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Properties

Label 16.8.25893015832...5017.7
Degree $16$
Signature $[8, 4]$
Discriminant $17^{15}\cdot 67^{6}$
Root discriminant $68.92$
Ramified primes $17, 67$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T841

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11279467, 70702018, -15288526, -16651844, 4095607, -971393, -267734, 146145, -16387, 17140, 577, 115, -169, -98, 16, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 16*x^14 - 98*x^13 - 169*x^12 + 115*x^11 + 577*x^10 + 17140*x^9 - 16387*x^8 + 146145*x^7 - 267734*x^6 - 971393*x^5 + 4095607*x^4 - 16651844*x^3 - 15288526*x^2 + 70702018*x + 11279467)
 
gp: K = bnfinit(x^16 - 4*x^15 + 16*x^14 - 98*x^13 - 169*x^12 + 115*x^11 + 577*x^10 + 17140*x^9 - 16387*x^8 + 146145*x^7 - 267734*x^6 - 971393*x^5 + 4095607*x^4 - 16651844*x^3 - 15288526*x^2 + 70702018*x + 11279467, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 16 x^{14} - 98 x^{13} - 169 x^{12} + 115 x^{11} + 577 x^{10} + 17140 x^{9} - 16387 x^{8} + 146145 x^{7} - 267734 x^{6} - 971393 x^{5} + 4095607 x^{4} - 16651844 x^{3} - 15288526 x^{2} + 70702018 x + 11279467 \)

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:  \(258930158322830089457985795017=17^{15}\cdot 67^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.92$
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:  $17, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{12989819896620557660964924028832967008286554961935900960081} a^{15} + \frac{5747708085420257308374135258944244830766170164437252163224}{12989819896620557660964924028832967008286554961935900960081} a^{14} - \frac{3543889689760933209059402208568696338644134228231045282219}{12989819896620557660964924028832967008286554961935900960081} a^{13} + \frac{5911409469667283559666350082643866583210892994063902686396}{12989819896620557660964924028832967008286554961935900960081} a^{12} - \frac{1462806530445609880973635024310393286412954109532134730618}{12989819896620557660964924028832967008286554961935900960081} a^{11} - \frac{4841845268001264387062499749468658125415044624610133837354}{12989819896620557660964924028832967008286554961935900960081} a^{10} + \frac{3719542667250509876264431012279121373763228454384475423806}{12989819896620557660964924028832967008286554961935900960081} a^{9} - \frac{1363577416705679301718699390406342429432567763821525677948}{12989819896620557660964924028832967008286554961935900960081} a^{8} - \frac{4512576181288643062709106345174944923836461804047237509088}{12989819896620557660964924028832967008286554961935900960081} a^{7} + \frac{3532668777893694791010471140717871793848443953666329756811}{12989819896620557660964924028832967008286554961935900960081} a^{6} - \frac{2131939316348627345895445395833188979808922910339338182423}{12989819896620557660964924028832967008286554961935900960081} a^{5} + \frac{5515865103378299834890650120987267367780403101486490771465}{12989819896620557660964924028832967008286554961935900960081} a^{4} - \frac{5695689007518167033944438786782564221321096797879230268236}{12989819896620557660964924028832967008286554961935900960081} a^{3} + \frac{28148442586082893550987821127346708788710386971176865317}{12989819896620557660964924028832967008286554961935900960081} a^{2} - \frac{1818504435307215588600701684331356517965630546877642383599}{12989819896620557660964924028832967008286554961935900960081} a + \frac{1491831632990557480275634830751279133516047387521244892697}{12989819896620557660964924028832967008286554961935900960081}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 93751163.0029 \) (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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ 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
17Data not computed
67Data not computed