalized defining polynomial

\( x^{34} + 225 x^{32} + 20590 x^{30} + 1030237 x^{28} + 31753524 x^{26} + 640400568 x^{24} + 8726161392 x^{22} + 81579185137 x^{20} + 525386382057 x^{18} + 2318506170329 x^{16} + 6901883479220 x^{14} + 13458854494998 x^{12} + 16357214524365 x^{10} + 11370254800894 x^{8} + 3800340397568 x^{6} + 373964654925 x^{4} + 1846678070 x^{2} + 383161 \)

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

Invariants

Degree:  $34$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 17]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-220674770153943459921555151520204246702049820735038763987383340493274252918604341182464=-\,2^{34}\cdot 239^{32}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $346.36$
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:  $2, 239$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(956=2^{2}\cdot 239\)
Dirichlet character group:    $\lbrace$$\chi_{956}(1,·)$, $\chi_{956}(261,·)$, $\chi_{956}(641,·)$, $\chi_{956}(371,·)$, $\chi_{956}(529,·)$, $\chi_{956}(275,·)$, $\chi_{956}(405,·)$, $\chi_{956}(279,·)$, $\chi_{956}(665,·)$, $\chi_{956}(545,·)$, $\chi_{956}(163,·)$, $\chi_{956}(549,·)$, $\chi_{956}(553,·)$, $\chi_{956}(71,·)$, $\chi_{956}(689,·)$, $\chi_{956}(51,·)$, $\chi_{956}(187,·)$, $\chi_{956}(245,·)$, $\chi_{956}(67,·)$, $\chi_{956}(455,·)$, $\chi_{956}(75,·)$, $\chi_{956}(845,·)$, $\chi_{956}(849,·)$, $\chi_{956}(211,·)$, $\chi_{956}(723,·)$, $\chi_{956}(479,·)$, $\chi_{956}(739,·)$, $\chi_{956}(101,·)$, $\chi_{956}(367,·)$, $\chi_{956}(753,·)$, $\chi_{956}(579,·)$, $\chi_{956}(883,·)$, $\chi_{956}(757,·)$, $\chi_{956}(933,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $\frac{1}{31831} a^{28} - \frac{10071}{31831} a^{26} - \frac{14887}{31831} a^{24} - \frac{5427}{31831} a^{22} - \frac{10485}{31831} a^{20} - \frac{13148}{31831} a^{18} - \frac{6896}{31831} a^{16} - \frac{2728}{31831} a^{14} - \frac{9903}{31831} a^{12} + \frac{9278}{31831} a^{10} + \frac{3778}{31831} a^{8} + \frac{5756}{31831} a^{6} - \frac{3077}{31831} a^{4} + \frac{4889}{31831} a^{2} - \frac{290}{31831}$, $\frac{1}{31831} a^{29} - \frac{10071}{31831} a^{27} - \frac{14887}{31831} a^{25} - \frac{5427}{31831} a^{23} - \frac{10485}{31831} a^{21} - \frac{13148}{31831} a^{19} - \frac{6896}{31831} a^{17} - \frac{2728}{31831} a^{15} - \frac{9903}{31831} a^{13} + \frac{9278}{31831} a^{11} + \frac{3778}{31831} a^{9} + \frac{5756}{31831} a^{7} - \frac{3077}{31831} a^{5} + \frac{4889}{31831} a^{3} - \frac{290}{31831} a$, $\frac{1}{542535235271} a^{30} + \frac{5920339}{542535235271} a^{28} + \frac{235615098938}{542535235271} a^{26} - \frac{16124855618}{542535235271} a^{24} - \frac{133695187091}{542535235271} a^{22} + \frac{181520543299}{542535235271} a^{20} + \frac{21384118500}{542535235271} a^{18} + \frac{183800383138}{542535235271} a^{16} + \frac{152352047878}{542535235271} a^{14} - \frac{53108313958}{542535235271} a^{12} - \frac{177160542020}{542535235271} a^{10} + \frac{152127465093}{542535235271} a^{8} - \frac{14222568458}{542535235271} a^{6} - \frac{170473130657}{542535235271} a^{4} - \frac{81043923785}{542535235271} a^{2} + \frac{25322589122}{542535235271}$, $\frac{1}{335829310632749} a^{31} + \frac{3005706755}{335829310632749} a^{29} + \frac{85042235980854}{335829310632749} a^{27} - \frac{15377042525976}{335829310632749} a^{25} + \frac{91008440516935}{335829310632749} a^{23} + \frac{124978907729587}{335829310632749} a^{21} + \frac{19716532965471}{335829310632749} a^{19} - \frac{111648646267126}{335829310632749} a^{17} - \frac{65539800233696}{335829310632749} a^{15} + \frac{157414662976889}{335829310632749} a^{13} - \frac{39619511347976}{335829310632749} a^{11} + \frac{119992367598941}{335829310632749} a^{9} - \frac{162326614832663}{335829310632749} a^{7} - \frac{41952930048949}{335829310632749} a^{5} + \frac{137740410270556}{335829310632749} a^{3} + \frac{111460178229579}{335829310632749} a$, $\frac{1}{39768326371372746838970526731280050790866297992928104049789607546660294787041990765282941} a^{32} - \frac{22775993140345734673824119231964140978948751354795076184682792314073328846126}{39768326371372746838970526731280050790866297992928104049789607546660294787041990765282941} a^{30} - \frac{282317095236900748292407188639261781316315721065851204530171183180033813782980945248}{39768326371372746838970526731280050790866297992928104049789607546660294787041990765282941} a^{28} + \frac{10197745405378194889411832242158405814817964560900825202758336749526623547847246973738426}{39768326371372746838970526731280050790866297992928104049789607546660294787041990765282941} a^{26} - \frac{6109316057121008439300553906642238115234981171219449423992027686449889516014308106567395}{39768326371372746838970526731280050790866297992928104049789607546660294787041990765282941} a^{24} + \frac{13852663896207825930356300289021936231276933036293228579509308076799884592129795101181871}{39768326371372746838970526731280050790866297992928104049789607546660294787041990765282941} a^{22} + \frac{19440634475038455424815352805372098323871738581085027826928122192831202547425832386070744}{39768326371372746838970526731280050790866297992928104049789607546660294787041990765282941} a^{20} + \