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Properties

Label 32.0.22462049622...1776.2
Degree $32$
Signature $[0, 16]$
Discriminant $2^{96}\cdot 17^{28}$
Root discriminant $95.44$
Ramified primes $2, 17$
Class number Not computed
Class group Not computed
Galois group $C_4\times C_8$ (as 32T43)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3243601, 0, -10774256, 0, 26615832, 0, -16686272, 0, 382501, 0, -2082176, 0, 6156800, 0, -3438736, 0, 376952, 0, 201200, 0, -27588, 0, -15672, 0, 4997, 0, 736, 0, -124, 0, -8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 8*x^30 - 124*x^28 + 736*x^26 + 4997*x^24 - 15672*x^22 - 27588*x^20 + 201200*x^18 + 376952*x^16 - 3438736*x^14 + 6156800*x^12 - 2082176*x^10 + 382501*x^8 - 16686272*x^6 + 26615832*x^4 - 10774256*x^2 + 3243601)
 
gp: K = bnfinit(x^32 - 8*x^30 - 124*x^28 + 736*x^26 + 4997*x^24 - 15672*x^22 - 27588*x^20 + 201200*x^18 + 376952*x^16 - 3438736*x^14 + 6156800*x^12 - 2082176*x^10 + 382501*x^8 - 16686272*x^6 + 26615832*x^4 - 10774256*x^2 + 3243601, 1)
 

Normalized defining polynomial

\( x^{32} - 8 x^{30} - 124 x^{28} + 736 x^{26} + 4997 x^{24} - 15672 x^{22} - 27588 x^{20} + 201200 x^{18} + 376952 x^{16} - 3438736 x^{14} + 6156800 x^{12} - 2082176 x^{10} + 382501 x^{8} - 16686272 x^{6} + 26615832 x^{4} - 10774256 x^{2} + 3243601 \)

