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Properties

Label 16.8.14266108229...8329.4
Degree $16$
Signature $[8, 4]$
Discriminant $13^{8}\cdot 53^{10}$
Root discriminant $43.12$
Ramified primes $13, 53$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4\wr C_2$ (as 16T42)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![211, -184, 1890, 1377, -4826, 2351, 4219, -6413, -973, 3913, -284, -1002, 150, 124, -27, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 27*x^14 + 124*x^13 + 150*x^12 - 1002*x^11 - 284*x^10 + 3913*x^9 - 973*x^8 - 6413*x^7 + 4219*x^6 + 2351*x^5 - 4826*x^4 + 1377*x^3 + 1890*x^2 - 184*x + 211)
 
gp: K = bnfinit(x^16 - 4*x^15 - 27*x^14 + 124*x^13 + 150*x^12 - 1002*x^11 - 284*x^10 + 3913*x^9 - 973*x^8 - 6413*x^7 + 4219*x^6 + 2351*x^5 - 4826*x^4 + 1377*x^3 + 1890*x^2 - 184*x + 211, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 27 x^{14} + 124 x^{13} + 150 x^{12} - 1002 x^{11} - 284 x^{10} + 3913 x^{9} - 973 x^{8} - 6413 x^{7} + 4219 x^{6} + 2351 x^{5} - 4826 x^{4} + 1377 x^{3} + 1890 x^{2} - 184 x + 211 \)

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:  \(142661082295126092995678329=13^{8}\cdot 53^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.12$
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:  $13, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{8041205879938884431127441} a^{15} + \frac{206016118478189652247435}{2680401959979628143709147} a^{14} - \frac{1995409824323939733066088}{8041205879938884431127441} a^{13} + \frac{3170702344458675232669970}{8041205879938884431127441} a^{12} - \frac{257579085092228692510810}{8041205879938884431127441} a^{11} + \frac{1217728249113629995928793}{2680401959979628143709147} a^{10} + \frac{1333159130281927689328109}{2680401959979628143709147} a^{9} - \frac{282159079307784525133486}{2680401959979628143709147} a^{8} + \frac{1945990287627962532463384}{8041205879938884431127441} a^{7} + \frac{752670222087231609623311}{8041205879938884431127441} a^{6} - \frac{144941974360175269090844}{2680401959979628143709147} a^{5} + \frac{3103612808295137256853847}{8041205879938884431127441} a^{4} - \frac{803087772945720405362719}{2680401959979628143709147} a^{3} + \frac{2832829082498381641771582}{8041205879938884431127441} a^{2} + \frac{2026897308819528169129001}{8041205879938884431127441} a - \frac{3059581055893392184146560}{8041205879938884431127441}$

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:  \( 13402915.644 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\wr C_2$ (as 16T42):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_4\wr C_2$
Character table for $C_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{53}) \), \(\Q(\sqrt{689}) \), 4.4.8957.1 x2, 4.4.36517.1 x2, \(\Q(\sqrt{13}, \sqrt{53})\), 8.4.11944081475573.3, 8.4.4252075997.1, 8.8.225360027841.1

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 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
53Data not computed