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Properties

Label 16.2.22173079161...8147.1
Degree $16$
Signature $[2, 7]$
Discriminant $-\,13^{8}\cdot 43^{7}$
Root discriminant $18.69$
Ramified primes $13, 43$
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![-1, 20, -78, 161, -196, 139, -18, -83, 123, -98, 51, -14, 1, 2, 0, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 2*x^13 + x^12 - 14*x^11 + 51*x^10 - 98*x^9 + 123*x^8 - 83*x^7 - 18*x^6 + 139*x^5 - 196*x^4 + 161*x^3 - 78*x^2 + 20*x - 1)
 
gp: K = bnfinit(x^16 - x^15 + 2*x^13 + x^12 - 14*x^11 + 51*x^10 - 98*x^9 + 123*x^8 - 83*x^7 - 18*x^6 + 139*x^5 - 196*x^4 + 161*x^3 - 78*x^2 + 20*x - 1, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 2 x^{13} + x^{12} - 14 x^{11} + 51 x^{10} - 98 x^{9} + 123 x^{8} - 83 x^{7} - 18 x^{6} + 139 x^{5} - 196 x^{4} + 161 x^{3} - 78 x^{2} + 20 x - 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-221730791619531718147=-\,13^{8}\cdot 43^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.69$
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:  $13, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{9} - \frac{1}{9} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{4}{9} a^{3} + \frac{1}{9} a^{2} + \frac{2}{9} a - \frac{2}{9}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{9} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{9}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{9} - \frac{1}{3} a^{7} - \frac{1}{9} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{9} a + \frac{1}{3}$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{10} - \frac{1}{9} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{9} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{873} a^{15} - \frac{46}{873} a^{14} + \frac{11}{291} a^{13} - \frac{28}{873} a^{12} - \frac{37}{291} a^{10} - \frac{95}{873} a^{9} + \frac{2}{291} a^{8} - \frac{244}{873} a^{7} - \frac{161}{873} a^{6} - \frac{16}{291} a^{5} + \frac{262}{873} a^{4} + \frac{37}{97} a^{3} - \frac{59}{291} a^{2} - \frac{358}{873} a - \frac{23}{291}$

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:  $8$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 13419.93728 \)
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

\(\Q(\sqrt{13}) \), 4.2.7267.1, 8.2.2270799427.1

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 $16$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43Data not computed