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Properties

Label 16.0.66202447602479769.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{14}\cdot 7^{12}$
Root discriminant $11.25$
Ramified primes $3, 7$
Class number $1$
Class group Trivial
Galois group $QD_{16}$ (as 16T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 24, -1, -16, 63, -26, -11, 57, -38, 16, 21, -16, 8, 3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 3*x^14 + 8*x^13 - 16*x^12 + 21*x^11 + 16*x^10 - 38*x^9 + 57*x^8 - 11*x^7 - 26*x^6 + 63*x^5 - 16*x^4 - x^3 + 24*x^2 - 5*x + 1)
 
gp: K = bnfinit(x^16 - 2*x^15 + 3*x^14 + 8*x^13 - 16*x^12 + 21*x^11 + 16*x^10 - 38*x^9 + 57*x^8 - 11*x^7 - 26*x^6 + 63*x^5 - 16*x^4 - x^3 + 24*x^2 - 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 3 x^{14} + 8 x^{13} - 16 x^{12} + 21 x^{11} + 16 x^{10} - 38 x^{9} + 57 x^{8} - 11 x^{7} - 26 x^{6} + 63 x^{5} - 16 x^{4} - x^{3} + 24 x^{2} - 5 x + 1 \)

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:  \(66202447602479769=3^{14}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $11.25$
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:  $3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{3}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{12} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{9} a^{4} - \frac{2}{9} a^{3} - \frac{1}{3} a^{2} - \frac{2}{9} a - \frac{2}{9}$, $\frac{1}{2115} a^{14} - \frac{46}{2115} a^{13} - \frac{10}{423} a^{12} - \frac{49}{423} a^{11} - \frac{301}{2115} a^{10} - \frac{55}{423} a^{9} - \frac{3}{235} a^{8} - \frac{56}{141} a^{7} + \frac{247}{705} a^{6} + \frac{65}{423} a^{5} - \frac{268}{2115} a^{4} + \frac{1}{9} a^{3} + \frac{122}{423} a^{2} - \frac{286}{2115} a + \frac{538}{2115}$, $\frac{1}{167085} a^{15} - \frac{2}{11139} a^{14} + \frac{913}{55695} a^{13} + \frac{637}{33417} a^{12} - \frac{1879}{18565} a^{11} + \frac{1001}{18565} a^{10} + \frac{12493}{167085} a^{9} + \frac{6626}{55695} a^{8} - \frac{5641}{18565} a^{7} - \frac{59729}{167085} a^{6} - \frac{4622}{18565} a^{5} + \frac{18154}{55695} a^{4} + \frac{5527}{33417} a^{3} - \frac{16582}{55695} a^{2} - \frac{7577}{18565} a - \frac{25232}{167085}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{84}{395} a^{15} + \frac{30}{79} a^{14} - \frac{186}{395} a^{13} - \frac{470}{237} a^{12} + \frac{1289}{395} a^{11} - \frac{3758}{1185} a^{10} - \frac{6406}{1185} a^{9} + \frac{3468}{395} a^{8} - \frac{3774}{395} a^{7} - \frac{844}{395} a^{6} + \frac{3617}{395} a^{5} - \frac{5448}{395} a^{4} + \frac{124}{237} a^{3} + \frac{1144}{395} a^{2} - \frac{6554}{1185} a + \frac{1334}{1185} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 134.265380275 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$SD_{16}$ (as 16T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $QD_{16}$
Character table for $QD_{16}$

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{-7})\), 4.2.1323.1 x2, 4.0.189.1 x2, 8.0.1750329.1, 8.2.257298363.1 x4

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

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
3Data not computed
$7$7.8.6.1$x^{8} + 35 x^{4} + 441$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
7.8.6.1$x^{8} + 35 x^{4} + 441$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$