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Properties

Label 16.0.30257271966...8144.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{62}\cdot 3^{8}$
Root discriminant $25.41$
Ramified primes $2, 3$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_8\times C_2$ (as 16T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 0, -32, 0, 216, 0, -288, 0, 270, 0, -128, 0, 44, 0, -8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 + 44*x^12 - 128*x^10 + 270*x^8 - 288*x^6 + 216*x^4 - 32*x^2 + 4)
 
gp: K = bnfinit(x^16 - 8*x^14 + 44*x^12 - 128*x^10 + 270*x^8 - 288*x^6 + 216*x^4 - 32*x^2 + 4, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{14} + 44 x^{12} - 128 x^{10} + 270 x^{8} - 288 x^{6} + 216 x^{4} - 32 x^{2} + 4 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(30257271966902092038144=2^{62}\cdot 3^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.41$
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:  $2, 3$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(96=2^{5}\cdot 3\)
Dirichlet character group:    $\lbrace$$\chi_{96}(1,·)$, $\chi_{96}(67,·)$, $\chi_{96}(65,·)$, $\chi_{96}(73,·)$, $\chi_{96}(11,·)$, $\chi_{96}(17,·)$, $\chi_{96}(83,·)$, $\chi_{96}(25,·)$, $\chi_{96}(89,·)$, $\chi_{96}(91,·)$, $\chi_{96}(35,·)$, $\chi_{96}(41,·)$, $\chi_{96}(43,·)$, $\chi_{96}(49,·)$, $\chi_{96}(19,·)$, $\chi_{96}(59,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{2} a^{13}$, $\frac{1}{475106} a^{14} - \frac{2524}{237553} a^{12} + \frac{23793}{475106} a^{10} + \frac{47417}{475106} a^{8} - \frac{1546}{237553} a^{6} - \frac{47553}{237553} a^{4} - \frac{23749}{237553} a^{2} - \frac{31768}{237553}$, $\frac{1}{475106} a^{15} - \frac{2524}{237553} a^{13} + \frac{23793}{475106} a^{11} + \frac{47417}{475106} a^{9} - \frac{1546}{237553} a^{7} - \frac{47553}{237553} a^{5} - \frac{23749}{237553} a^{3} - \frac{31768}{237553} a$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{9662}{237553} a^{14} + \frac{75411}{237553} a^{12} - \frac{411768}{237553} a^{10} + \frac{2331343}{475106} a^{8} - \frac{2432304}{237553} a^{6} + \frac{2434698}{237553} a^{4} - \frac{1927144}{237553} a^{2} + \frac{285433}{237553} \) (order $6$)
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:  \( 95624.3098505 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_8$ (as 16T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\zeta_{16})^+\), 4.0.18432.2, 8.0.339738624.2, 8.0.2147483648.1, 8.8.173946175488.1

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

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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
3Data not computed