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Properties

Label 16.0.396154108207169536.5
Degree $16$
Signature $[0, 8]$
Discriminant $2^{36}\cdot 7^{8}$
Root discriminant $12.59$
Ramified primes $2, 7$
Class number $1$
Class group Trivial
Galois group $D_4:D_4$ (as 16T141)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -2, 0, 7, 0, -22, 0, 34, 0, -22, 0, 7, 0, -2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^14 + 7*x^12 - 22*x^10 + 34*x^8 - 22*x^6 + 7*x^4 - 2*x^2 + 1)
 
gp: K = bnfinit(x^16 - 2*x^14 + 7*x^12 - 22*x^10 + 34*x^8 - 22*x^6 + 7*x^4 - 2*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{14} + 7 x^{12} - 22 x^{10} + 34 x^{8} - 22 x^{6} + 7 x^{4} - 2 x^{2} + 1 \)

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:  \(396154108207169536=2^{36}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12.59$
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:  $2, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{14} a^{12} + \frac{3}{14} a^{10} - \frac{1}{2} a^{8} - \frac{2}{7} a^{6} - \frac{1}{2} a^{4} + \frac{3}{14} a^{2} + \frac{1}{14}$, $\frac{1}{14} a^{13} + \frac{3}{14} a^{11} - \frac{1}{2} a^{9} - \frac{2}{7} a^{7} - \frac{1}{2} a^{5} + \frac{3}{14} a^{3} + \frac{1}{14} a$, $\frac{1}{196} a^{14} - \frac{1}{28} a^{13} + \frac{1}{49} a^{12} - \frac{3}{28} a^{11} - \frac{1}{49} a^{10} + \frac{1}{4} a^{9} + \frac{45}{196} a^{8} + \frac{1}{7} a^{7} - \frac{39}{196} a^{6} + \frac{1}{4} a^{5} + \frac{20}{49} a^{4} - \frac{3}{28} a^{3} - \frac{13}{49} a^{2} - \frac{1}{28} a - \frac{27}{196}$, $\frac{1}{196} a^{15} - \frac{3}{196} a^{13} - \frac{1}{28} a^{12} - \frac{25}{196} a^{11} - \frac{3}{28} a^{10} + \frac{47}{98} a^{9} + \frac{1}{4} a^{8} - \frac{11}{196} a^{7} + \frac{1}{7} a^{6} - \frac{67}{196} a^{5} + \frac{1}{4} a^{4} - \frac{73}{196} a^{3} - \frac{3}{28} a^{2} - \frac{17}{98} a - \frac{1}{28}$

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:  \( \frac{27}{49} a^{14} - \frac{85}{98} a^{12} + \frac{351}{98} a^{10} - \frac{1049}{98} a^{8} + \frac{725}{49} a^{6} - \frac{727}{98} a^{4} + \frac{307}{98} a^{2} - \frac{93}{98} \) (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:  \( 480.586770627 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4:D_4$ (as 16T141):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 16 conjugacy class representatives for $D_4:D_4$
Character table for $D_4:D_4$

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{7}) \), 4.0.1568.1, 4.0.392.1, \(\Q(i, \sqrt{7})\), 8.0.314703872.2 x2, 8.0.78675968.1 x2, 8.0.9834496.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 siblings: data not computed
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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.8.24.6$x^{8} + 44 x^{4} + 100$$8$$1$$24$$D_4$$[2, 3, 4]$
2.8.12.14$x^{8} + 12 x^{4} + 144$$4$$2$$12$$D_4$$[2, 2]^{2}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$