frac{4674643171892959808938164426433736869049977056964008401398679341790748940411356209528897}{39768326371372746838970526731280050790866297992928104049789607546660294787041990765282941} a^{18} + \frac{10399423085377752773358652453134356715975547884029607211472781806706846166737218384650366}{39768326371372746838970526731280050790866297992928104049789607546660294787041990765282941} a^{16} + \frac{10921769240092109310611571514922771288589720083758684277980125816885986565518428023172446}{39768326371372746838970526731280050790866297992928104049789607546660294787041990765282941} a^{14} - \frac{19379615360003023045762911729661877710760291715663574042813295214753147592754365241391532}{39768326371372746838970526731280050790866297992928104049789607546660294787041990765282941} a^{12} + \frac{10080671478136307578567480338662611983259424209980181249072203199226054129830115541976928}{39768326371372746838970526731280050790866297992928104049789607546660294787041990765282941} a^{10} + \frac{10611721718487368460263121772992941730688923797066941124275484206419472974985789286021676}{39768326371372746838970526731280050790866297992928104049789607546660294787041990765282941} a^{8} - \frac{8252295183721901537701869363331042680746778053057118426079903748464564657336705880022834}{39768326371372746838970526731280050790866297992928104049789607546660294787041990765282941} a^{6} + \frac{13263013300773660977077049867959214766054278043618554510268944072415488190015667292034415}{39768326371372746838970526731280050790866297992928104049789607546660294787041990765282941} a^{4} + \frac{5790468103024300810424174389508780748895515427369479969387726564775397235073329635307873}{39768326371372746838970526731280050790866297992928104049789607546660294787041990765282941} a^{2} - \frac{4353786971844434391063473496331534000941872778193504513289723018960965604015997472177}{64246084606418007817399881633731907578136184156588213327608412837900314680197077165239}$, $\frac{1}{39768326371372746838970526731280050790866297992928104049789607546660294787041990765282941} a^{33} - \frac{39685076832186073190640370623727791630336202824999157857199335569131378798}{39768326371372746838970526731280050790866297992928104049789607546660294787041990765282941} a^{31} + \frac{591005492331622427265662238133163293829499357166610877939139395294528717828680099598}{39768326371372746838970526731280050790866297992928104049789607546660294787041990765282941} a^{29} - \frac{1521758005427719059259060999969726496504884534278140330547463707281390413338349253196194}{39768326371372746838970526731280050790866297992928104049789607546660294787041990765282941} a^{27} + \frac{12336983372878402598113471952713387285592422860902116996561936038284007359171524094298699}{39768326371372746838970526731280050790866297992928104049789607546660294787041990765282941} a^{25} - \frac{16453165077513682063756158870222902814917244520619223610070365866405949525289734322266753}{39768326371372746838970526731280050790866297992928104049789607546660294787041990765282941} a^{23} - \frac{10775574151025763615070635758478044650915911766751057958423389851411997889671614667460320}{39768326371372746838970526731280050790866297992928104049789607546660294787041990765282941} a^{21} - \frac{12132836035256013778263935997780171075064336556161621628525359742328298185843905657480097}{39768326371372746838970526731280050790866297992928104049789607546660294787041990765282941} a^{19} + \frac{5115446369157396557202404082302139484803292146183580919632485469235396062191890087459394}{39768326371372746838970526731280050790866297992928104049789607546660294787041990765282941} a^{17} + \frac{17188931606215517534769744922442028393122207904605135029664978834021025422606658523431243}{39768326371372746838970526731280050790866297992928104049789607546660294787041990765282941} a^{15} + \frac{12315442700304607612131529979225818790082865611734445363899160624596630838971109958627296}{39768326371372746838970526731280050790866297992928104049789607546660294787041990765282941} a^{13} - \frac{5467878923077897718750315358970811373952078803577845468993055546182872138152313177256174}{39768326371372746838970526731280050790866297992928104049789607546660294787041990765282941} a^{11} + \frac{18274821298159026315960367907746185935160923215950548913706970009688087826974161149189250}{39768326371372746838970526731280050790866297992928104049789607546660294787041990765282941} a^{9} + \frac{8846832391584150005811308476707618240972270947100907362782695232040112894674657766435361}{39768326371372746838970526731280050790866297992928104049789607546660294787041990765282941} a^{7} - \frac{17170965231383016999863346254502670245217871354094376533921910413595038959081500706332781}{39768326371372746838970526731280050790866297992928104049789607546660294787041990765282941} a^{5} - \frac{15890410183970371758763874526046666470682663529052653427936219406198956672267351988863436}{39768326371372746838970526731280050790866297992928104049789607546660294787041990765282941} a^{3} + \frac{5890019259788383246795604760034113124678309297937951276098360774954991129589003425489385}{39768326371372746838970526731280050790866297992928104049789607546660294787041990765282941} a$

