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Properties

Label 16.0.95038694188...8441.2
Degree $16$
Signature $[0, 8]$
Discriminant $13^{14}\cdot 17^{6}$
Root discriminant $27.30$
Ramified primes $13, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4\wr C_2$ (as 16T28)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![27, 90, 162, 208, 307, 579, 811, 704, 330, 28, 31, 33, 60, 13, 6, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 6*x^14 + 13*x^13 + 60*x^12 + 33*x^11 + 31*x^10 + 28*x^9 + 330*x^8 + 704*x^7 + 811*x^6 + 579*x^5 + 307*x^4 + 208*x^3 + 162*x^2 + 90*x + 27)
 
gp: K = bnfinit(x^16 - x^15 + 6*x^14 + 13*x^13 + 60*x^12 + 33*x^11 + 31*x^10 + 28*x^9 + 330*x^8 + 704*x^7 + 811*x^6 + 579*x^5 + 307*x^4 + 208*x^3 + 162*x^2 + 90*x + 27, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 6 x^{14} + 13 x^{13} + 60 x^{12} + 33 x^{11} + 31 x^{10} + 28 x^{9} + 330 x^{8} + 704 x^{7} + 811 x^{6} + 579 x^{5} + 307 x^{4} + 208 x^{3} + 162 x^{2} + 90 x + 27 \)

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:  \(95038694188787201488441=13^{14}\cdot 17^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.30$
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:  $13, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{9} - \frac{1}{9} a^{8} + \frac{2}{9} a^{7} - \frac{1}{9} a^{6} - \frac{1}{9} a^{5} - \frac{2}{9} a^{4} - \frac{1}{9} a^{3} - \frac{4}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{1}{9} a^{8} - \frac{4}{9} a^{7} - \frac{4}{9} a^{6} + \frac{4}{9} a^{5} - \frac{4}{9} a^{4} + \frac{2}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{567} a^{14} - \frac{11}{567} a^{13} - \frac{1}{63} a^{12} + \frac{92}{567} a^{11} - \frac{50}{567} a^{10} - \frac{5}{567} a^{9} - \frac{32}{567} a^{8} + \frac{31}{81} a^{7} - \frac{4}{21} a^{6} - \frac{5}{189} a^{5} - \frac{92}{567} a^{4} - \frac{22}{81} a^{3} + \frac{3}{7} a^{2} + \frac{23}{63} a - \frac{1}{21}$, $\frac{1}{3498453071379} a^{15} + \frac{1366803722}{3498453071379} a^{14} - \frac{95480202887}{3498453071379} a^{13} + \frac{31965109175}{3498453071379} a^{12} + \frac{25414905175}{166593003399} a^{11} + \frac{348345798095}{3498453071379} a^{10} + \frac{507339268805}{3498453071379} a^{9} + \frac{143339424749}{3498453071379} a^{8} - \frac{1357439084816}{3498453071379} a^{7} - \frac{301517250803}{1166151023793} a^{6} + \frac{283577465134}{3498453071379} a^{5} - \frac{188539524569}{388717007931} a^{4} + \frac{34184415326}{3498453071379} a^{3} + \frac{86584678972}{388717007931} a^{2} - \frac{3122586307}{9039930417} a + \frac{47647982864}{129572335977}$

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:  \( -1 \) (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:  \( 408026.14455 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\wr C_2$ (as 16T28):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_4\wr C_2$
Character table for $C_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{13}) \), 4.0.2197.1, 4.4.37349.1, 4.0.2873.1, 8.0.1394947801.1

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 8 siblings: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
13Data not computed
$17$17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$