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Properties

Label 16.6.52166656000...0000.1
Degree $16$
Signature $[6, 5]$
Discriminant $-\,2^{32}\cdot 5^{14}\cdot 199$
Root discriminant $22.77$
Ramified primes $2, 5, 199$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1604

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -16, 64, -36, -14, 32, -114, 172, -168, 88, -34, -12, 31, -24, 14, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 14*x^14 - 24*x^13 + 31*x^12 - 12*x^11 - 34*x^10 + 88*x^9 - 168*x^8 + 172*x^7 - 114*x^6 + 32*x^5 - 14*x^4 - 36*x^3 + 64*x^2 - 16*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 + 14*x^14 - 24*x^13 + 31*x^12 - 12*x^11 - 34*x^10 + 88*x^9 - 168*x^8 + 172*x^7 - 114*x^6 + 32*x^5 - 14*x^4 - 36*x^3 + 64*x^2 - 16*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 14 x^{14} - 24 x^{13} + 31 x^{12} - 12 x^{11} - 34 x^{10} + 88 x^{9} - 168 x^{8} + 172 x^{7} - 114 x^{6} + 32 x^{5} - 14 x^{4} - 36 x^{3} + 64 x^{2} - 16 x + 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-5216665600000000000000=-\,2^{32}\cdot 5^{14}\cdot 199\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.77$
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:  $2, 5, 199$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{42012296877701} a^{15} + \frac{484632787536}{42012296877701} a^{14} - \frac{7749062395439}{42012296877701} a^{13} - \frac{648391435261}{2211173519879} a^{12} + \frac{13599187807198}{42012296877701} a^{11} + \frac{4557531754481}{42012296877701} a^{10} + \frac{1018404014563}{2211173519879} a^{9} + \frac{4834006265137}{42012296877701} a^{8} - \frac{2843909611600}{42012296877701} a^{7} + \frac{11781713347026}{42012296877701} a^{6} + \frac{9210052667605}{42012296877701} a^{5} - \frac{8578980940184}{42012296877701} a^{4} + \frac{6128286044778}{42012296877701} a^{3} + \frac{11477960823418}{42012296877701} a^{2} + \frac{12811050166243}{42012296877701} a + \frac{19825925414151}{42012296877701}$

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:  $10$
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:  \( 46308.7258106 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1604:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 73 conjugacy class representatives for t16n1604 are not computed
Character table for t16n1604 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.8.5120000000.1

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 $16$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
5Data not computed
$199$$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
199.2.1.1$x^{2} - 199$$2$$1$$1$$C_2$$[\ ]_{2}$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$