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Properties

Label 16.4.50581108116...4889.1
Degree $16$
Signature $[4, 6]$
Discriminant $3^{6}\cdot 97^{12}$
Root discriminant $46.67$
Ramified primes $3, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4\wr C_2$ (as 16T28)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2281, 3960, 7273, -2167, -12165, 5149, 5725, -3845, 493, 255, -295, 191, -117, 31, 5, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 5*x^14 + 31*x^13 - 117*x^12 + 191*x^11 - 295*x^10 + 255*x^9 + 493*x^8 - 3845*x^7 + 5725*x^6 + 5149*x^5 - 12165*x^4 - 2167*x^3 + 7273*x^2 + 3960*x - 2281)
 
gp: K = bnfinit(x^16 - 4*x^15 + 5*x^14 + 31*x^13 - 117*x^12 + 191*x^11 - 295*x^10 + 255*x^9 + 493*x^8 - 3845*x^7 + 5725*x^6 + 5149*x^5 - 12165*x^4 - 2167*x^3 + 7273*x^2 + 3960*x - 2281, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 5 x^{14} + 31 x^{13} - 117 x^{12} + 191 x^{11} - 295 x^{10} + 255 x^{9} + 493 x^{8} - 3845 x^{7} + 5725 x^{6} + 5149 x^{5} - 12165 x^{4} - 2167 x^{3} + 7273 x^{2} + 3960 x - 2281 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(505811081165674302215084889=3^{6}\cdot 97^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.67$
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:  $3, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} + \frac{1}{7} a^{11} + \frac{1}{7} a^{9} + \frac{3}{7} a^{8} + \frac{2}{7} a^{7} + \frac{1}{7} a^{6} + \frac{1}{7} a^{4} + \frac{1}{7} a^{3} + \frac{3}{7} a^{2} + \frac{3}{7} a + \frac{3}{7}$, $\frac{1}{7} a^{13} - \frac{1}{7} a^{11} + \frac{1}{7} a^{10} + \frac{2}{7} a^{9} - \frac{1}{7} a^{8} - \frac{1}{7} a^{7} - \frac{1}{7} a^{6} + \frac{1}{7} a^{5} + \frac{2}{7} a^{3} - \frac{3}{7}$, $\frac{1}{14} a^{14} - \frac{1}{14} a^{13} - \frac{1}{14} a^{12} - \frac{5}{14} a^{11} + \frac{1}{14} a^{10} - \frac{3}{14} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{5}{14} a^{6} - \frac{1}{14} a^{5} - \frac{5}{14} a^{4} + \frac{5}{14} a^{3} - \frac{1}{2} a^{2} - \frac{3}{14} a + \frac{3}{14}$, $\frac{1}{11341246984177883374674434194} a^{15} + \frac{20123850433241942642677395}{1620178140596840482096347742} a^{14} - \frac{247409755652015932435648865}{11341246984177883374674434194} a^{13} - \frac{786078984820363626773517907}{11341246984177883374674434194} a^{12} + \frac{4287999397625126331265654709}{11341246984177883374674434194} a^{11} + \frac{764158774227632228562774477}{11341246984177883374674434194} a^{10} - \frac{81073948609341745529319545}{241303127322933688822860302} a^{9} - \frac{4107758209633064111396492805}{11341246984177883374674434194} a^{8} - \frac{1284302229358823285628319679}{11341246984177883374674434194} a^{7} - \frac{3547230693910172396732035905}{11341246984177883374674434194} a^{6} - \frac{5019531622836811217691933031}{11341246984177883374674434194} a^{5} + \frac{1502566688550428122220298453}{11341246984177883374674434194} a^{4} + \frac{960955909812437853603284321}{11341246984177883374674434194} a^{3} + \frac{4361818807922954254184534021}{11341246984177883374674434194} a^{2} + \frac{5398098287387717484173121501}{11341246984177883374674434194} a + \frac{2632916073518631958606783825}{5670623492088941687337217097}$

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:  $9$
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:  \( 9364641.51422 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\wr C_2$ (as 16T28):

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

\(\Q(\sqrt{97}) \), 4.2.28227.1, 4.4.912673.1, 4.2.2738019.1, 8.4.7496748044361.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
97Data not computed