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Properties

Label 16.8.14266108229...8329.5
Degree $16$
Signature $[8, 4]$
Discriminant $13^{8}\cdot 53^{10}$
Root discriminant $43.12$
Ramified primes $13, 53$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2.D_4$ (as 16T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, 0, -2912, 0, 14277, 0, -19670, 0, 10547, 0, -2637, 0, 367, 0, -30, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 30*x^14 + 367*x^12 - 2637*x^10 + 10547*x^8 - 19670*x^6 + 14277*x^4 - 2912*x^2 + 256)
 
gp: K = bnfinit(x^16 - 30*x^14 + 367*x^12 - 2637*x^10 + 10547*x^8 - 19670*x^6 + 14277*x^4 - 2912*x^2 + 256, 1)
 

Normalized defining polynomial

\( x^{16} - 30 x^{14} + 367 x^{12} - 2637 x^{10} + 10547 x^{8} - 19670 x^{6} + 14277 x^{4} - 2912 x^{2} + 256 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(142661082295126092995678329=13^{8}\cdot 53^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.12$
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 not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{26} a^{8} - \frac{1}{26} a^{6} - \frac{1}{2} a^{5} - \frac{5}{26} a^{4} + \frac{1}{26} a^{2} - \frac{6}{13}$, $\frac{1}{26} a^{9} - \frac{1}{26} a^{7} - \frac{1}{2} a^{6} - \frac{5}{26} a^{5} + \frac{1}{26} a^{3} - \frac{6}{13} a$, $\frac{1}{26} a^{10} - \frac{3}{13} a^{6} + \frac{9}{26} a^{4} - \frac{1}{2} a^{3} - \frac{11}{26} a^{2} - \frac{1}{2} a - \frac{6}{13}$, $\frac{1}{26} a^{11} - \frac{3}{13} a^{7} + \frac{9}{26} a^{5} - \frac{1}{2} a^{4} - \frac{11}{26} a^{3} - \frac{1}{2} a^{2} - \frac{6}{13} a$, $\frac{1}{338} a^{12} + \frac{2}{169} a^{10} + \frac{3}{169} a^{8} - \frac{27}{338} a^{6} - \frac{1}{2} a^{5} - \frac{113}{338} a^{4} - \frac{1}{2} a^{3} - \frac{61}{169} a^{2} + \frac{21}{169}$, $\frac{1}{1352} a^{13} - \frac{11}{676} a^{11} + \frac{19}{1352} a^{9} - \frac{53}{1352} a^{7} - \frac{581}{1352} a^{5} - \frac{165}{676} a^{3} - \frac{1}{2} a^{2} - \frac{647}{1352} a - \frac{1}{2}$, $\frac{1}{345344064} a^{14} - \frac{45239}{172672032} a^{12} - \frac{3551953}{345344064} a^{10} - \frac{235889}{26564928} a^{8} - \frac{17824541}{345344064} a^{6} + \frac{854747}{4427488} a^{4} + \frac{103711}{681152} a^{2} - \frac{1}{2} a - \frac{1532897}{21584004}$, $\frac{1}{2762752512} a^{15} - \frac{1}{690688128} a^{14} + \frac{465625}{1381376256} a^{13} - \frac{465625}{345344064} a^{12} - \frac{12747505}{2762752512} a^{11} + \frac{12747505}{690688128} a^{10} + \frac{42911203}{2762752512} a^{9} + \frac{10218653}{690688128} a^{8} + \frac{26136943}{212519424} a^{7} - \frac{3658927}{53129856} a^{6} + \frac{168968687}{460458752} a^{5} + \frac{16985809}{115114688} a^{4} + \frac{37961719}{920917504} a^{3} + \frac{94862921}{230229376} a^{2} + \frac{24393451}{172672032} a - \frac{21072835}{43168008}$

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:  $11$
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:  \( 31553758.4088 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.D_4$ (as 16T33):

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

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{53}) \), \(\Q(\sqrt{689}) \), 4.4.8957.1 x2, 4.4.36517.1 x2, \(\Q(\sqrt{13}, \sqrt{53})\), 8.4.11944081475573.2 x2, 8.8.225360027841.1, 8.4.225360027841.2 x2

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 8 siblings: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
53Data not computed