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Properties

Label 16.0.36046552944...0000.9
Degree $16$
Signature $[0, 8]$
Discriminant $2^{64}\cdot 5^{8}\cdot 29^{8}$
Root discriminant $192.67$
Ramified primes $2, 5, 29$
Class number $48456000$ (GRH)
Class group $[5, 30, 323040]$ (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![17072903371969, 0, 5015306502144, 0, 731398864896, 0, 40633270272, 0, 1108546560, 0, 16422912, 0, 134784, 0, 576, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 576*x^14 + 134784*x^12 + 16422912*x^10 + 1108546560*x^8 + 40633270272*x^6 + 731398864896*x^4 + 5015306502144*x^2 + 17072903371969)
 
gp: K = bnfinit(x^16 + 576*x^14 + 134784*x^12 + 16422912*x^10 + 1108546560*x^8 + 40633270272*x^6 + 731398864896*x^4 + 5015306502144*x^2 + 17072903371969, 1)
 

Normalized defining polynomial

\( x^{16} + 576 x^{14} + 134784 x^{12} + 16422912 x^{10} + 1108546560 x^{8} + 40633270272 x^{6} + 731398864896 x^{4} + 5015306502144 x^{2} + 17072903371969 \)

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:  \(3604655294407338976901437849600000000=2^{64}\cdot 5^{8}\cdot 29^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $192.67$
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:  $2, 5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4640=2^{5}\cdot 5\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{4640}(1,·)$, $\chi_{4640}(579,·)$, $\chi_{4640}(581,·)$, $\chi_{4640}(1159,·)$, $\chi_{4640}(1161,·)$, $\chi_{4640}(1739,·)$, $\chi_{4640}(1741,·)$, $\chi_{4640}(2319,·)$, $\chi_{4640}(2321,·)$, $\chi_{4640}(2899,·)$, $\chi_{4640}(2901,·)$, $\chi_{4640}(3479,·)$, $\chi_{4640}(3481,·)$, $\chi_{4640}(4059,·)$, $\chi_{4640}(4061,·)$, $\chi_{4640}(4639,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{199801} a^{8} + \frac{288}{199801} a^{6} + \frac{25920}{199801} a^{4} - \frac{52708}{199801} a^{2} - \frac{37385}{199801}$, $\frac{1}{825565144537} a^{9} - \frac{128763552371}{825565144537} a^{7} - \frac{251374607205}{825565144537} a^{5} + \frac{63460341310}{825565144537} a^{3} + \frac{316710921347}{825565144537} a$, $\frac{1}{825565144537} a^{10} + \frac{360}{825565144537} a^{8} - \frac{317902924842}{825565144537} a^{6} - \frac{20733176323}{117937877791} a^{4} - \frac{23949313533}{48562655561} a^{2} + \frac{10876}{199801}$, $\frac{1}{825565144537} a^{11} - \frac{194672165354}{825565144537} a^{7} + \frac{363125605006}{825565144537} a^{5} - \frac{19576736375}{117937877791} a^{3} - \frac{43002792002}{825565144537} a$, $\frac{1}{825565144537} a^{12} - \frac{85536}{825565144537} a^{8} + \frac{290254764074}{825565144537} a^{6} - \frac{90891682209}{825565144537} a^{4} + \frac{5660678749}{35894136719} a^{2} + \frac{88726}{199801}$, $\frac{1}{825565144537} a^{13} + \frac{235632426335}{825565144537} a^{7} + \frac{174895897076}{825565144537} a^{5} + \frac{183124572612}{825565144537} a^{3} - \frac{368239402401}{825565144537} a$, $\frac{1}{825565144537} a^{14} + \frac{455036}{825565144537} a^{8} + \frac{9230014998}{825565144537} a^{6} + \frac{5798599446}{35894136719} a^{4} + \frac{345234555200}{825565144537} a^{2} + \frac{77725}{199801}$, $\frac{1}{825565144537} a^{15} + \frac{7372660770}{117937877791} a^{7} + \frac{14522983811}{117937877791} a^{5} + \frac{222991833226}{825565144537} a^{3} + \frac{229804851238}{825565144537} a$

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

Class group and class number

$C_{5}\times C_{30}\times C_{323040}$, which has order $48456000$ (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:  \( 15753.94986242651 \) (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

\(\Q(\sqrt{-145}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-290}) \), \(\Q(\sqrt{2}, \sqrt{-145})\), \(\Q(\zeta_{16})^+\), 4.0.43059200.3, 8.0.7416378818560000.32, \(\Q(\zeta_{32})^+\), 8.0.949296488775680000.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
5Data not computed
29Data not computed