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Properties

Label 16.8.14256966527...0352.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{50}\cdot 3^{8}\cdot 193$
Root discriminant $21.00$
Ramified primes $2, 3, 193$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1769

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, -16, 20, 136, -206, -88, 208, 24, -37, -80, -12, 32, 28, -8, -8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 - 8*x^13 + 28*x^12 + 32*x^11 - 12*x^10 - 80*x^9 - 37*x^8 + 24*x^7 + 208*x^6 - 88*x^5 - 206*x^4 + 136*x^3 + 20*x^2 - 16*x - 2)
 
gp: K = bnfinit(x^16 - 8*x^14 - 8*x^13 + 28*x^12 + 32*x^11 - 12*x^10 - 80*x^9 - 37*x^8 + 24*x^7 + 208*x^6 - 88*x^5 - 206*x^4 + 136*x^3 + 20*x^2 - 16*x - 2, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{14} - 8 x^{13} + 28 x^{12} + 32 x^{11} - 12 x^{10} - 80 x^{9} - 37 x^{8} + 24 x^{7} + 208 x^{6} - 88 x^{5} - 206 x^{4} + 136 x^{3} + 20 x^{2} - 16 x - 2 \)

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:  \(1425696652737330020352=2^{50}\cdot 3^{8}\cdot 193\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.00$
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:  $2, 3, 193$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} + \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{5555721678428} a^{15} + \frac{327564169565}{2777860839214} a^{14} - \frac{493258260461}{5555721678428} a^{13} - \frac{123067437737}{2777860839214} a^{12} + \frac{307227866639}{5555721678428} a^{11} + \frac{356407019395}{1388930419607} a^{10} - \frac{748537808211}{5555721678428} a^{9} - \frac{34984823426}{1388930419607} a^{8} - \frac{571324425618}{1388930419607} a^{7} - \frac{654923623594}{1388930419607} a^{6} + \frac{497166095453}{1388930419607} a^{5} - \frac{194000307958}{1388930419607} a^{4} + \frac{232311378401}{2777860839214} a^{3} - \frac{405604193720}{1388930419607} a^{2} + \frac{740980153113}{2777860839214} a - \frac{322148010484}{1388930419607}$

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:  \( 125590.485375 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1769:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16384
The 220 conjugacy class representatives for t16n1769 are not computed
Character table for t16n1769 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}, \sqrt{3})\), 8.6.2717908992.1

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 R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.25.71$x^{8} + 4 x^{6} + 2 x^{4} + 4 x^{2} + 16 x + 14$$8$$1$$25$$(((C_4 \times C_2): C_2):C_2):C_2$$[2, 3, 7/2, 4, 17/4]^{2}$
2.8.25.71$x^{8} + 4 x^{6} + 2 x^{4} + 4 x^{2} + 16 x + 14$$8$$1$$25$$(((C_4 \times C_2): C_2):C_2):C_2$$[2, 3, 7/2, 4, 17/4]^{2}$
3Data not computed
193Data not computed