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Properties

Label 16.0.63384657313...2576.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 7^{8}$
Root discriminant $14.97$
Ramified primes $2, 7$
Class number $2$
Class group $[2]$
Galois group $C_2\wr C_2^2$ (as 16T127)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -6, -12, -6, 52, 166, 280, 346, 280, 166, 52, -6, -12, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^14 - 12*x^13 - 6*x^12 + 52*x^11 + 166*x^10 + 280*x^9 + 346*x^8 + 280*x^7 + 166*x^6 + 52*x^5 - 6*x^4 - 12*x^3 - 6*x^2 + 1)
 
gp: K = bnfinit(x^16 - 6*x^14 - 12*x^13 - 6*x^12 + 52*x^11 + 166*x^10 + 280*x^9 + 346*x^8 + 280*x^7 + 166*x^6 + 52*x^5 - 6*x^4 - 12*x^3 - 6*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{14} - 12 x^{13} - 6 x^{12} + 52 x^{11} + 166 x^{10} + 280 x^{9} + 346 x^{8} + 280 x^{7} + 166 x^{6} + 52 x^{5} - 6 x^{4} - 12 x^{3} - 6 x^{2} + 1 \)

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:  \(6338465731314712576=2^{40}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.97$
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:  $2, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} + \frac{1}{4} a^{4} - \frac{1}{4}$, $\frac{1}{12} a^{13} - \frac{1}{6} a^{10} - \frac{1}{4} a^{9} - \frac{1}{6} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{5}{12} a^{5} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a + \frac{1}{6}$, $\frac{1}{2088} a^{14} + \frac{7}{1044} a^{13} + \frac{5}{696} a^{12} - \frac{41}{522} a^{11} + \frac{293}{2088} a^{10} + \frac{245}{1044} a^{9} - \frac{401}{2088} a^{8} - \frac{32}{261} a^{7} - \frac{923}{2088} a^{6} + \frac{419}{1044} a^{5} + \frac{119}{2088} a^{4} - \frac{215}{522} a^{3} - \frac{169}{696} a^{2} + \frac{181}{1044} a - \frac{173}{2088}$, $\frac{1}{6264} a^{15} + \frac{1}{6264} a^{14} - \frac{167}{6264} a^{13} + \frac{685}{6264} a^{12} - \frac{185}{6264} a^{11} + \frac{1379}{6264} a^{10} + \frac{353}{2088} a^{9} + \frac{259}{6264} a^{8} + \frac{1361}{6264} a^{7} - \frac{245}{2088} a^{6} - \frac{1379}{6264} a^{5} - \frac{83}{216} a^{4} - \frac{1333}{6264} a^{3} + \frac{167}{6264} a^{2} - \frac{2269}{6264} a - \frac{1405}{6264}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( \frac{2603}{3132} a^{15} + \frac{427}{6264} a^{14} - \frac{16363}{3132} a^{13} - \frac{65093}{6264} a^{12} - \frac{12943}{3132} a^{11} + \frac{285299}{6264} a^{10} + \frac{147775}{1044} a^{9} + \frac{49667}{216} a^{8} + \frac{28883}{108} a^{7} + \frac{420373}{2088} a^{6} + \frac{322061}{3132} a^{5} + \frac{147251}{6264} a^{4} - \frac{16391}{3132} a^{3} - \frac{40093}{6264} a^{2} - \frac{2171}{3132} a + \frac{275}{6264} \) (order $8$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3964.06469644 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_2^2$ (as 16T127):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 16 conjugacy class representatives for $C_2\wr C_2^2$
Character table for $C_2\wr C_2^2$

Intermediate fields

\(\Q(\sqrt{7}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{-2}, \sqrt{-7})\), \(\Q(i, \sqrt{7})\), \(\Q(\sqrt{2}, \sqrt{-7})\), \(\Q(\sqrt{-2}, \sqrt{7})\), \(\Q(\zeta_{8})\), \(\Q(\sqrt{2}, \sqrt{7})\), \(\Q(i, \sqrt{14})\), 8.0.78675968.3, 8.0.314703872.5, 8.0.157351936.1

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
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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.8.22.102$x^{8} + 8 x^{7} + 16 x^{5} + 144$$8$$1$$22$$D_4\times C_2$$[2, 3, 7/2]^{2}$
2.8.18.53$x^{8} + 2 x^{6} + 4 x^{3} + 2$$8$$1$$18$$D_4\times C_2$$[2, 2, 3]^{2}$
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$