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Properties

Label 16.0.16620815899...6144.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{10}$
Root discriminant $15.90$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
Galois group $C_2\wr C_2^2$ (as 16T149)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 16, 32, 96, 176, 16, -128, -192, -2, 72, 96, 8, -8, -16, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 16*x^13 - 8*x^12 + 8*x^11 + 96*x^10 + 72*x^9 - 2*x^8 - 192*x^7 - 128*x^6 + 16*x^5 + 176*x^4 + 96*x^3 + 32*x^2 + 16*x + 4)
 
gp: K = bnfinit(x^16 - 16*x^13 - 8*x^12 + 8*x^11 + 96*x^10 + 72*x^9 - 2*x^8 - 192*x^7 - 128*x^6 + 16*x^5 + 176*x^4 + 96*x^3 + 32*x^2 + 16*x + 4, 1)
 

Normalized defining polynomial

\( x^{16} - 16 x^{13} - 8 x^{12} + 8 x^{11} + 96 x^{10} + 72 x^{9} - 2 x^{8} - 192 x^{7} - 128 x^{6} + 16 x^{5} + 176 x^{4} + 96 x^{3} + 32 x^{2} + 16 x + 4 \)

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:  \(16620815899787526144=2^{48}\cdot 3^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.90$
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 not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{6} a^{11} + \frac{1}{6} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{6} a^{12} + \frac{1}{6} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{6} a^{13} + \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{1002} a^{14} - \frac{10}{167} a^{13} + \frac{53}{1002} a^{11} - \frac{5}{334} a^{10} - \frac{13}{1002} a^{9} + \frac{50}{501} a^{8} + \frac{62}{501} a^{7} - \frac{7}{167} a^{6} - \frac{31}{167} a^{5} - \frac{46}{501} a^{4} + \frac{76}{167} a^{3} + \frac{205}{501} a^{2} - \frac{223}{501} a + \frac{176}{501}$, $\frac{1}{601533666} a^{15} - \frac{15025}{300766833} a^{14} + \frac{18711991}{601533666} a^{13} - \frac{13216919}{200511222} a^{12} - \frac{32688893}{601533666} a^{11} + \frac{16329017}{200511222} a^{10} - \frac{25794923}{200511222} a^{9} + \frac{8341390}{100255611} a^{8} - \frac{61027000}{300766833} a^{7} - \frac{100734460}{300766833} a^{6} - \frac{24582944}{300766833} a^{5} - \frac{4079326}{33418537} a^{4} + \frac{14558692}{300766833} a^{3} - \frac{77666090}{300766833} a^{2} - \frac{102029221}{300766833} a + \frac{55247737}{300766833}$

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:  \( -\frac{27472}{200111} a^{15} + \frac{27064}{600333} a^{14} - \frac{22096}{600333} a^{13} + \frac{1335158}{600333} a^{12} + \frac{216262}{600333} a^{11} - \frac{175864}{200111} a^{10} - \frac{7766422}{600333} a^{9} - \frac{7012205}{1200666} a^{8} + \frac{303568}{600333} a^{7} + \frac{15465296}{600333} a^{6} + \frac{5785076}{600333} a^{5} - \frac{541364}{200111} a^{4} - \frac{13611296}{600333} a^{3} - \frac{1381640}{200111} a^{2} - \frac{826940}{200111} a - \frac{628819}{600333} \) (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:  \( 3959.00387644 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_2^2$ (as 16T149):

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

\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-2}, \sqrt{-3})\), 4.0.3072.2, 4.0.3072.1, 8.0.84934656.1, 8.0.254803968.1 x2, 8.0.1358954496.2 x2

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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$