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Properties

Label 16.0.16620815899...6144.2
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![9, 24, 28, 48, 50, -40, -96, -80, -15, 24, 48, 40, 6, -16, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 - 16*x^13 + 6*x^12 + 40*x^11 + 48*x^10 + 24*x^9 - 15*x^8 - 80*x^7 - 96*x^6 - 40*x^5 + 50*x^4 + 48*x^3 + 28*x^2 + 24*x + 9)
 
gp: K = bnfinit(x^16 - 4*x^14 - 16*x^13 + 6*x^12 + 40*x^11 + 48*x^10 + 24*x^9 - 15*x^8 - 80*x^7 - 96*x^6 - 40*x^5 + 50*x^4 + 48*x^3 + 28*x^2 + 24*x + 9, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{14} - 16 x^{13} + 6 x^{12} + 40 x^{11} + 48 x^{10} + 24 x^{9} - 15 x^{8} - 80 x^{7} - 96 x^{6} - 40 x^{5} + 50 x^{4} + 48 x^{3} + 28 x^{2} + 24 x + 9 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{11} + \frac{1}{9} a^{9} + \frac{1}{9} a^{8} + \frac{2}{9} a^{6} + \frac{2}{9} a^{5} + \frac{1}{3} a^{4} + \frac{2}{9} a^{3} + \frac{2}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{11} + \frac{1}{9} a^{10} - \frac{1}{9} a^{8} - \frac{1}{9} a^{7} + \frac{1}{3} a^{6} + \frac{4}{9} a^{5} + \frac{2}{9} a^{4} + \frac{1}{3} a^{3} + \frac{4}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{333} a^{14} - \frac{1}{111} a^{13} - \frac{4}{333} a^{12} - \frac{14}{333} a^{11} - \frac{3}{37} a^{10} - \frac{49}{333} a^{9} - \frac{43}{333} a^{8} + \frac{13}{111} a^{7} - \frac{41}{333} a^{6} - \frac{49}{333} a^{5} - \frac{15}{37} a^{4} + \frac{103}{333} a^{3} + \frac{35}{111} a^{2} + \frac{3}{37} a - \frac{4}{37}$, $\frac{1}{1033765533} a^{15} + \frac{921943}{1033765533} a^{14} - \frac{18589657}{344588511} a^{13} + \frac{43718378}{1033765533} a^{12} + \frac{107432639}{1033765533} a^{11} + \frac{10212219}{114862837} a^{10} + \frac{2659105}{114862837} a^{9} + \frac{46088770}{344588511} a^{8} - \frac{1977770}{114862837} a^{7} - \frac{51819875}{1033765533} a^{6} - \frac{102267821}{1033765533} a^{5} + \frac{26595338}{344588511} a^{4} + \frac{384340577}{1033765533} a^{3} - \frac{93642487}{1033765533} a^{2} + \frac{13731968}{114862837} a + \frac{28881725}{114862837}$

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:  \( \frac{3057248}{27939609} a^{15} - \frac{483692}{9313203} a^{14} - \frac{3674240}{9313203} a^{13} - \frac{43490606}{27939609} a^{12} + \frac{12297016}{9313203} a^{11} + \frac{32031032}{9313203} a^{10} + \frac{11291528}{3104401} a^{9} + \frac{15252641}{9313203} a^{8} - \frac{4626032}{3104401} a^{7} - \frac{207038488}{27939609} a^{6} - \frac{60769000}{9313203} a^{5} - \frac{20979962}{9313203} a^{4} + \frac{138540544}{27939609} a^{3} + \frac{7027332}{3104401} a^{2} + \frac{25002824}{9313203} a + \frac{6591364}{3104401} \) (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:  \( 11028.2859854 \)
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

\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-2}, \sqrt{-3})\), 4.0.3072.1, 4.0.3072.2, 8.0.84934656.1, 8.0.254803968.2 x2, 8.0.1358954496.1 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{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}$