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Properties

Label 16.0.28249111657...3664.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 17^{14}$
Root discriminant $33.74$
Ramified primes $2, 17$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^2 : C_8$ (as 16T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2209, 0, -6030, 0, 5870, 0, -240, 0, 587, 0, -64, 0, 54, 0, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^14 + 54*x^12 - 64*x^10 + 587*x^8 - 240*x^6 + 5870*x^4 - 6030*x^2 + 2209)
 
gp: K = bnfinit(x^16 - 6*x^14 + 54*x^12 - 64*x^10 + 587*x^8 - 240*x^6 + 5870*x^4 - 6030*x^2 + 2209, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{14} + 54 x^{12} - 64 x^{10} + 587 x^{8} - 240 x^{6} + 5870 x^{4} - 6030 x^{2} + 2209 \)

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:  \(2824911165797606216433664=2^{24}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.74$
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:  $2, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{12} - \frac{1}{4} a^{11} + \frac{1}{8} a^{10} + \frac{1}{4} a^{7} + \frac{3}{8} a^{6} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{3}{8} a^{2} + \frac{1}{8}$, $\frac{1}{8} a^{13} + \frac{1}{8} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{3}{8} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2} + \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{4291187787688} a^{14} - \frac{7322056951}{536398473461} a^{12} - \frac{955173450889}{4291187787688} a^{10} - \frac{1}{4} a^{9} + \frac{9267665395}{4291187787688} a^{8} + \frac{77406554401}{4291187787688} a^{6} + \frac{1}{4} a^{5} - \frac{701343469655}{4291187787688} a^{4} - \frac{45389648923}{1072796946922} a^{2} - \frac{1}{4} a + \frac{1729605644679}{4291187787688}$, $\frac{1}{201685826021336} a^{15} - \frac{7322056951}{25210728252667} a^{13} + \frac{35519922744459}{201685826021336} a^{11} - \frac{1}{4} a^{10} - \frac{27883452954577}{201685826021336} a^{9} + \frac{58008441688189}{201685826021336} a^{7} + \frac{1}{4} a^{6} + \frac{40064940513381}{201685826021336} a^{5} + \frac{10437075408028}{25210728252667} a^{3} - \frac{1}{4} a^{2} - \frac{94822119578301}{201685826021336} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 131202.513168 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_8$ (as 16T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2^2 : C_8$
Character table for $C_2^2 : C_8$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 4.0.39304.1, 4.0.2312.1, 8.0.105046700288.1, 8.8.420186801152.1, 8.0.1544804416.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
17Data not computed