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Properties

Label 16.8.13735096585...9809.5
Degree $16$
Signature $[8, 4]$
Discriminant $13^{8}\cdot 17^{14}$
Root discriminant $43.01$
Ramified primes $13, 17$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^5.C_2.C_2$ (as 16T258)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2704, 14248, 4716, -33566, -6231, 30021, -81, -13429, 1579, 2854, -817, -281, 230, 18, -28, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 28*x^14 + 18*x^13 + 230*x^12 - 281*x^11 - 817*x^10 + 2854*x^9 + 1579*x^8 - 13429*x^7 - 81*x^6 + 30021*x^5 - 6231*x^4 - 33566*x^3 + 4716*x^2 + 14248*x + 2704)
 
gp: K = bnfinit(x^16 - x^15 - 28*x^14 + 18*x^13 + 230*x^12 - 281*x^11 - 817*x^10 + 2854*x^9 + 1579*x^8 - 13429*x^7 - 81*x^6 + 30021*x^5 - 6231*x^4 - 33566*x^3 + 4716*x^2 + 14248*x + 2704, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 28 x^{14} + 18 x^{13} + 230 x^{12} - 281 x^{11} - 817 x^{10} + 2854 x^{9} + 1579 x^{8} - 13429 x^{7} - 81 x^{6} + 30021 x^{5} - 6231 x^{4} - 33566 x^{3} + 4716 x^{2} + 14248 x + 2704 \)

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:  \(137350965859713069141239809=13^{8}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.01$
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:  $13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{188} a^{14} - \frac{1}{4} a^{13} - \frac{13}{94} a^{12} - \frac{2}{47} a^{11} - \frac{9}{94} a^{10} - \frac{33}{188} a^{9} - \frac{87}{188} a^{8} + \frac{11}{94} a^{7} - \frac{77}{188} a^{6} + \frac{27}{188} a^{5} + \frac{27}{188} a^{4} + \frac{69}{188} a^{3} - \frac{45}{188} a^{2} - \frac{14}{47} a - \frac{9}{47}$, $\frac{1}{149954294044795853660845582616} a^{15} - \frac{286827909998990066610983067}{149954294044795853660845582616} a^{14} + \frac{16007961348207820819731914243}{74977147022397926830422791308} a^{13} - \frac{9728963779219917141059233841}{74977147022397926830422791308} a^{12} - \frac{13900816335408464966300583373}{74977147022397926830422791308} a^{11} + \frac{957970930553204377007818339}{149954294044795853660845582616} a^{10} + \frac{12184437590660369909448262821}{149954294044795853660845582616} a^{9} - \frac{13852716821817838731553064559}{37488573511198963415211395654} a^{8} - \frac{16807262439093965170613006213}{149954294044795853660845582616} a^{7} - \frac{23962266461689739728137765}{245424376505394195844264456} a^{6} - \frac{18297743792772033253178658835}{149954294044795853660845582616} a^{5} - \frac{52199242452523542990219732993}{149954294044795853660845582616} a^{4} - \frac{70421795173419030779281575697}{149954294044795853660845582616} a^{3} + \frac{1138544140758241381296474983}{2883736423938381801170107358} a^{2} - \frac{11702782610754515921480797377}{37488573511198963415211395654} a - \frac{689618912912856057973689567}{1441868211969190900585053679}$

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:  $11$
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:  \( 9958822.6725 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2.C_2$ (as 16T258):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^5.C_2.C_2$
Character table for $C_2^5.C_2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 8.4.53030239093.1, 8.8.11719682839553.1, 8.4.5334402749.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$13$13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17Data not computed