↑

Properties

Label 16.0.32914193623...8601.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 59^{14}$
Root discriminant $80.79$
Ramified primes $3, 59$
Class number $27$ (GRH)
Class group $[3, 3, 3]$ (GRH)
Galois group $QD_{16}$ (as 16T12)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![143392768, 211959808, 75011072, -29006080, -26235328, -2423392, 3196416, 794904, -45536, -72966, -10425, 2465, 824, 29, -28, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 28*x^14 + 29*x^13 + 824*x^12 + 2465*x^11 - 10425*x^10 - 72966*x^9 - 45536*x^8 + 794904*x^7 + 3196416*x^6 - 2423392*x^5 - 26235328*x^4 - 29006080*x^3 + 75011072*x^2 + 211959808*x + 143392768)
 
gp: K = bnfinit(x^16 - 5*x^15 - 28*x^14 + 29*x^13 + 824*x^12 + 2465*x^11 - 10425*x^10 - 72966*x^9 - 45536*x^8 + 794904*x^7 + 3196416*x^6 - 2423392*x^5 - 26235328*x^4 - 29006080*x^3 + 75011072*x^2 + 211959808*x + 143392768, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 28 x^{14} + 29 x^{13} + 824 x^{12} + 2465 x^{11} - 10425 x^{10} - 72966 x^{9} - 45536 x^{8} + 794904 x^{7} + 3196416 x^{6} - 2423392 x^{5} - 26235328 x^{4} - 29006080 x^{3} + 75011072 x^{2} + 211959808 x + 143392768 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3291419362365438198446597168601=3^{12}\cdot 59^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $80.79$
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:  $3, 59$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{1}{4} a^{4} - \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{9} - \frac{1}{16} a^{8} + \frac{1}{16} a^{6} - \frac{3}{16} a^{4} - \frac{1}{16} a^{3} + \frac{1}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{32} a^{10} - \frac{1}{32} a^{9} + \frac{1}{32} a^{7} + \frac{5}{32} a^{5} - \frac{1}{32} a^{4} + \frac{1}{16} a^{3} + \frac{1}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{11} - \frac{1}{64} a^{10} - \frac{3}{64} a^{8} - \frac{1}{16} a^{7} + \frac{1}{64} a^{6} + \frac{11}{64} a^{5} - \frac{5}{32} a^{4} - \frac{1}{8} a^{3} + \frac{1}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{8448} a^{12} - \frac{31}{8448} a^{11} + \frac{1}{384} a^{10} - \frac{131}{8448} a^{9} + \frac{47}{4224} a^{8} + \frac{17}{8448} a^{7} - \frac{763}{8448} a^{6} - \frac{7}{2112} a^{5} + \frac{35}{192} a^{4} + \frac{103}{528} a^{3} + \frac{169}{528} a^{2} - \frac{1}{33} a + \frac{7}{33}$, $\frac{1}{435139584} a^{13} + \frac{18965}{435139584} a^{12} + \frac{533213}{217569792} a^{11} - \frac{987755}{435139584} a^{10} - \frac{5532871}{217569792} a^{9} + \frac{218539}{39558144} a^{8} + \frac{18418313}{435139584} a^{7} - \frac{5576981}{54392448} a^{6} - \frac{12073043}{108784896} a^{5} + \frac{1450189}{27196224} a^{4} + \frac{354797}{2472384} a^{3} + \frac{271285}{3399528} a^{2} + \frac{108781}{424941} a - \frac{6310}{141647}$, $\frac{1}{157520529408} a^{14} - \frac{169}{157520529408} a^{13} - \frac{756925}{13126710784} a^{12} - \frac{200836663}{157520529408} a^{11} + \frac{18102999}{1193337344} a^{10} - \frac{120157417}{14320048128} a^{9} - \frac{1263646277}{157520529408} a^{8} + \frac{28669269}{2386674688} a^{7} + \frac{169866665}{39380132352} a^{6} - \frac{4467554969}{19690066176} a^{5} + \frac{734331727}{3281677696} a^{4} - \frac{314075557}{4922516544} a^{3} - \frac{772951}{2596264} a^{2} + \frac{125622527}{307657284} a + \frac{31652848}{76914321}$, $\frac{1}{8474576310377190324998455296} a^{15} - \frac{22916279013700781}{8474576310377190324998455296} a^{14} + \frac{727950776898201163}{2118644077594297581249613824} a^{13} - \frac{474056013740367882138467}{8474576310377190324998455296} a^{12} + \frac{677491292374561859128661}{529661019398574395312403456} a^{11} + \frac{30436354475357167810256779}{2824858770125730108332818432} a^{10} + \frac{146673977317082287742068031}{8474576310377190324998455296} a^{9} - \frac{114445460891970340684075375}{4237288155188595162499227648} a^{8} + \frac{10301767709468631957201921}{176553673132858131770801152} a^{7} + \frac{42427480227932154532889975}{1059322038797148790624806912} a^{6} + \frac{19237456207575397884252911}{132415254849643598828100864} a^{5} - \frac{20330482660048725651078325}{88276836566429065885400576} a^{4} + \frac{20557932512764884263083}{488617176566950549181184} a^{3} + \frac{3708872857727589065681913}{11034604570803633235675072} a^{2} + \frac{1742512032463470393050621}{8275953428102724926756304} a - \frac{84244201482943634139984}{172415696418806769307423}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}$, which has order $27$ (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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{23334861610667887}{225034555096449462943744} a^{15} + \frac{452962179942571889}{675103665289348388831232} a^{14} + \frac{107092420291690197}{56258638774112365735936} a^{13} - \frac{3958088725153454137}{675103665289348388831232} a^{12} - \frac{6395631024352236149}{84387958161168548603904} a^{11} - \frac{31802395606712753039}{225034555096449462943744} a^{10} + \frac{859993839186684066965}{675103665289348388831232} a^{9} + \frac{1878335590302236732879}{337551832644674194415616} a^{8} - \frac{576779072740385679}{159825678335546493568} a^{7} - \frac{6336397354757037426967}{84387958161168548603904} a^{6} - \frac{1118455996084671751211}{5274247385073034287744} a^{5} + \frac{3937303510010280724781}{7032329846764045716992} a^{4} + \frac{6336761583215832343}{3538575904108040448} a^{3} + \frac{19810964373074576251}{239738517503319740352} a^{2} - \frac{1667818117330302938635}{219760307711376428656} a - \frac{352335055177326930802}{41205057695883080373} \) (order $6$)
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:  \( 383035476.079 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$SD_{16}$ (as 16T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $QD_{16}$
Character table for $QD_{16}$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{177}) \), \(\Q(\sqrt{-59}) \), \(\Q(\sqrt{-3}, \sqrt{-59})\), 4.2.1848411.1 x2, 4.0.616137.1 x2, 8.0.3416623224921.1, 8.2.1814226932433051.1 x4

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ R

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
$3$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
59Data not computed