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Properties

Label 32.0.30565847394...1872.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{32}\cdot 193^{31}$
Root discriminant $327.46$
Ramified primes $2, 193$
Class number Not computed
Class group Not computed
Galois group $C_{32}$ (as 32T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![780515353, 0, 15477116409, 0, 114648303766, 0, 419409562413, 0, 844182228126, 0, 1008442843815, 0, 750483998897, 0, 358606414532, 0, 112150227566, 0, 23224085650, 0, 3201313195, 0, 293049077, 0, 17579019, 0, 671061, 0, 15440, 0, 193, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 193*x^30 + 15440*x^28 + 671061*x^26 + 17579019*x^24 + 293049077*x^22 + 3201313195*x^20 + 23224085650*x^18 + 112150227566*x^16 + 358606414532*x^14 + 750483998897*x^12 + 1008442843815*x^10 + 844182228126*x^8 + 419409562413*x^6 + 114648303766*x^4 + 15477116409*x^2 + 780515353)
 
gp: K = bnfinit(x^32 + 193*x^30 + 15440*x^28 + 671061*x^26 + 17579019*x^24 + 293049077*x^22 + 3201313195*x^20 + 23224085650*x^18 + 112150227566*x^16 + 358606414532*x^14 + 750483998897*x^12 + 1008442843815*x^10 + 844182228126*x^8 + 419409562413*x^6 + 114648303766*x^4 + 15477116409*x^2 + 780515353, 1)
 

Normalized defining polynomial

\( x^{32} + 193 x^{30} + 15440 x^{28} + 671061 x^{26} + 17579019 x^{24} + 293049077 x^{22} + 3201313195 x^{20} + 23224085650 x^{18} + 112150227566 x^{16} + 358606414532 x^{14} + 750483998897 x^{12} + 1008442843815 x^{10} + 844182228126 x^{8} + 419409562413 x^{6} + 114648303766 x^{4} + 15477116409 x^{2} + 780515353 \)

