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Properties

Label 16.0.50787142148...1281.4
Degree $16$
Signature $[0, 8]$
Discriminant $13^{8}\cdot 53^{8}$
Root discriminant $26.25$
Ramified primes $13, 53$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_{8}$ (as 16T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8281, 0, 2678, 0, 2729, 0, 1833, 0, 1160, 0, 354, 0, 53, 0, 2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 2*x^14 + 53*x^12 + 354*x^10 + 1160*x^8 + 1833*x^6 + 2729*x^4 + 2678*x^2 + 8281)
 
gp: K = bnfinit(x^16 + 2*x^14 + 53*x^12 + 354*x^10 + 1160*x^8 + 1833*x^6 + 2729*x^4 + 2678*x^2 + 8281, 1)
 

Normalized defining polynomial

\( x^{16} + 2 x^{14} + 53 x^{12} + 354 x^{10} + 1160 x^{8} + 1833 x^{6} + 2729 x^{4} + 2678 x^{2} + 8281 \)

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:  \(50787142148496295121281=13^{8}\cdot 53^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.25$
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:  $13, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{6} a^{8} - \frac{1}{6} a^{6} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{6} a^{2} - \frac{1}{2}$, $\frac{1}{42} a^{9} + \frac{5}{42} a^{7} - \frac{10}{21} a^{5} - \frac{1}{2} a^{4} - \frac{1}{42} a^{3} + \frac{3}{14} a$, $\frac{1}{42} a^{10} - \frac{1}{21} a^{8} + \frac{1}{42} a^{6} - \frac{1}{2} a^{5} - \frac{1}{42} a^{4} - \frac{1}{2} a^{3} - \frac{2}{7} a^{2} + \frac{1}{6}$, $\frac{1}{42} a^{11} - \frac{1}{14} a^{7} - \frac{1}{6} a^{6} + \frac{5}{14} a^{5} + \frac{1}{6} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{14} a - \frac{1}{3}$, $\frac{1}{4158} a^{12} - \frac{47}{4158} a^{10} + \frac{2}{99} a^{8} - \frac{1}{6} a^{7} - \frac{235}{4158} a^{6} - \frac{1}{3} a^{5} - \frac{85}{462} a^{4} + \frac{1}{3} a^{3} + \frac{410}{2079} a^{2} - \frac{1}{3} a + \frac{50}{297}$, $\frac{1}{4158} a^{13} - \frac{47}{4158} a^{11} - \frac{5}{1386} a^{9} + \frac{328}{2079} a^{7} - \frac{1}{6} a^{6} - \frac{19}{462} a^{5} + \frac{1}{6} a^{4} + \frac{113}{2079} a^{3} - \frac{1}{6} a^{2} - \frac{1577}{4158} a + \frac{1}{6}$, $\frac{1}{1318539222} a^{14} + \frac{40435}{439513074} a^{12} - \frac{10930165}{1318539222} a^{10} + \frac{40797700}{659269611} a^{8} - \frac{1}{6} a^{7} - \frac{133260683}{1318539222} a^{6} + \frac{1}{6} a^{5} + \frac{16947377}{50713047} a^{4} - \frac{1}{6} a^{3} + \frac{136414009}{439513074} a^{2} + \frac{1}{6} a - \frac{86362}{658611}$, $\frac{1}{1318539222} a^{15} + \frac{40435}{439513074} a^{13} - \frac{10930165}{1318539222} a^{11} - \frac{12585973}{1318539222} a^{9} - \frac{82327237}{659269611} a^{7} + \frac{21777191}{50713047} a^{5} + \frac{94555621}{439513074} a^{3} - \frac{989531}{9220554} a - \frac{1}{2}$

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:  \( -1 \) (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:  \( 82825.5955779 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_8$ (as 16T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $D_{8}$
Character table for $D_{8}$

Intermediate fields

\(\Q(\sqrt{689}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{53}) \), \(\Q(\sqrt{13}, \sqrt{53})\), 4.4.8957.1 x2, 4.4.36517.1 x2, 8.8.225360027841.1, 8.0.4252075997.1 x4, 8.0.17335386757.1 x4

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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}$
$53$53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$