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Properties

Label 16.16.6394994603...0625.2
Degree $16$
Signature $[16, 0]$
Discriminant $3^{12}\cdot 5^{12}\cdot 149^{4}$
Root discriminant $26.63$
Ramified primes $3, 5, 149$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\wr C_4$ (as 16T158)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1231, 307, -6358, -1882, 12527, 4010, -12117, -3786, 6355, 1761, -1887, -415, 317, 47, -28, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 28*x^14 + 47*x^13 + 317*x^12 - 415*x^11 - 1887*x^10 + 1761*x^9 + 6355*x^8 - 3786*x^7 - 12117*x^6 + 4010*x^5 + 12527*x^4 - 1882*x^3 - 6358*x^2 + 307*x + 1231)
 
gp: K = bnfinit(x^16 - 2*x^15 - 28*x^14 + 47*x^13 + 317*x^12 - 415*x^11 - 1887*x^10 + 1761*x^9 + 6355*x^8 - 3786*x^7 - 12117*x^6 + 4010*x^5 + 12527*x^4 - 1882*x^3 - 6358*x^2 + 307*x + 1231, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 28 x^{14} + 47 x^{13} + 317 x^{12} - 415 x^{11} - 1887 x^{10} + 1761 x^{9} + 6355 x^{8} - 3786 x^{7} - 12117 x^{6} + 4010 x^{5} + 12527 x^{4} - 1882 x^{3} - 6358 x^{2} + 307 x + 1231 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(63949946033164306640625=3^{12}\cdot 5^{12}\cdot 149^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.63$
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:  $3, 5, 149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{11} + \frac{1}{9} a^{10} + \frac{4}{9} a^{9} - \frac{2}{9} a^{8} + \frac{2}{9} a^{7} - \frac{1}{9} a^{6} + \frac{2}{9} a^{5} + \frac{4}{9} a^{4} + \frac{4}{9} a^{3} + \frac{1}{9} a^{2} + \frac{2}{9} a + \frac{1}{9}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{10} - \frac{4}{9} a^{9} - \frac{1}{3} a^{8} + \frac{4}{9} a^{7} + \frac{4}{9} a^{6} - \frac{1}{3} a^{5} + \frac{2}{9} a^{4} - \frac{1}{9} a^{3} - \frac{1}{3} a + \frac{4}{9}$, $\frac{1}{27} a^{14} + \frac{1}{27} a^{12} + \frac{4}{27} a^{11} + \frac{1}{9} a^{10} + \frac{4}{27} a^{9} + \frac{8}{27} a^{8} - \frac{2}{9} a^{7} - \frac{13}{27} a^{6} - \frac{8}{27} a^{5} - \frac{2}{27} a^{3} + \frac{13}{27} a^{2} - \frac{2}{9} a + \frac{1}{27}$, $\frac{1}{231867167247} a^{15} + \frac{2718055580}{231867167247} a^{14} + \frac{3457628662}{231867167247} a^{13} + \frac{505454887}{77289055749} a^{12} - \frac{5068667203}{231867167247} a^{11} - \frac{28649661020}{231867167247} a^{10} - \frac{8674745030}{231867167247} a^{9} + \frac{45901211689}{231867167247} a^{8} + \frac{99772078970}{231867167247} a^{7} - \frac{27364864564}{231867167247} a^{6} - \frac{41002900576}{231867167247} a^{5} - \frac{76448365445}{231867167247} a^{4} - \frac{25405003049}{77289055749} a^{3} + \frac{22563649331}{231867167247} a^{2} + \frac{110940343960}{231867167247} a - \frac{89079492634}{231867167247}$

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:  $15$
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:  \( 1367021.17451 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T158):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 4.4.33525.1, 4.4.18625.1, 8.8.1697203125.1 x2, 8.8.50576653125.1 x2, 8.8.28098140625.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.8.6.3$x^{8} - 3 x^{4} + 18$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
3.8.6.3$x^{8} - 3 x^{4} + 18$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$149$149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.4.2.1$x^{4} + 745 x^{2} + 199809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
149.4.2.1$x^{4} + 745 x^{2} + 199809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$