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Properties

Label 16.8.60215786657...2784.4
Degree $16$
Signature $[8, 4]$
Discriminant $2^{32}\cdot 3^{8}\cdot 17^{6}\cdot 97^{4}$
Root discriminant $62.91$
Ramified primes $2, 3, 17, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T657)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4071673, 4480320, 364190, 1685116, 371076, -18480, 148310, -134932, 18137, -1816, -3358, 1932, -296, 48, 8, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 8*x^14 + 48*x^13 - 296*x^12 + 1932*x^11 - 3358*x^10 - 1816*x^9 + 18137*x^8 - 134932*x^7 + 148310*x^6 - 18480*x^5 + 371076*x^4 + 1685116*x^3 + 364190*x^2 + 4480320*x + 4071673)
 
gp: K = bnfinit(x^16 - 8*x^15 + 8*x^14 + 48*x^13 - 296*x^12 + 1932*x^11 - 3358*x^10 - 1816*x^9 + 18137*x^8 - 134932*x^7 + 148310*x^6 - 18480*x^5 + 371076*x^4 + 1685116*x^3 + 364190*x^2 + 4480320*x + 4071673, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 8 x^{14} + 48 x^{13} - 296 x^{12} + 1932 x^{11} - 3358 x^{10} - 1816 x^{9} + 18137 x^{8} - 134932 x^{7} + 148310 x^{6} - 18480 x^{5} + 371076 x^{4} + 1685116 x^{3} + 364190 x^{2} + 4480320 x + 4071673 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(60215786657648562405359222784=2^{32}\cdot 3^{8}\cdot 17^{6}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.91$
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:  $2, 3, 17, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{776} a^{14} + \frac{13}{776} a^{13} - \frac{12}{97} a^{12} - \frac{79}{776} a^{11} + \frac{93}{776} a^{10} + \frac{53}{388} a^{9} - \frac{109}{776} a^{8} + \frac{165}{776} a^{7} + \frac{227}{776} a^{6} - \frac{29}{194} a^{5} + \frac{155}{776} a^{4} + \frac{183}{776} a^{3} + \frac{33}{97} a^{2} + \frac{23}{776} a + \frac{141}{776}$, $\frac{1}{39999356283120964720867422541244702455626664} a^{15} + \frac{3594917835000265409332489547374848594331}{19999678141560482360433711270622351227813332} a^{14} + \frac{3119100299812053531774554657757698415518923}{39999356283120964720867422541244702455626664} a^{13} - \frac{1117875154057254699182657925712097720028383}{39999356283120964720867422541244702455626664} a^{12} + \frac{421757496039358276452146723982397990586410}{4999919535390120590108427817655587806953333} a^{11} + \frac{9547325460541514212850150272878364992836387}{39999356283120964720867422541244702455626664} a^{10} + \frac{6914402466677263366439478700344612199112841}{39999356283120964720867422541244702455626664} a^{9} + \frac{610831872009260949157984962920150722737427}{9999839070780241180216855635311175613906666} a^{8} + \frac{5309116369376511895323253069898491478003641}{19999678141560482360433711270622351227813332} a^{7} + \frac{2513329181244425704654606891852031765457683}{39999356283120964720867422541244702455626664} a^{6} - \frac{15968598271876218046704389438288673665707457}{39999356283120964720867422541244702455626664} a^{5} - \frac{8575403393956604336177837941470840977844089}{19999678141560482360433711270622351227813332} a^{4} + \frac{1034376762219180624440820990512187754086245}{39999356283120964720867422541244702455626664} a^{3} + \frac{4836154380440621696724485239342812064629651}{39999356283120964720867422541244702455626664} a^{2} - \frac{7208040360971090531594746004127667002415701}{19999678141560482360433711270622351227813332} a + \frac{10899301849433014402591613126944426634714857}{39999356283120964720867422541244702455626664}$

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:  $11$
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:  \( 391920555.371 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T657):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), 4.4.9792.1, 4.4.4352.1, \(\Q(\sqrt{2}, \sqrt{3})\), 8.8.1534132224.1

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
3Data not computed
$17$17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
97Data not computed