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Properties

Label 16.0.90115968700...6849.4
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 13^{8}\cdot 17^{14}$
Root discriminant $74.50$
Ramified primes $3, 13, 17$
Class number $36864$ (GRH)
Class group $[4, 4, 12, 192]$ (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![333123229, -120478059, 190235724, -61397739, 50660274, -14377568, 8155895, -2002980, 864452, -179463, 61600, -10382, 2875, -362, 80, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 80*x^14 - 362*x^13 + 2875*x^12 - 10382*x^11 + 61600*x^10 - 179463*x^9 + 864452*x^8 - 2002980*x^7 + 8155895*x^6 - 14377568*x^5 + 50660274*x^4 - 61397739*x^3 + 190235724*x^2 - 120478059*x + 333123229)
 
gp: K = bnfinit(x^16 - 6*x^15 + 80*x^14 - 362*x^13 + 2875*x^12 - 10382*x^11 + 61600*x^10 - 179463*x^9 + 864452*x^8 - 2002980*x^7 + 8155895*x^6 - 14377568*x^5 + 50660274*x^4 - 61397739*x^3 + 190235724*x^2 - 120478059*x + 333123229, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 80 x^{14} - 362 x^{13} + 2875 x^{12} - 10382 x^{11} + 61600 x^{10} - 179463 x^{9} + 864452 x^{8} - 2002980 x^{7} + 8155895 x^{6} - 14377568 x^{5} + 50660274 x^{4} - 61397739 x^{3} + 190235724 x^{2} - 120478059 x + 333123229 \)

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:  \(901159687005577446635674386849=3^{8}\cdot 13^{8}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.50$
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:  $3, 13, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(663=3\cdot 13\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{663}(1,·)$, $\chi_{663}(196,·)$, $\chi_{663}(389,·)$, $\chi_{663}(586,·)$, $\chi_{663}(77,·)$, $\chi_{663}(274,·)$, $\chi_{663}(467,·)$, $\chi_{663}(662,·)$, $\chi_{663}(155,·)$, $\chi_{663}(157,·)$, $\chi_{663}(545,·)$, $\chi_{663}(38,·)$, $\chi_{663}(625,·)$, $\chi_{663}(118,·)$, $\chi_{663}(506,·)$, $\chi_{663}(508,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{707559898229858732670008955739610191519885486} a^{15} + \frac{36983344400399505453322399944877912128434943}{707559898229858732670008955739610191519885486} a^{14} - \frac{32661980995904241614345408389761890545851615}{353779949114929366335004477869805095759942743} a^{13} - \frac{55085233924641028840090826757588725097081557}{707559898229858732670008955739610191519885486} a^{12} - \frac{20185437521173527746559328558331742009727301}{707559898229858732670008955739610191519885486} a^{11} - \frac{9843281883067889561341654817599318069100380}{353779949114929366335004477869805095759942743} a^{10} - \frac{176248807541448750574288111536341038247491945}{707559898229858732670008955739610191519885486} a^{9} - \frac{92932416474324531291129951133692315463085887}{707559898229858732670008955739610191519885486} a^{8} + \frac{51838610213749262899044504543248591451469612}{353779949114929366335004477869805095759942743} a^{7} + \frac{294554209885873440803188818915441893006537227}{707559898229858732670008955739610191519885486} a^{6} - \frac{251624163184129360520070841071525240158267089}{707559898229858732670008955739610191519885486} a^{5} - \frac{35950523758550727538713218743786368430232975}{353779949114929366335004477869805095759942743} a^{4} + \frac{353690848254412348422557690309477665159069847}{707559898229858732670008955739610191519885486} a^{3} + \frac{306770531952130665069425078615590185814993915}{707559898229858732670008955739610191519885486} a^{2} + \frac{94829680869449560763165755624188060198352536}{353779949114929366335004477869805095759942743} a - \frac{100813092612244984235238364066932046344429772}{353779949114929366335004477869805095759942743}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{12}\times C_{192}$, which has order $36864$ (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:  \( 3640.012213375973 \) (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

\(\Q(\sqrt{-39}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-663}) \), \(\Q(\sqrt{17}, \sqrt{-39})\), 4.4.4913.1, 4.0.7472673.1, 8.0.55840841764929.8, \(\Q(\zeta_{17})^+\), 8.0.949294310003793.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R 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
3Data not computed
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17Data not computed