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

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 16]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(305658473945657028577927750534391762814465590760665601328732812917726751722831872=2^{32}\cdot 193^{31}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $327.46$
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:  $2, 193$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(772=2^{2}\cdot 193\)
Dirichlet character group:    $\lbrace$$\chi_{772}(1,·)$, $\chi_{772}(363,·)$, $\chi_{772}(389,·)$, $\chi_{772}(769,·)$, $\chi_{772}(9,·)$, $\chi_{772}(651,·)$, $\chi_{772}(529,·)$, $\chi_{772}(23,·)$, $\chi_{772}(413,·)$, $\chi_{772}(129,·)$, $\chi_{772}(555,·)$, $\chi_{772}(257,·)$, $\chi_{772}(429,·)$, $\chi_{772}(385,·)$, $\chi_{772}(305,·)$, $\chi_{772}(179,·)$, $\chi_{772}(571,·)$, $\chi_{772}(151,·)$, $\chi_{772}(319,·)$, $\chi_{772}(67,·)$, $\chi_{772}(455,·)$, $\chi_{772}(587,·)$, $\chi_{772}(207,·)$, $\chi_{772}(81,·)$, $\chi_{772}(729,·)$, $\chi_{772}(603,·)$, $\chi_{772}(745,·)$, $\chi_{772}(235,·)$, $\chi_{772}(629,·)$, $\chi_{772}(377,·)$, $\chi_{772}(507,·)$, $\chi_{772}(703,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{7} a^{24} + \frac{1}{7} a^{22} + \frac{1}{7} a^{20} - \frac{2}{7} a^{16} + \frac{1}{7} a^{14} - \frac{1}{7} a^{10} - \frac{3}{7} a^{8} + \frac{3}{7} a^{6} - \frac{1}{7} a^{4} + \frac{2}{7} a^{2} + \frac{1}{7}$, $\frac{1}{7} a^{25} + \frac{1}{7} a^{23} + \frac{1}{7} a^{21} - \frac{2}{7} a^{17} + \frac{1}{7} a^{15} - \frac{1}{7} a^{11} - \frac{3}{7} a^{9} + \frac{3}{7} a^{7} - \frac{1}{7} a^{5} + \frac{2}{7} a^{3} + \frac{1}{7} a$, $\frac{1}{7} a^{26} - \frac{1}{7} a^{20} - \frac{2}{7} a^{18} + \frac{3}{7} a^{16} - \frac{1}{7} a^{14} - \frac{1}{7} a^{12} - \frac{2}{7} a^{10} - \frac{1}{7} a^{8} + \frac{3}{7} a^{6} + \frac{3}{7} a^{4} - \frac{1}{7} a^{2} - \frac{1}{7}$, $\frac{1}{7} a^{27} - \frac{1}{7} a^{21} - \frac{2}{7} a^{19} + \frac{3}{7} a^{17} - \frac{1}{7} a^{15} - \frac{1}{7} a^{13} - \frac{2}{7} a^{11} - \frac{1}{7} a^{9} + \frac{3}{7} a^{7} + \frac{3}{7} a^{5} - \frac{1}{7} a^{3} - \frac{1}{7} a$, $\frac{1}{763} a^{28} - \frac{33}{763} a^{26} + \frac{4}{763} a^{24} + \frac{157}{763} a^{22} + \frac{27}{109} a^{20} + \frac{328}{763} a^{18} - \frac{129}{763} a^{16} + \frac{43}{763} a^{14} - \frac{228}{763} a^{12} - \frac{114}{763} a^{10} + \frac{59}{763} a^{8} - \frac{13}{109} a^{6} - \frac{83}{763} a^{4} + \frac{110}{763} a^{2} - \frac{264}{763}$, $\frac{1}{763} a^{29} - \frac{33}{763} a^{27} + \frac{4}{763} a^{25} + \frac{157}{763} a^{23} + \frac{27}{109} a^{21} + \frac{328}{763} a^{19} - \frac{129}{763} a^{17} + \frac{43}{763} a^{15} - \frac{228}{763} a^{13} - \frac{114}{763} a^{11} + \frac{59}{763} a^{9} - \frac{13}{109} a^{7} - \frac{83}{763} a^{5} + \frac{110}{763} a^{3} - \frac{264}{763} a$, $\frac{1}{59455465513955339241498930965910895623034990285551039991729495124935568234050827499} a^{30} - \frac{4662056140937809989050490912662613038749833811146616142805731412620967599051151}{59455465513955339241498930965910895623034990285551039991729495124935568234050827499} a^{28} - \frac{770277668950269552028918341396402069402132950915708525527375724142847162462716922}{59455465513955339241498930965910895623034990285551039991729495124935568234050827499} a^{26} + \frac{3632724510423946424839693607775455289455665541098341568867664043546024085892183693}{59455465513955339241498930965910895623034990285551039991729495124935568234050827499} a^{24} + \frac{11522765651474090984427450103760636744289651310182184254069492598500043605081363811}{59455465513955339241498930965910895623034990285551039991729495124935568234050827499} a^{22} + \frac{24251561059322479983449814765127680313142186793567486994995147437636585725978538284}{59455465513955339241498930965910895623034990285551039991729495124935568234050827499} a^{20} - \frac{1275764779056126743184916375067953679035618212903688366950359933305639287304242769}{8493637930565048463071275852272985089004998612221577141675642160705081176292975357} a^{18} - \frac{16307367118260329318889971188804688518020362807851001765829442878858288595987999700}{59455465513955339241498930965910895623034990285551039991729495124935568234050827499} a^{16} - \frac{24568298887201786765243842778104990295655436977672827249216307798348906512617513705}{59455465513955339241498930965910895623034990285551039991729495124935568234050827499} a^{14} - \frac{25041784402544580200131315565782857320978452072559544824924351068650918206341395487}{59455465513955339241498930965910895623034990285551039991729495124935568234050827499} a^{12} + \frac{22175211652845879892021170920189519130026858332109517682251961477611052383595076028}{59455465513955339