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:  $16$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{7851960463838949188564327543523114561834178706398711639820112}{249052077132259010574001170287709238939638465895127445133402058018461} a^{33} + \frac{1766697331251013024404923927105974582575257550738201076824534127}{249052077132259010574001170287709238939638465895127445133402058018461} a^{31} + \frac{161673267099683429170277306263665978296012227490898126152480601289}{249052077132259010574001170287709238939638465895127445133402058018461} a^{29} + \frac{8089508426406953366181220853916469456708923918987263746268112054825}{249052077132259010574001170287709238939638465895127445133402058018461} a^{27} + \frac{249333832256825331323732848786194231226437079976084018287332145226184}{249052077132259010574001170287709238939638465895127445133402058018461} a^{25} + \frac{5028597769266582943893783267891816303955740862198429365347451524504521}{249052077132259010574001170287709238939638465895127445133402058018461} a^{23} + \frac{68521465120763839677288473117674458491396335654954998893223201908623947}{249052077132259010574001170287709238939638465895127445133402058018461} a^{21} + \frac{640610937152151688043395488628509971174093781940895058262839465775576142}{249052077132259010574001170287709238939638465895127445133402058018461} a^{19} + \frac{4125821957814019168338928928141008036936422376855433897126339538951155984}{249052077132259010574001170287709238939638465895127445133402058018461} a^{17} + \frac{18208098496321289319371152792389416366278537557557135459761412784196828189}{249052077132259010574001170287709238939638465895127445133402058018461} a^{15} + \frac{54207806289975966761719987689407020988154949551406607896879528076123028660}{249052077132259010574001170287709238939638465895127445133402058018461} a^{13} + \frac{105721606411555365571594658515067291880859352074264807985512329751426516505}{249052077132259010574001170287709238939638465895127445133402058018461} a^{11} + \frac{128520719486586793604415705241725611204920247861180085226823938558286305146}{249052077132259010574001170287709238939638465895127445133402058018461} a^{9} + \frac{89382063431424533381259868656795681587387013006285601591332339429184935953}{249052077132259010574001170287709238939638465895127445133402058018461} a^{7} + \frac{29912723220641463530027890711794975063079113065396552718413554732667519204}{249052077132259010574001170287709238939638465895127445133402058018461} a^{5} + \frac{2961407886941473105805568212863147452870115566638053717451419123006602530}{249052077132259010574001170287709238939638465895127445133402058018461} a^{3} + \frac{17547917385867482099470116922355808775393772717492839299768439726401569}{249052077132259010574001170287709238939638465895127445133402058018461} a \) (order $4$)
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

\(\Q(\sqrt{-1}) \), 17.17.113335617496346216833223278514633468161.1

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 $34$ $17^{2}$ $34$ $34$ $17^{2}$ $17^{2}$ $34$ $34$ $17^{2}$ $34$ $17^{2}$ $17^{2}$ $34$ $34$ $17^{2}$ $34$

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

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
2Data not computed
239Data not computed