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Properties

Label 16.0.13872985106...9121.9
Degree $16$
Signature $[0, 8]$
Discriminant $17^{12}\cdot 47^{8}$
Root discriminant $57.40$
Ramified primes $17, 47$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T172)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![168437, -814322, 1466315, -1154488, 299347, 37860, 48569, -51128, -13366, 18420, -825, -2516, 381, 162, -34, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 34*x^14 + 162*x^13 + 381*x^12 - 2516*x^11 - 825*x^10 + 18420*x^9 - 13366*x^8 - 51128*x^7 + 48569*x^6 + 37860*x^5 + 299347*x^4 - 1154488*x^3 + 1466315*x^2 - 814322*x + 168437)
 
gp: K = bnfinit(x^16 - 4*x^15 - 34*x^14 + 162*x^13 + 381*x^12 - 2516*x^11 - 825*x^10 + 18420*x^9 - 13366*x^8 - 51128*x^7 + 48569*x^6 + 37860*x^5 + 299347*x^4 - 1154488*x^3 + 1466315*x^2 - 814322*x + 168437, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 34 x^{14} + 162 x^{13} + 381 x^{12} - 2516 x^{11} - 825 x^{10} + 18420 x^{9} - 13366 x^{8} - 51128 x^{7} + 48569 x^{6} + 37860 x^{5} + 299347 x^{4} - 1154488 x^{3} + 1466315 x^{2} - 814322 x + 168437 \)

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:  \(13872985106194361214049869121=17^{12}\cdot 47^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.40$
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:  $17, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{12} a^{8} - \frac{1}{6} a^{7} + \frac{1}{12} a^{6} + \frac{1}{12} a^{5} + \frac{1}{3} a^{4} + \frac{1}{12} a^{3} + \frac{1}{12} a^{2} + \frac{1}{12} a + \frac{5}{12}$, $\frac{1}{12} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{5}{12} a + \frac{1}{3}$, $\frac{1}{12} a^{10} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{6} a^{2} + \frac{1}{12} a + \frac{1}{4}$, $\frac{1}{12} a^{11} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{12} a^{3} + \frac{1}{3} a^{2} + \frac{1}{4}$, $\frac{1}{72} a^{12} + \frac{1}{72} a^{10} - \frac{1}{36} a^{9} - \frac{1}{72} a^{8} + \frac{7}{36} a^{7} + \frac{5}{72} a^{6} - \frac{5}{36} a^{5} - \frac{1}{4} a^{4} - \frac{1}{6} a^{3} + \frac{1}{12} a^{2} + \frac{5}{36} a + \frac{23}{72}$, $\frac{1}{72} a^{13} + \frac{1}{72} a^{11} - \frac{1}{36} a^{10} - \frac{1}{72} a^{9} + \frac{1}{36} a^{8} - \frac{7}{72} a^{7} + \frac{7}{36} a^{6} + \frac{1}{12} a^{5} - \frac{1}{3} a^{4} - \frac{1}{12} a^{3} - \frac{1}{36} a^{2} - \frac{25}{72} a + \frac{1}{6}$, $\frac{1}{144} a^{14} - \frac{1}{144} a^{13} - \frac{1}{48} a^{11} + \frac{5}{144} a^{9} - \frac{1}{72} a^{8} - \frac{5}{144} a^{7} + \frac{29}{144} a^{6} - \frac{25}{72} a^{5} - \frac{1}{3} a^{4} - \frac{7}{72} a^{3} + \frac{49}{144} a^{2} + \frac{11}{48} a + \frac{31}{144}$, $\frac{1}{17023895295409133092803312} a^{15} - \frac{7761246168761378894927}{5674631765136377697601104} a^{14} + \frac{15470604198750199756409}{2837315882568188848800552} a^{13} + \frac{3632187259748303280563}{1891543921712125899200368} a^{12} - \frac{41664647401524745236595}{8511947647704566546401656} a^{11} + \frac{31672285684912074766769}{5674631765136377697601104} a^{10} + \frac{2763267200625022416944}{1063993455963070818300207} a^{9} + \frac{165318219617709286928077}{17023895295409133092803312} a^{8} + \frac{929869158624743594339219}{17023895295409133092803312} a^{7} - \frac{522283219696482799564087}{4255973823852283273200828} a^{6} + \frac{1114803912167329452613249}{4255973823852283273200828} a^{5} - \frac{1038647822368447092004189}{8511947647704566546401656} a^{4} + \frac{3477015860963055931869065}{17023895295409133092803312} a^{3} + \frac{5436942284411650929912125}{17023895295409133092803312} a^{2} - \frac{425341035983925590842399}{1891543921712125899200368} a + \frac{4210371383889693029465081}{8511947647704566546401656}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  \( 3414476.88793 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T172):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.0.10852817.2, 4.2.13583.1, 4.2.230911.1, 8.0.3136464113.1, 8.4.6928449225617.1, 8.0.117783636835489.3

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$17$17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$47$47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$