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Properties

Label 16.0.36093282486...0449.1
Degree $16$
Signature $[0, 8]$
Discriminant $11^{8}\cdot 17^{14}$
Root discriminant $39.57$
Ramified primes $11, 17$
Class number $289$ (GRH)
Class group $[17, 17]$ (GRH)
Galois group $C_8\times C_2$ (as 16T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![153851, -83057, 155680, -84235, 84686, -43150, 31149, -14574, 8268, -3427, 1652, -610, 229, -68, 24, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 24*x^14 - 68*x^13 + 229*x^12 - 610*x^11 + 1652*x^10 - 3427*x^9 + 8268*x^8 - 14574*x^7 + 31149*x^6 - 43150*x^5 + 84686*x^4 - 84235*x^3 + 155680*x^2 - 83057*x + 153851)
 
gp: K = bnfinit(x^16 - 6*x^15 + 24*x^14 - 68*x^13 + 229*x^12 - 610*x^11 + 1652*x^10 - 3427*x^9 + 8268*x^8 - 14574*x^7 + 31149*x^6 - 43150*x^5 + 84686*x^4 - 84235*x^3 + 155680*x^2 - 83057*x + 153851, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 24 x^{14} - 68 x^{13} + 229 x^{12} - 610 x^{11} + 1652 x^{10} - 3427 x^{9} + 8268 x^{8} - 14574 x^{7} + 31149 x^{6} - 43150 x^{5} + 84686 x^{4} - 84235 x^{3} + 155680 x^{2} - 83057 x + 153851 \)

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:  \(36093282486485263170800449=11^{8}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.57$
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:  $11, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(187=11\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{187}(1,·)$, $\chi_{187}(67,·)$, $\chi_{187}(76,·)$, $\chi_{187}(144,·)$, $\chi_{187}(21,·)$, $\chi_{187}(87,·)$, $\chi_{187}(89,·)$, $\chi_{187}(155,·)$, $\chi_{187}(32,·)$, $\chi_{187}(98,·)$, $\chi_{187}(100,·)$, $\chi_{187}(166,·)$, $\chi_{187}(43,·)$, $\chi_{187}(111,·)$, $\chi_{187}(120,·)$, $\chi_{187}(186,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{13802} a^{14} - \frac{981}{13802} a^{13} + \frac{827}{13802} a^{12} - \frac{6297}{13802} a^{11} - \frac{5805}{13802} a^{10} + \frac{1801}{13802} a^{9} + \frac{1997}{13802} a^{8} + \frac{1817}{13802} a^{7} - \frac{2085}{13802} a^{6} + \frac{4409}{13802} a^{5} - \frac{725}{13802} a^{4} + \frac{753}{13802} a^{3} + \frac{15}{206} a^{2} - \frac{6795}{13802} a - \frac{6563}{13802}$, $\frac{1}{9372523636405670211392627698} a^{15} + \frac{135318817825228358218445}{4686261818202835105696313849} a^{14} - \frac{2153054006965375201896716463}{9372523636405670211392627698} a^{13} + \frac{2029247652046227814676377107}{9372523636405670211392627698} a^{12} + \frac{1437420464009289871097819769}{4686261818202835105696313849} a^{11} + \frac{710576144967424822656874753}{9372523636405670211392627698} a^{10} - \frac{3597385380185689874084263927}{9372523636405670211392627698} a^{9} - \frac{1837200686840751521445445980}{4686261818202835105696313849} a^{8} + \frac{3302069108083373722027226635}{9372523636405670211392627698} a^{7} + \frac{315003385576434820840899731}{9372523636405670211392627698} a^{6} - \frac{1206279126078223146203943910}{4686261818202835105696313849} a^{5} - \frac{4617775304147148139314078497}{9372523636405670211392627698} a^{4} - \frac{2704874032501126325708396837}{9372523636405670211392627698} a^{3} - \frac{919389224458360855686647018}{4686261818202835105696313849} a^{2} + \frac{573790673476742018139897935}{9372523636405670211392627698} a + \frac{1667815491161000476499243274}{4686261818202835105696313849}$

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

Class group and class number

$C_{17}\times C_{17}$, which has order $289$ (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:  \( 3640.01221338 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_8$ (as 16T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{-187}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-11}, \sqrt{17})\), 4.4.4913.1, 4.0.594473.1, 8.0.353398147729.1, \(\Q(\zeta_{17})^+\), 8.0.6007768511393.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
11Data not computed
17Data not computed