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Properties

Label 16.0.43104723599...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{8}\cdot 17^{14}$
Root discriminant $53.35$
Ramified primes $2, 5, 17$
Class number $128$ (GRH)
Class group $[8, 16]$ (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![2209, 0, 25265, 0, 54752, 0, 49767, 0, 22483, 0, 5303, 0, 660, 0, 41, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 41*x^14 + 660*x^12 + 5303*x^10 + 22483*x^8 + 49767*x^6 + 54752*x^4 + 25265*x^2 + 2209)
 
gp: K = bnfinit(x^16 + 41*x^14 + 660*x^12 + 5303*x^10 + 22483*x^8 + 49767*x^6 + 54752*x^4 + 25265*x^2 + 2209, 1)
 

Normalized defining polynomial

\( x^{16} + 41 x^{14} + 660 x^{12} + 5303 x^{10} + 22483 x^{8} + 49767 x^{6} + 54752 x^{4} + 25265 x^{2} + 2209 \)

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:  \(4310472359920663782400000000=2^{16}\cdot 5^{8}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.35$
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:  $2, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(340=2^{2}\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{340}(1,·)$, $\chi_{340}(259,·)$, $\chi_{340}(9,·)$, $\chi_{340}(331,·)$, $\chi_{340}(81,·)$, $\chi_{340}(339,·)$, $\chi_{340}(21,·)$, $\chi_{340}(151,·)$, $\chi_{340}(239,·)$, $\chi_{340}(229,·)$, $\chi_{340}(291,·)$, $\chi_{340}(101,·)$, $\chi_{340}(111,·)$, $\chi_{340}(49,·)$, $\chi_{340}(189,·)$, $\chi_{340}(319,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{13} a^{10} - \frac{6}{13} a^{8} - \frac{3}{13} a^{6} + \frac{5}{13} a^{4} - \frac{4}{13} a^{2} - \frac{2}{13}$, $\frac{1}{13} a^{11} - \frac{6}{13} a^{9} - \frac{3}{13} a^{7} + \frac{5}{13} a^{5} - \frac{4}{13} a^{3} - \frac{2}{13} a$, $\frac{1}{13} a^{12} + \frac{1}{13}$, $\frac{1}{13} a^{13} + \frac{1}{13} a$, $\frac{1}{178120267} a^{14} + \frac{3999756}{178120267} a^{12} - \frac{1411964}{178120267} a^{10} + \frac{197427}{178120267} a^{8} + \frac{30910618}{178120267} a^{6} - \frac{66952510}{178120267} a^{4} + \frac{79400055}{178120267} a^{2} - \frac{48222918}{178120267}$, $\frac{1}{8371652549} a^{15} - \frac{133015834}{8371652549} a^{13} - \frac{289144703}{8371652549} a^{11} + \frac{3329676264}{8371652549} a^{9} - \frac{3558897840}{8371652549} a^{7} + \frac{2234909402}{8371652549} a^{5} - \frac{289002805}{643973273} a^{3} + \frac{2349549907}{8371652549} a$

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

Class group and class number

$C_{8}\times C_{16}$, which has order $128$ (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:  \( 81485.0410293661 \) (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

\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-85}) \), \(\Q(\sqrt{-5}, \sqrt{17})\), 4.4.4913.1, 4.0.1965200.1, 8.0.3862011040000.2, 8.8.256461670625.1, 8.0.105046700288.1

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
5Data not computed
17Data not computed