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Properties

Label 16.16.1699206967...7344.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{50}\cdot 3^{12}\cdot 73^{4}$
Root discriminant $58.13$
Ramified primes $2, 3, 73$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2.C_2\wr C_2^2$ (as 16T394)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2642, 20192, -48704, 15032, 89604, -70432, -70712, 58512, 34678, -18768, -9856, 2208, 1256, -80, -64, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 64*x^14 - 80*x^13 + 1256*x^12 + 2208*x^11 - 9856*x^10 - 18768*x^9 + 34678*x^8 + 58512*x^7 - 70712*x^6 - 70432*x^5 + 89604*x^4 + 15032*x^3 - 48704*x^2 + 20192*x - 2642)
 
gp: K = bnfinit(x^16 - 64*x^14 - 80*x^13 + 1256*x^12 + 2208*x^11 - 9856*x^10 - 18768*x^9 + 34678*x^8 + 58512*x^7 - 70712*x^6 - 70432*x^5 + 89604*x^4 + 15032*x^3 - 48704*x^2 + 20192*x - 2642, 1)
 

Normalized defining polynomial

\( x^{16} - 64 x^{14} - 80 x^{13} + 1256 x^{12} + 2208 x^{11} - 9856 x^{10} - 18768 x^{9} + 34678 x^{8} + 58512 x^{7} - 70712 x^{6} - 70432 x^{5} + 89604 x^{4} + 15032 x^{3} - 48704 x^{2} + 20192 x - 2642 \)

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:  \(16992069679396728584320057344=2^{50}\cdot 3^{12}\cdot 73^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.13$
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:  $2, 3, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{227733587283317981729856885} a^{15} - \frac{25642460322731076506696624}{227733587283317981729856885} a^{14} - \frac{2158358138235842664097437}{25303731920368664636650765} a^{13} - \frac{37842161262937418796403508}{227733587283317981729856885} a^{12} + \frac{8983281753617020002498797}{25303731920368664636650765} a^{11} - \frac{28471877396493066904199993}{75911195761105993909952295} a^{10} + \frac{14771980330059749122754395}{45546717456663596345971377} a^{9} + \frac{101516393674107076248710222}{227733587283317981729856885} a^{8} - \frac{310192741916174295218952}{5060746384073732927330153} a^{7} + \frac{10643331207301150526673098}{25303731920368664636650765} a^{6} - \frac{20697890461692212593952965}{45546717456663596345971377} a^{5} + \frac{12441899605911653127054271}{75911195761105993909952295} a^{4} + \frac{12933913585601333905036189}{75911195761105993909952295} a^{3} - \frac{103259929508719146796147276}{227733587283317981729856885} a^{2} + \frac{5098559752422300643104887}{15182239152221198781990459} a + \frac{50012901768958726863176717}{227733587283317981729856885}$

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:  \( 1667919879.14 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2.C_2\wr C_2^2$ (as 16T394):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 17 conjugacy class representatives for $C_2.C_2\wr C_2^2$
Character table for $C_2.C_2\wr C_2^2$

Intermediate fields

\(\Q(\sqrt{6}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), 4.4.27648.1 x2, 4.4.13824.1 x2, \(\Q(\sqrt{2}, \sqrt{3})\), 8.8.3057647616.1

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

Sibling fields

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$3.8.6.2$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$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
73Data not computed