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Properties

Label 16.8.29972682638...5248.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{16}\cdot 17^{15}\cdot 19993^{4}$
Root discriminant $338.68$
Ramified primes $2, 17, 19993$
Class number $16$ (GRH)
Class group $[2, 2, 4]$ (GRH)
Galois group 16T841

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2716193998733560817, 0, 390453736425761706, 0, -2580057810918438, 0, -18614015293632, 0, 87007049988, 0, 311883224, 0, -583899, 0, -1020, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 1020*x^14 - 583899*x^12 + 311883224*x^10 + 87007049988*x^8 - 18614015293632*x^6 - 2580057810918438*x^4 + 390453736425761706*x^2 + 2716193998733560817)
 
gp: K = bnfinit(x^16 - 1020*x^14 - 583899*x^12 + 311883224*x^10 + 87007049988*x^8 - 18614015293632*x^6 - 2580057810918438*x^4 + 390453736425761706*x^2 + 2716193998733560817, 1)
 

Normalized defining polynomial

\( x^{16} - 1020 x^{14} - 583899 x^{12} + 311883224 x^{10} + 87007049988 x^{8} - 18614015293632 x^{6} - 2580057810918438 x^{4} + 390453736425761706 x^{2} + 2716193998733560817 \)

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:  \(29972682638651894650701256229375114805248=2^{16}\cdot 17^{15}\cdot 19993^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $338.68$
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:  $2, 17, 19993$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{39986} a^{10} - \frac{510}{19993} a^{8} + \frac{15891}{39986} a^{6} - \frac{3788}{19993} a^{4} + \frac{13113}{39986} a^{2} - \frac{1}{2}$, $\frac{1}{39986} a^{11} - \frac{510}{19993} a^{9} + \frac{15891}{39986} a^{7} - \frac{3788}{19993} a^{5} + \frac{13113}{39986} a^{3} - \frac{1}{2} a$, $\frac{1}{799440098} a^{12} - \frac{510}{399720049} a^{10} - \frac{583899}{799440098} a^{8} + \frac{155941612}{399720049} a^{6} + \frac{267799355}{799440098} a^{4} - \frac{12593}{39986} a^{2}$, $\frac{1}{799440098} a^{13} - \frac{510}{399720049} a^{11} - \frac{583899}{799440098} a^{9} + \frac{155941612}{399720049} a^{7} + \frac{267799355}{799440098} a^{5} - \frac{12593}{39986} a^{3}$, $\frac{1}{9102851086208716918083616520253316993714226371098266} a^{14} + \frac{1905274390415824182626353573911857931427851}{4551425543104358459041808260126658496857113185549133} a^{12} + \frac{50516777518893516773305837345872173391174262453}{9102851086208716918083616520253316993714226371098266} a^{10} + \frac{1199702103828044399315637184959230848722557961771129}{9102851086208716918083616520253316993714226371098266} a^{8} - \frac{710015753262730191439944719765983457881975899723526}{4551425543104358459041808260126658496857113185549133} a^{6} - \frac{221620866687200978204367683744305357670668315403}{455301909978928470868984970752429199905678305962} a^{4} + \frac{5328481886658681560101278310458235071182412}{11386533036035824310233205890872535385026717} a^{2} - \frac{46536566730291383032531220524616295849}{1139051971793710229603681877744464100938}$, $\frac{1}{9102851086208716918083616520253316993714226371098266} a^{15} + \frac{1905274390415824182626353573911857931427851}{4551425543104358459041808260126658496857113185549133} a^{13} + \frac{50516777518893516773305837345872173391174262453}{9102851086208716918083616520253316993714226371098266} a^{11} + \frac{1199702103828044399315637184959230848722557961771129}{9102851086208716918083616520253316993714226371098266} a^{9} - \frac{710015753262730191439944719765983457881975899723526}{4551425543104358459041808260126658496857113185549133} a^{7} - \frac{221620866687200978204367683744305357670668315403}{455301909978928470868984970752429199905678305962} a^{5} + \frac{5328481886658681560101278310458235071182412}{11386533036035824310233205890872535385026717} a^{3} - \frac{46536566730291383032531220524616295849}{1139051971793710229603681877744464100938} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}$, which has order $16$ (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:  \( 9606579923990 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T841:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n841
Character table for t16n841 is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

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 $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
2.8.8.5$x^{8} + 2 x^{7} + 8 x^{2} + 16$$2$$4$$8$$C_8:C_2$$[2, 2]^{4}$
17Data not computed
19993Data not computed