241498930965910895623034990285551039991729495124935568234050827499} a^{10} + \frac{23002219393640307912637991420189713398460269477915092566200783801323873310603581996}{59455465513955339241498930965910895623034990285551039991729495124935568234050827499} a^{8} - \frac{8456755268764160512005248083478964793120433448824658648163859502721702192917088698}{59455465513955339241498930965910895623034990285551039991729495124935568234050827499} a^{6} - \frac{2028723398098003761597007090168383923130117323238989841570400465167130041890313724}{59455465513955339241498930965910895623034990285551039991729495124935568234050827499} a^{4} + \frac{17642691412373400353369756999367493492809625125592334175822480640227499248256262386}{59455465513955339241498930965910895623034990285551039991729495124935568234050827499} a^{2} - \frac{21017546857873862580620473314236702860240962601728601603708201416229499553696577226}{59455465513955339241498930965910895623034990285551039991729495124935568234050827499}$, $\frac{1}{119564941148564187214654350172446811097923365464243141423368014696245427718676214100489} a^{31} + \frac{66152205861746458017052355000644808401321285228355082680028021740501268194536142226}{119564941148564187214654350172446811097923365464243141423368014696245427718676214100489} a^{29} - \frac{5513452987740733329368335433653803100433242745993259514836016409293693010619770438307}{119564941148564187214654350172446811097923365464243141423368014696245427718676214100489} a^{27} + \frac{3767639023574773496518370966710475196016955234025053041123915544141695025367138804285}{119564941148564187214654350172446811097923365464243141423368014696245427718676214100489} a^{25} + \frac{4635376442358094435137565746396993268150827364771070634791836636411338767959552964396}{17080705878366312459236335738920973013989052209177591631909716385177918245525173442927} a^{23} - \frac{8157563451033175286994863426323491722519398544207072083355090413404413406554325984307}{17080705878366312459236335738920973013989052209177591631909716385177918245525173442927} a^{21} + \frac{26268593168188234065662704270980513779930536424887627469938550015874654924191127445696}{119564941148564187214654350172446811097923365464243141423368014696245427718676214100489} a^{19} - \frac{51519857619254314565684815816248750014646752865056253340708939877762755661377590276396}{119564941148564187214654350172446811097923365464243141423368014696245427718676214100489} a^{17} - \frac{31672486614442705574734914642721726183127473293536349739858653541129386329466577123271}{119564941148564187214654350172446811097923365464243141423368014696245427718676214100489} a^{15} + \frac{177349335236692494663804892569230006819936554611439453339105449241751945312973153826}{17080705878366312459236335738920973013989052209177591631909716385177918245525173442927} a^{13} - \frac{6143738386234779393373152367049138592088207363574945125129768340286349161540354961116}{119564941148564187214654350172446811097923365464243141423368014696245427718676214100489} a^{11} - \frac{5058254261413963252967856793779159063419492319293200160294231320225855370505169841004}{17080705878366312459236335738920973013989052209177591631909716385177918245525173442927} a^{9} + \frac{40915605106436505674622168487697579759830743705642705337441565062770098045424055112134}{119564941148564187214654350172446811097923365464243141423368014696245427718676214100489} a^{7} - \frac{4248912150335137888496360888732687272460612009339376965843958207063860955545079429750}{17080705878366312459236335738920973013989052209177591631909716385177918245525173442927} a^{5} - \frac{1580953475060488702581558904554580112937810067807670556296432555386054522143031714209}{119564941148564187214654350172446811097923365464243141423368014696245427718676214100489} a^{3} - \frac{1323037695314859135694913576209918426274777859955951835184059622179726759871855524383}{119564941148564187214654350172446811097923365464243141423368014696245427718676214100489} 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:  $15$
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

$C_{32}$ (as 32T33):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 32
The 32 conjugacy class representatives for $C_{32}$
Character table for $C_{32}$ is not computed

Intermediate fields

\(\Q(\sqrt{193}) \), 4.4.7189057.1, 8.8.9974730326005057.1, 16.16.19202582299769315484813817587637057.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{4}$ $32$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ $32$ $32$ $32$ $32$ $16^{2}$ $32$ $16^{2}$ $32$ $32$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ $32$ $32$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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
193Data not computed