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Properties

Label 16.0.16672722350...1257.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 73^{11}$
Root discriminant $43.54$
Ramified primes $3, 73$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T289)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![96091, 1089775, 3313449, 779238, -198259, 511794, 19031, -129668, 113688, -41890, 20123, -5493, 1505, -159, 54, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 54*x^14 - 159*x^13 + 1505*x^12 - 5493*x^11 + 20123*x^10 - 41890*x^9 + 113688*x^8 - 129668*x^7 + 19031*x^6 + 511794*x^5 - 198259*x^4 + 779238*x^3 + 3313449*x^2 + 1089775*x + 96091)
 
gp: K = bnfinit(x^16 - x^15 + 54*x^14 - 159*x^13 + 1505*x^12 - 5493*x^11 + 20123*x^10 - 41890*x^9 + 113688*x^8 - 129668*x^7 + 19031*x^6 + 511794*x^5 - 198259*x^4 + 779238*x^3 + 3313449*x^2 + 1089775*x + 96091, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 54 x^{14} - 159 x^{13} + 1505 x^{12} - 5493 x^{11} + 20123 x^{10} - 41890 x^{9} + 113688 x^{8} - 129668 x^{7} + 19031 x^{6} + 511794 x^{5} - 198259 x^{4} + 779238 x^{3} + 3313449 x^{2} + 1089775 x + 96091 \)

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:  \(166727223505134651779581257=3^{12}\cdot 73^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.54$
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:  $3, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{15944001882826891605015855890443385776973598393931372} a^{15} + \frac{382922998971708717499424257466800239755003577867824}{3986000470706722901253963972610846444243399598482843} a^{14} - \frac{1122162430397489422591043183758630814873239490453105}{15944001882826891605015855890443385776973598393931372} a^{13} - \frac{74179332772970290521644425634308067194089415392095}{3986000470706722901253963972610846444243399598482843} a^{12} - \frac{3605999852240305250868449342709503015219688617384263}{15944001882826891605015855890443385776973598393931372} a^{11} + \frac{785960053830846822542657350974223149103320234373225}{15944001882826891605015855890443385776973598393931372} a^{10} + \frac{1851718725235681098066010288369895179069318693027353}{15944001882826891605015855890443385776973598393931372} a^{9} - \frac{535030461436822705885771514327225193302086091198766}{3986000470706722901253963972610846444243399598482843} a^{8} + \frac{2625236144701137815134365435149032575006115823320355}{7972000941413445802507927945221692888486799196965686} a^{7} + \frac{816049289331053448863211557476204853152235429170459}{3986000470706722901253963972610846444243399598482843} a^{6} + \frac{105061697109574702479959279718216659898074452004347}{3986000470706722901253963972610846444243399598482843} a^{5} + \frac{2253918642367193964619467637807081059437267830457077}{15944001882826891605015855890443385776973598393931372} a^{4} + \frac{5941754990037734709961629213879043684427042421377319}{15944001882826891605015855890443385776973598393931372} a^{3} - \frac{3313157193663124789083071711513192314829251769677745}{15944001882826891605015855890443385776973598393931372} a^{2} - \frac{325787143072384129437550697684476244844193974165846}{3986000470706722901253963972610846444243399598482843} a + \frac{573850710078032489184202238393437427059244022906079}{3986000470706722901253963972610846444243399598482843}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( -\frac{24044038594749408129669881}{2288826930690371296340103643804} a^{15} + \frac{28202609004671488318998647}{2288826930690371296340103643804} a^{14} - \frac{1302186431797809253272988501}{2288826930690371296340103643804} a^{13} + \frac{1011824049938208728120283256}{572206732672592824085025910951} a^{12} - \frac{9207391087410548145199112410}{572206732672592824085025910951} a^{11} + \frac{69137658173772289220924683053}{1144413465345185648170051821902} a^{10} - \frac{253106229127918654084804218135}{1144413465345185648170051821902} a^{9} + \frac{544415766585550010555832283013}{1144413465345185648170051821902} a^{8} - \frac{1450833688481110115579626311213}{1144413465345185648170051821902} a^{7} + \frac{3571008388413820000724812061217}{2288826930690371296340103643804} a^{6} - \frac{947166634141291610920057177257}{2288826930690371296340103643804} a^{5} - \frac{3093118903937836236966404090620}{572206732672592824085025910951} a^{4} + \frac{7044821845814271717624831659623}{2288826930690371296340103643804} a^{3} - \frac{19910447726642962643749898762207}{2288826930690371296340103643804} a^{2} - \frac{76028999015782248435995045332733}{2288826930690371296340103643804} a - \frac{11611918822440583865138265344083}{2288826930690371296340103643804} \) (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:  \( 3000100.42501 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T289):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 44 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.657.1, 8.0.31510377.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ $16$ $16$ $16$

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
3Data not computed
$73$73.8.7.8$x^{8} + 5703125$$8$$1$$7$$C_8$$[\ ]_{8}$
73.8.4.2$x^{8} - 389017 x^{2} + 369177133$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$