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Properties

Label 16.8.70704738646...3209.2
Degree $16$
Signature $[8, 4]$
Discriminant $67^{8}\cdot 89^{15}$
Root discriminant $550.29$
Ramified primes $67, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_2\times OD_{16}).D_4$ (as 16T591)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5144433664, 14280858112, 15103522624, 4871115840, -1949248180, -1501317600, -174579123, 43053882, 12101655, 4115620, 715973, -79038, -9147, 1320, -165, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 165*x^14 + 1320*x^13 - 9147*x^12 - 79038*x^11 + 715973*x^10 + 4115620*x^9 + 12101655*x^8 + 43053882*x^7 - 174579123*x^6 - 1501317600*x^5 - 1949248180*x^4 + 4871115840*x^3 + 15103522624*x^2 + 14280858112*x + 5144433664)
 
gp: K = bnfinit(x^16 - 2*x^15 - 165*x^14 + 1320*x^13 - 9147*x^12 - 79038*x^11 + 715973*x^10 + 4115620*x^9 + 12101655*x^8 + 43053882*x^7 - 174579123*x^6 - 1501317600*x^5 - 1949248180*x^4 + 4871115840*x^3 + 15103522624*x^2 + 14280858112*x + 5144433664, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 165 x^{14} + 1320 x^{13} - 9147 x^{12} - 79038 x^{11} + 715973 x^{10} + 4115620 x^{9} + 12101655 x^{8} + 43053882 x^{7} - 174579123 x^{6} - 1501317600 x^{5} - 1949248180 x^{4} + 4871115840 x^{3} + 15103522624 x^{2} + 14280858112 x + 5144433664 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(70704738646532877581735658786300135551283209=67^{8}\cdot 89^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $550.29$
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:  $67, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{8} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{8} - \frac{1}{4} a^{6} + \frac{1}{8} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{3}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{9} - \frac{1}{8} a^{6} + \frac{1}{8} a^{3}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{11} - \frac{1}{16} a^{9} - \frac{1}{8} a^{8} + \frac{1}{16} a^{7} - \frac{1}{8} a^{6} - \frac{1}{16} a^{5} - \frac{1}{8} a^{4} - \frac{1}{16} a^{3} - \frac{1}{4} a$, $\frac{1}{313984} a^{14} - \frac{2479}{156992} a^{13} + \frac{8675}{313984} a^{12} + \frac{303}{7136} a^{11} - \frac{19147}{313984} a^{10} - \frac{11877}{156992} a^{9} - \frac{16035}{313984} a^{8} - \frac{4709}{39248} a^{7} + \frac{76215}{313984} a^{6} - \frac{8485}{156992} a^{5} + \frac{4421}{313984} a^{4} + \frac{35525}{78496} a^{3} - \frac{34761}{78496} a^{2} + \frac{7745}{19624} a + \frac{245}{4906}$, $\frac{1}{200428920475136179304434074395238117202045096025610988711618048} a^{15} + \frac{87076359231306196896443029106249988481931055413120597967}{100214460237568089652217037197619058601022548012805494355809024} a^{14} + \frac{2346480192776539875696158152580091727952730366571428820566635}{200428920475136179304434074395238117202045096025610988711618048} a^{13} - \frac{344949740966881771883960729124798804252915768463667370733771}{25053615059392022413054259299404764650255637003201373588952256} a^{12} + \frac{3904235506741358343303885907404800341410862719078036950521461}{200428920475136179304434074395238117202045096025610988711618048} a^{11} - \frac{4185401049907167295329818521197630044946290932797917301219823}{100214460237568089652217037197619058601022548012805494355809024} a^{10} + \frac{15754188153761197169350125788008276245014219651451532339733909}{200428920475136179304434074395238117202045096025610988711618048} a^{9} + \frac{1407270349778588421879771448006037053387141230161283045162105}{50107230118784044826108518598809529300511274006402747177904512} a^{8} - \frac{15084417397605981695575305116802992928518196442937447680287577}{200428920475136179304434074395238117202045096025610988711618048} a^{7} - \frac{22860703755238531316319059597200056859798287281258883410490675}{100214460237568089652217037197619058601022548012805494355809024} a^{6} + \frac{3626474898169506498104008445778292695556984687057557807964925}{200428920475136179304434074395238117202045096025610988711618048} a^{5} + \frac{46000979633961419640284814147473875686375254755257203173291}{569400342258909600296687711350108287505809931890940308839824} a^{4} - \frac{334836315226600655359355766987655422693805241970126827135899}{4555202738071276802373501690800866300046479455127522470718592} a^{3} - \frac{1099268141884046576940429631702196070576234499257048117487459}{3131701882424002801631782412425595581281954625400171698619032} a^{2} + \frac{454507778450766411017837378160927810744888420647352690908585}{1565850941212001400815891206212797790640977312700085849309516} a - \frac{356350498666534346300971106640487771366586617371002102909751}{782925470606000700407945603106398895320488656350042924654758}$

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:  $11$
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:  \( 118751746524000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times OD_{16}).D_4$ (as 16T591):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 28 conjugacy class representatives for $(C_2\times OD_{16}).D_4$
Character table for $(C_2\times OD_{16}).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{89}) \), 4.4.704969.1, 8.8.891310981471327238009.1

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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
$67$67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89Data not computed