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Properties

Label 16.0.14958734309...5296.5
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{12}$
Root discriminant $18.24$
Ramified primes $2, 3$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2:D_4$ (as 16T43)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 4, 0, 22, 0, 76, 0, 118, 0, 76, 0, 22, 0, 4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 4*x^14 + 22*x^12 + 76*x^10 + 118*x^8 + 76*x^6 + 22*x^4 + 4*x^2 + 1)
 
gp: K = bnfinit(x^16 + 4*x^14 + 22*x^12 + 76*x^10 + 118*x^8 + 76*x^6 + 22*x^4 + 4*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{16} + 4 x^{14} + 22 x^{12} + 76 x^{10} + 118 x^{8} + 76 x^{6} + 22 x^{4} + 4 x^{2} + 1 \)

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:  \(149587343098087735296=2^{48}\cdot 3^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.24$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{6} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{2} + \frac{1}{6}$, $\frac{1}{6} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{3} + \frac{1}{6} a$, $\frac{1}{6} a^{10} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{2} a^{2} - \frac{1}{3}$, $\frac{1}{6} a^{11} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{2} a^{3} - \frac{1}{3} a$, $\frac{1}{18} a^{12} + \frac{2}{9} a^{6} + \frac{1}{6} a^{4} - \frac{1}{3} a^{2} - \frac{1}{9}$, $\frac{1}{18} a^{13} + \frac{2}{9} a^{7} + \frac{1}{6} a^{5} - \frac{1}{3} a^{3} - \frac{1}{9} a$, $\frac{1}{54} a^{14} - \frac{1}{54} a^{12} + \frac{2}{27} a^{8} - \frac{1}{54} a^{6} + \frac{1}{6} a^{4} + \frac{11}{27} a^{2} + \frac{10}{27}$, $\frac{1}{54} a^{15} - \frac{1}{54} a^{13} + \frac{2}{27} a^{9} - \frac{1}{54} a^{7} + \frac{1}{6} a^{5} + \frac{11}{27} a^{3} + \frac{10}{27} a$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{1}{3} a^{14} + \frac{19}{18} a^{12} + \frac{19}{3} a^{10} + \frac{59}{3} a^{8} + \frac{185}{9} a^{6} + \frac{5}{6} a^{4} - \frac{7}{3} a^{2} + \frac{5}{9} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 30299.4171562 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:D_4$ (as 16T43):

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 $C_2^2:D_4$
Character table for $C_2^2:D_4$

Intermediate fields

\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-3}) \), 4.0.6144.1 x2, 4.2.18432.2 x2, 4.0.1728.1, \(\Q(\sqrt{-2}, \sqrt{-3})\), 4.0.432.1, 8.0.47775744.3, 8.0.339738624.10, 8.0.3057647616.2

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 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$3$3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$