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Properties

Label 16.0.83646323707...5776.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 41^{7}$
Root discriminant $20.31$
Ramified primes $2, 41$
Class number $1$
Class group Trivial
Galois group $D_{16}$ (as 16T56)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![41, 0, 44, 0, 62, 0, 0, 0, 19, 0, 0, 0, 6, 0, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 + 6*x^12 + 19*x^8 + 62*x^4 + 44*x^2 + 41)
 
gp: K = bnfinit(x^16 - 4*x^14 + 6*x^12 + 19*x^8 + 62*x^4 + 44*x^2 + 41, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{14} + 6 x^{12} + 19 x^{8} + 62 x^{4} + 44 x^{2} + 41 \)

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:  \(836463237075121995776=2^{32}\cdot 41^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.31$
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:  $2, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4} + \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5} + \frac{1}{4} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7} + \frac{1}{4} a^{3}$, $\frac{1}{4} a^{12} + \frac{1}{4}$, $\frac{1}{4} a^{13} + \frac{1}{4} a$, $\frac{1}{166244} a^{14} + \frac{4547}{166244} a^{12} - \frac{3975}{166244} a^{10} - \frac{5595}{83122} a^{8} + \frac{28115}{166244} a^{6} - \frac{7477}{83122} a^{4} + \frac{10663}{83122} a^{2} - \frac{73387}{166244}$, $\frac{1}{166244} a^{15} + \frac{4547}{166244} a^{13} - \frac{3975}{166244} a^{11} - \frac{5595}{83122} a^{9} + \frac{28115}{166244} a^{7} - \frac{7477}{83122} a^{5} + \frac{10663}{83122} a^{3} - \frac{73387}{166244} a$

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:  \( -\frac{59}{3197} a^{14} + \frac{275}{3197} a^{12} - \frac{909}{6394} a^{10} + \frac{59}{6394} a^{8} - \frac{2281}{6394} a^{6} + \frac{3025}{6394} a^{4} - \frac{6823}{6394} a^{2} - \frac{1007}{6394} \) (order $4$)
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:  \( 16401.7169889 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{16}$ (as 16T56):

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 $D_{16}$
Character table for $D_{16}$

Intermediate fields

\(\Q(\sqrt{-1}) \), 4.0.656.1, 8.0.17643776.1

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 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 R $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.16.5$x^{8} + 4 x^{6} + 40 x^{2} + 4$$4$$2$$16$$D_4$$[2, 3]^{2}$
2.8.16.5$x^{8} + 4 x^{6} + 40 x^{2} + 4$$4$$2$$16$$D_4$$[2, 3]^{2}$
$41$41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$