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Properties

Label 16.8.34820097123...9824.4
Degree $16$
Signature $[8, 4]$
Discriminant $2^{40}\cdot 3^{8}\cdot 13^{6}$
Root discriminant $25.64$
Ramified primes $2, 3, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times D_4:D_4$ (as 16T265)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 0, -64, 0, 80, 0, -24, 0, 44, 0, -24, 0, -4, 0, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 - 4*x^12 - 24*x^10 + 44*x^8 - 24*x^6 + 80*x^4 - 64*x^2 + 4)
 
gp: K = bnfinit(x^16 - 4*x^14 - 4*x^12 - 24*x^10 + 44*x^8 - 24*x^6 + 80*x^4 - 64*x^2 + 4, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{14} - 4 x^{12} - 24 x^{10} + 44 x^{8} - 24 x^{6} + 80 x^{4} - 64 x^{2} + 4 \)

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:  \(34820097123453788749824=2^{40}\cdot 3^{8}\cdot 13^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.64$
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:  $2, 3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{216} a^{12} + \frac{11}{108} a^{10} + \frac{1}{12} a^{8} - \frac{25}{54} a^{6} - \frac{1}{2} a^{5} - \frac{5}{12} a^{4} + \frac{5}{27} a^{2} + \frac{19}{54}$, $\frac{1}{216} a^{13} - \frac{5}{216} a^{11} + \frac{1}{12} a^{9} + \frac{1}{27} a^{7} - \frac{1}{2} a^{6} - \frac{5}{12} a^{5} - \frac{1}{2} a^{4} + \frac{47}{108} a^{3} - \frac{4}{27} a$, $\frac{1}{2376} a^{14} + \frac{1}{594} a^{12} + \frac{5}{44} a^{10} - \frac{131}{1188} a^{8} - \frac{1}{2} a^{7} + \frac{29}{132} a^{6} + \frac{37}{594} a^{4} + \frac{217}{594} a^{2} + \frac{31}{66}$, $\frac{1}{2376} a^{15} + \frac{1}{594} a^{13} - \frac{1}{88} a^{11} - \frac{131}{1188} a^{9} - \frac{37}{132} a^{7} + \frac{37}{594} a^{5} - \frac{1}{2} a^{4} - \frac{457}{1188} a^{3} - \frac{1}{33} a$

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:  $11$
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:  \( 430273.178782 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_4:D_4$ (as 16T265):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_2\times D_4:D_4$
Character table for $C_2\times D_4:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), 4.4.7488.1, 4.4.29952.1, \(\Q(\sqrt{2}, \sqrt{3})\), 8.4.11662589952.4, 8.4.46650359808.6, 8.8.3588489216.1

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

Sibling fields

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 R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{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}$