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Properties

Label 16.8.11654783511...6241.1
Degree $16$
Signature $[8, 4]$
Discriminant $13^{12}\cdot 29^{8}$
Root discriminant $36.87$
Ramified primes $13, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2.D_4$ (as 16T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-576, -3360, 992, 6520, -5048, -6162, 5383, 1850, -1774, 51, -77, 83, 53, -33, 2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 2*x^14 - 33*x^13 + 53*x^12 + 83*x^11 - 77*x^10 + 51*x^9 - 1774*x^8 + 1850*x^7 + 5383*x^6 - 6162*x^5 - 5048*x^4 + 6520*x^3 + 992*x^2 - 3360*x - 576)
 
gp: K = bnfinit(x^16 - 2*x^15 + 2*x^14 - 33*x^13 + 53*x^12 + 83*x^11 - 77*x^10 + 51*x^9 - 1774*x^8 + 1850*x^7 + 5383*x^6 - 6162*x^5 - 5048*x^4 + 6520*x^3 + 992*x^2 - 3360*x - 576, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 2 x^{14} - 33 x^{13} + 53 x^{12} + 83 x^{11} - 77 x^{10} + 51 x^{9} - 1774 x^{8} + 1850 x^{7} + 5383 x^{6} - 6162 x^{5} - 5048 x^{4} + 6520 x^{3} + 992 x^{2} - 3360 x - 576 \)

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:  \(11654783511381160590876241=13^{12}\cdot 29^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.87$
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:  $13, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{6} a$, $\frac{1}{12} a^{10} - \frac{1}{12} a^{9} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} + \frac{5}{12} a^{2} - \frac{1}{6} a$, $\frac{1}{12} a^{11} - \frac{1}{12} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} + \frac{1}{6} a^{3} + \frac{1}{4} a^{2} - \frac{1}{6} a$, $\frac{1}{72} a^{12} + \frac{1}{36} a^{11} - \frac{1}{36} a^{10} - \frac{1}{72} a^{9} - \frac{1}{24} a^{8} - \frac{1}{8} a^{7} + \frac{5}{24} a^{6} + \frac{1}{24} a^{5} + \frac{1}{36} a^{4} + \frac{5}{36} a^{3} + \frac{11}{72} a^{2} - \frac{13}{36} a - \frac{1}{6}$, $\frac{1}{144} a^{13} - \frac{1}{24} a^{11} + \frac{1}{48} a^{10} + \frac{11}{144} a^{9} + \frac{11}{48} a^{8} - \frac{1}{48} a^{7} - \frac{3}{16} a^{6} + \frac{17}{36} a^{5} + \frac{7}{24} a^{4} + \frac{3}{16} a^{3} - \frac{1}{12} a^{2} - \frac{1}{18} a + \frac{1}{6}$, $\frac{1}{145728} a^{14} - \frac{85}{36432} a^{13} - \frac{181}{72864} a^{12} + \frac{659}{145728} a^{11} - \frac{707}{48576} a^{10} + \frac{1747}{48576} a^{9} + \frac{6551}{48576} a^{8} + \frac{3079}{48576} a^{7} - \frac{8491}{36432} a^{6} + \frac{11255}{72864} a^{5} + \frac{1}{13248} a^{4} + \frac{295}{2277} a^{3} - \frac{333}{4048} a^{2} - \frac{25}{66} a - \frac{265}{1012}$, $\frac{1}{3161500833034368} a^{15} + \frac{4220154683}{1580750416517184} a^{14} - \frac{275904944855}{526916805505728} a^{13} - \frac{15227346880369}{3161500833034368} a^{12} + \frac{14191788313255}{1053833611011456} a^{11} + \frac{127705446040471}{3161500833034368} a^{10} + \frac{216942698160391}{3161500833034368} a^{9} - \frac{2827875056255}{15272950884224} a^{8} - \frac{7180571555}{634584671424} a^{7} + \frac{780734529811543}{1580750416517184} a^{6} + \frac{134117729733023}{351277870337152} a^{5} - \frac{447339078375385}{1580750416517184} a^{4} - \frac{33129623514013}{263458402752864} a^{3} + \frac{165953502262909}{395187604129296} a^{2} + \frac{76874340142937}{197593802064648} a - \frac{8294874976735}{32932300344108}$

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:  \( 19014964.3925 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.D_4$ (as 16T33):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^2.D_4$
Character table for $C_2^2.D_4$

Intermediate fields

\(\Q(\sqrt{377}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{29}) \), 4.4.1847677.1 x2, 4.4.63713.1 x2, \(\Q(\sqrt{13}, \sqrt{29})\), 8.4.262608484333.1 x2, 8.4.3413910296329.1 x2, 8.8.3413910296329.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$