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Properties

Label 16.0.168110601127133184.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{12}\cdot 13^{6}$
Root discriminant $11.93$
Ramified primes $2, 3, 13$
Class number $1$
Class group Trivial
Galois group $C_4^2:C_2^2.C_2$ (as 16T382)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 1, -14, 25, -12, -20, 60, -89, 68, 16, -98, 113, -74, 31, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 31*x^14 - 74*x^13 + 113*x^12 - 98*x^11 + 16*x^10 + 68*x^9 - 89*x^8 + 60*x^7 - 20*x^6 - 12*x^5 + 25*x^4 - 14*x^3 + x^2 + 1)
 
gp: K = bnfinit(x^16 - 8*x^15 + 31*x^14 - 74*x^13 + 113*x^12 - 98*x^11 + 16*x^10 + 68*x^9 - 89*x^8 + 60*x^7 - 20*x^6 - 12*x^5 + 25*x^4 - 14*x^3 + x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 31 x^{14} - 74 x^{13} + 113 x^{12} - 98 x^{11} + 16 x^{10} + 68 x^{9} - 89 x^{8} + 60 x^{7} - 20 x^{6} - 12 x^{5} + 25 x^{4} - 14 x^{3} + x^{2} + 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(168110601127133184=2^{16}\cdot 3^{12}\cdot 13^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $11.93$
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:  $2, 3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{10511123} a^{15} - \frac{648492}{10511123} a^{14} - \frac{2833948}{10511123} a^{13} + \frac{5189438}{10511123} a^{12} + \frac{737310}{1501589} a^{11} + \frac{729737}{1501589} a^{10} + \frac{10264}{10511123} a^{9} - \frac{2498849}{10511123} a^{8} - \frac{672102}{1501589} a^{7} + \frac{4725025}{10511123} a^{6} + \frac{4864733}{10511123} a^{5} + \frac{1831206}{10511123} a^{4} - \frac{3159631}{10511123} a^{3} - \frac{5101492}{10511123} a^{2} - \frac{3381522}{10511123} a + \frac{899019}{10511123}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{26526}{79031} a^{15} + \frac{169394}{79031} a^{14} - \frac{508366}{79031} a^{13} + \frac{847288}{79031} a^{12} - \frac{584430}{79031} a^{11} - \frac{613227}{79031} a^{10} + \frac{1658582}{79031} a^{9} - \frac{1281087}{79031} a^{8} + \frac{181774}{79031} a^{7} + \frac{356184}{79031} a^{6} - \frac{564944}{79031} a^{5} + \frac{490267}{79031} a^{4} - \frac{161656}{79031} a^{3} - \frac{233578}{79031} a^{2} + \frac{122378}{79031} a + \frac{68194}{79031} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 213.428367804 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2:C_2^2.C_2$ (as 16T382):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_4^2:C_2^2.C_2$
Character table for $C_4^2:C_2^2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.5616.2, 4.0.117.1, 4.0.432.1, 8.0.31539456.2

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 R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
3Data not computed
$13$13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$