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:  \(2246204962214211251170300217479976898657250038323286379149131776=2^{96}\cdot 17^{28}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $95.44$
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:  $2, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(272=2^{4}\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{272}(1,·)$, $\chi_{272}(259,·)$, $\chi_{272}(263,·)$, $\chi_{272}(137,·)$, $\chi_{272}(15,·)$, $\chi_{272}(151,·)$, $\chi_{272}(33,·)$, $\chi_{272}(35,·)$, $\chi_{272}(169,·)$, $\chi_{272}(171,·)$, $\chi_{272}(53,·)$, $\chi_{272}(189,·)$, $\chi_{272}(67,·)$, $\chi_{272}(203,·)$, $\chi_{272}(77,·)$, $\chi_{272}(81,·)$, $\chi_{272}(213,·)$, $\chi_{272}(87,·)$, $\chi_{272}(89,·)$, $\chi_{272}(93,·)$, $\chi_{272}(223,·)$, $\chi_{272}(225,·)$, $\chi_{272}(123,·)$, $\chi_{272}(229,·)$, $\chi_{272}(111,·)$, $\chi_{272}(115,·)$, $\chi_{272}(117,·)$, $\chi_{272}(247,·)$, $\chi_{272}(217,·)$, $\chi_{272}(251,·)$, $\chi_{272}(253,·)$, $\chi_{272}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{4} a^{24} + \frac{1}{4}$, $\frac{1}{4} a^{25} + \frac{1}{4} a$, $\frac{1}{188} a^{26} - \frac{7}{94} a^{24} + \frac{8}{47} a^{22} + \frac{9}{47} a^{20} - \frac{14}{47} a^{18} + \frac{15}{47} a^{16} - \frac{17}{47} a^{14} + \frac{20}{47} a^{12} + \frac{2}{47} a^{10} + \frac{18}{47} a^{8} + \frac{22}{47} a^{6} - \frac{17}{47} a^{4} - \frac{11}{188} a^{2} - \frac{39}{94}$, $\frac{1}{338588} a^{27} - \frac{5855}{84647} a^{25} - \frac{35289}{84647} a^{23} - \frac{32797}{84647} a^{21} + \frac{20196}{84647} a^{19} + \frac{3869}{84647} a^{17} + \frac{22308}{84647} a^{15} + \frac{11206}{84647} a^{13} - \frac{39337}{84647} a^{11} - \frac{35420}{84647} a^{9} + \frac{27752}{84647} a^{7} + \frac{39886}{84647} a^{5} + \frac{38905}{338588} a^{3} + \frac{10297}{84647} a$, $\frac{1}{444566044} a^{28} + \frac{150832}{111141511} a^{26} - \frac{622895}{17098694} a^{24} - \frac{40220311}{111141511} a^{22} + \frac{5491634}{111141511} a^{20} - \frac{27139002}{111141511} a^{18} + \frac{13486584}{111141511} a^{16} + \frac{3323245}{111141511} a^{14} + \frac{43561072}{111141511} a^{12} - \frac{12933}{84647} a^{10} - \frac{44134569}{111141511} a^{8} + \frac{17806751}{111141511} a^{6} + \frac{3129421}{444566044} a^{4} + \frac{39933064}{111141511} a^{2} - \frac{11927}{123422}$, $\frac{1}{444566044} a^{29} - \frac{163}{111141511} a^{27} + \frac{1080341}{34197388} a^{25} + \frac{45601308}{111141511} a^{23} + \frac{31034736}{111141511} a^{21} + \frac{447128}{111141511} a^{19} + \frac{10659695}{111141511} a^{17} - \frac{22139764}{111141511} a^{15} + \frac{54993363}{111141511} a^{13} - \frac{32371}{84647} a^{11} + \frac{9666919}{111141511} a^{9} + \frac{819616}{2364713} a^{7} + \frac{112575849}{444566044} a^{5} - \frac{55168839}{111141511} a^{3} + \frac{87150681}{444566044} a$, $\frac{1}{2706538595135248791566307062992096314495269591322580516} a^{30} - \frac{600487548852727940477907727917675789758642113}{1353269297567624395783153531496048157247634795661290258} a^{28} - \frac{37458470103965598184745781558172935868261981994895}{208195276548865291658946697153238178038097660870967732} a^{26} - \frac{324739941360682433099820124315344238226709280286357703}{2706538595135248791566307062992096314495269591322580516} a^{24} - \frac{92155560990404062312762394252119162804776663197383721}{676634648783812197891576765748024078623817397830645129} a^{22} + \frac{125933975581520629901308803254805101091065275327434632}{676634648783812197891576765748024078623817397830645129} a^{20} - \frac{228199875049020571162462232208124573866952602861023771}{676634648783812197891576765748024078623817397830645129} a^{18} + \frac{327988367990323730983693076167851847868967002900204751}{676634648783812197891576765748024078623817397830645129} a^{16} + \frac{107644422330533918808471548554243044614029632814620108}{676634648783812197891576765748024078623817397830645129} a^{14} + \frac{8163532059898341318315793536150536562811018503014018}{52048819137216322914736674288309544509524415217741933} a^{12} - \frac{74782012744973850591682423093023032201010160120946673}{676634648783812197891576765748024078623817397830645129} a^{10} - \frac{15244715789237413399200802015096195785017218083719742}{676634648783812197891576765748024078623817397830645129} a^{8} - \frac{396251917918477859695144951377819180113569120327133987}{2706538595135248791566307062992096314495269591322580516} a^{6} + \frac{556176832633228654026892313078525653602742531369985587}{1353269297567624395783153531496048157247634795661290258} a^{4} - \frac{536021386190466792040473948633710046757769635024141039}{2706538595135248791566307062992096314495269591322580516} a^{2} - \frac{33843220613181253124649784056895166915632435584483}{115599820404700328516905439840776334279898756730132}$, $\frac{1}{2706538595135248791566307062992096314495269591322580516} a^{31} - \frac{600487548852727940477907727917675789758642113}{1353269297567624395783153531496048157247634795661290258} a^{29} + \frac{12495565347323910637001193285697404043313225396}{52048819137216322914736674288309544509524415217741933} a^{27} - \frac{120887159595913021800440646515660341306482566873693425}{1353269297567624395783153531496048157247634795661290258} a^{25} + \frac{293080225615598055340417979201098605351420624995373030}{676634648783812197891576765748024078623817397830645129} a^{23} - \frac{303610361335876705691231618546745123830425760265410320}{676634648783812197891576765748024078623817397830645129} a^{21} + \frac{146684193596805322923792061033819419887105879110712915}{676634648783812197891576765748024078623817397830645129} a^{19} + \frac{184647050259142232140358350029966656688153818996473627}{676634648783812197891576765748024078623817397830645129} a^{17} + \frac{159091261551919382033750507812743440021107128442793960}{676634648783812197891576765748024078623817397830645129} a^{15} + \frac{12092696168334698229076568858398200265196939011022228}{52048819137216322914736674288309544509524415217741933} a^{13} - \frac{310121725845783269922856730042915844690097877459210560}{676634648783812197891576765748024078623817397830645129} a^{11} + \frac{306114176438565320066098253315890891871186865080333272}{676634648783812197891576765748024078623817397830645129} a^{9} - \frac{398426178370344989278313981352696760528535726730313491}{2706538595135248791566307062992096314495269591322580516} a^{7} + \frac{208982610330298851396878901575463786898111713297564349}{1353269297567624395783153531496048157247634795661290258} a^{5} - \frac{127832285393372614110845386430121587147317549216125804}{676634648783812197891576765748024078623817397830645129} a^{3} + \frac{39314189497528605691596002271703101940352613774745489}{104097638274432645829473348576619089019048830435483866} a$

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

$C_4\times C_8$ (as 32T43):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_4\times C_8$
Character table for $C_4\times C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{34}) \), 4.0.2048.2, \(\Q(\sqrt{2}, \sqrt{17})\), 4.0.591872.5, 4.4.314432.1, 4.4.4913.1, 4.0.10061824.1, 4.0.10061824.2, 8.0.350312464384.1, 8.8.98867482624.1, 8.0.101240302206976.1, 8.8.1721085137518592.1, 8.8.1721085137518592.2, 8.0.105046700288.1, 8.0.1680747204608.1, 16.0.10249598790959829536343064576.2, 16.16.2962134050587390736003145662464.1, 16.0.723177258444187191407017984.3

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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
$17$17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$