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Properties

Label 16.16.1998750580...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{16}\cdot 5^{8}\cdot 29^{6}\cdot 139^{4}\cdot 181^{6}$
Root discriminant $381.33$
Ramified primes $2, 5, 29, 139, 181$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T868

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10285190654933041, 0, -2822816545662095, 0, 230917087201933, 0, -7216691979704, 0, 80387048099, 0, -430086456, 0, 1211156, 0, -1736, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 1736*x^14 + 1211156*x^12 - 430086456*x^10 + 80387048099*x^8 - 7216691979704*x^6 + 230917087201933*x^4 - 2822816545662095*x^2 + 10285190654933041)
 
gp: K = bnfinit(x^16 - 1736*x^14 + 1211156*x^12 - 430086456*x^10 + 80387048099*x^8 - 7216691979704*x^6 + 230917087201933*x^4 - 2822816545662095*x^2 + 10285190654933041, 1)
 

Normalized defining polynomial

\( x^{16} - 1736 x^{14} + 1211156 x^{12} - 430086456 x^{10} + 80387048099 x^{8} - 7216691979704 x^{6} + 230917087201933 x^{4} - 2822816545662095 x^{2} + 10285190654933041 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(199875058073008840373089541044249600000000=2^{16}\cdot 5^{8}\cdot 29^{6}\cdot 139^{4}\cdot 181^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $381.33$
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:  $2, 5, 29, 139, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{4031} a^{10} + \frac{210}{4031} a^{8} - \frac{646}{4031} a^{6} + \frac{1645}{4031} a^{4} + \frac{5}{139} a^{2}$, $\frac{1}{4031} a^{11} + \frac{210}{4031} a^{9} - \frac{646}{4031} a^{7} + \frac{1645}{4031} a^{5} + \frac{5}{139} a^{3}$, $\frac{1}{30816995} a^{12} + \frac{3268}{30816995} a^{10} - \frac{13426656}{30816995} a^{8} + \frac{2041053}{30816995} a^{6} + \frac{1437428}{6163399} a^{4} - \frac{661}{7645} a^{2} + \frac{6}{55}$, $\frac{1}{30816995} a^{13} + \frac{3268}{30816995} a^{11} - \frac{13426656}{30816995} a^{9} + \frac{2041053}{30816995} a^{7} + \frac{1437428}{6163399} a^{5} - \frac{661}{7645} a^{3} + \frac{6}{55} a$, $\frac{1}{10679128075171499684454177445275477482585} a^{14} - \frac{2868536367707199908898369750206}{10679128075171499684454177445275477482585} a^{12} + \frac{113001077194640601337343072535768137}{970829825015590880404925222297770680235} a^{10} + \frac{292763368409889923257841923007992013864}{821471390397807668034936726559652114045} a^{8} + \frac{1022228473151274501811413387581109418653}{10679128075171499684454177445275477482585} a^{6} - \frac{7191680251077226689555924074958684799}{76828259533607911398950916872485449515} a^{4} - \frac{9054221437054025105376400966354893}{19059354883058276209117071910812565} a^{2} - \frac{288589354062958783302521838716}{757556138282852108951749748035}$, $\frac{1}{10679128075171499684454177445275477482585} a^{15} - \frac{2868536367707199908898369750206}{10679128075171499684454177445275477482585} a^{13} + \frac{113001077194640601337343072535768137}{970829825015590880404925222297770680235} a^{11} + \frac{292763368409889923257841923007992013864}{821471390397807668034936726559652114045} a^{9} + \frac{1022228473151274501811413387581109418653}{10679128075171499684454177445275477482585} a^{7} - \frac{7191680251077226689555924074958684799}{76828259533607911398950916872485449515} a^{5} - \frac{9054221437054025105376400966354893}{19059354883058276209117071910812565} a^{3} - \frac{288589354062958783302521838716}{757556138282852108951749748035} a$

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:  $15$
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:  \( 3446021166400000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T868:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 53 conjugacy class representatives for t16n868 are not computed
Character table for t16n868 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.4525.1, 4.4.131225.1, 4.4.725.1, 8.8.17220000625.2

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.8.9$x^{8} + 6 x^{6} + 4 x^{5} + 16$$2$$4$$8$$((C_8 : C_2):C_2):C_2$$[2, 2, 2, 2]^{4}$
2.8.8.10$x^{8} + 2 x^{6} + 8 x^{3} + 16$$2$$4$$8$$((C_8 : C_2):C_2):C_2$$[2, 2, 2, 2]^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.8.6.1$x^{8} - 203 x^{4} + 68121$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$139$$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
139.2.1.1$x^{2} - 139$$2$$1$$1$$C_2$$[\ ]_{2}$
139.2.1.1$x^{2} - 139$$2$$1$$1$$C_2$$[\ ]_{2}$
139.2.1.1$x^{2} - 139$$2$$1$$1$$C_2$$[\ ]_{2}$
139.2.1.1$x^{2} - 139$$2$$1$$1$$C_2$$[\ ]_{2}$
$181$181.4.2.2$x^{4} - 181 x^{2} + 589698$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
181.4.0.1$x^{4} - x + 54$$1$$4$$0$$C_4$$[\ ]^{4}$
181.8.4.1$x^{8} + 3538188 x^{4} - 5929741 x^{2} + 3129693580836$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$