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Properties

Label 16.0.14876562920...4896.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{64}\cdot 73^{8}$
Root discriminant $136.70$
Ramified primes $2, 73$
Class number $2812032$ (GRH)
Class group $[6, 468672]$ (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![96031192321, 0, 39182082048, 0, 11428107264, 0, 1269789696, 0, 69284160, 0, 2052864, 0, 33696, 0, 288, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 288*x^14 + 33696*x^12 + 2052864*x^10 + 69284160*x^8 + 1269789696*x^6 + 11428107264*x^4 + 39182082048*x^2 + 96031192321)
 
gp: K = bnfinit(x^16 + 288*x^14 + 33696*x^12 + 2052864*x^10 + 69284160*x^8 + 1269789696*x^6 + 11428107264*x^4 + 39182082048*x^2 + 96031192321, 1)
 

Normalized defining polynomial

\( x^{16} + 288 x^{14} + 33696 x^{12} + 2052864 x^{10} + 69284160 x^{8} + 1269789696 x^{6} + 11428107264 x^{4} + 39182082048 x^{2} + 96031192321 \)

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:  \(14876562920830399091974866674384896=2^{64}\cdot 73^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $136.70$
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:  $2, 73$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2336=2^{5}\cdot 73\)
Dirichlet character group:    $\lbrace$$\chi_{2336}(1,·)$, $\chi_{2336}(583,·)$, $\chi_{2336}(585,·)$, $\chi_{2336}(1167,·)$, $\chi_{2336}(1169,·)$, $\chi_{2336}(1751,·)$, $\chi_{2336}(1753,·)$, $\chi_{2336}(2335,·)$, $\chi_{2336}(291,·)$, $\chi_{2336}(293,·)$, $\chi_{2336}(875,·)$, $\chi_{2336}(877,·)$, $\chi_{2336}(1459,·)$, $\chi_{2336}(1461,·)$, $\chi_{2336}(2043,·)$, $\chi_{2336}(2045,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{26677} a^{8} + \frac{144}{26677} a^{6} + \frac{6480}{26677} a^{4} + \frac{13281}{26677} a^{2} - \frac{3464}{26677}$, $\frac{1}{8266908853} a^{9} - \frac{2405358238}{8266908853} a^{7} + \frac{2800477909}{8266908853} a^{5} + \frac{1614158520}{8266908853} a^{3} - \frac{2007020882}{8266908853} a$, $\frac{1}{8266908853} a^{10} + \frac{180}{8266908853} a^{8} + \frac{1961918275}{8266908853} a^{6} - \frac{436198514}{1180986979} a^{4} + \frac{222320584}{8266908853} a^{2} + \frac{2848}{26677}$, $\frac{1}{8266908853} a^{11} - \frac{3219768094}{8266908853} a^{7} - \frac{2857973185}{8266908853} a^{5} - \frac{140629023}{1180986979} a^{3} - \frac{1597666900}{8266908853} a$, $\frac{1}{8266908853} a^{12} - \frac{21384}{8266908853} a^{8} - \frac{2161342713}{8266908853} a^{6} - \frac{2703667333}{8266908853} a^{4} + \frac{3405800894}{8266908853} a^{2} - \frac{3687}{26677}$, $\frac{1}{8266908853} a^{13} - \frac{1635020739}{8266908853} a^{7} - \frac{2771792409}{8266908853} a^{5} - \frac{2039777554}{8266908853} a^{3} + \frac{2513663345}{8266908853} a$, $\frac{1}{8266908853} a^{14} - \frac{6625}{1180986979} a^{8} + \frac{170152589}{1180986979} a^{6} + \frac{383408639}{1180986979} a^{4} - \frac{561365202}{8266908853} a^{2} - \frac{2319}{26677}$, $\frac{1}{8266908853} a^{15} - \frac{270866514}{1180986979} a^{7} + \frac{244115674}{1180986979} a^{5} - \frac{819664117}{8266908853} a^{3} + \frac{814740586}{8266908853} a$

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

Class group and class number

$C_{6}\times C_{468672}$, which has order $2812032$ (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

\(\Q(\sqrt{-146}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-73}) \), \(\Q(\sqrt{2}, \sqrt{-73})\), \(\Q(\zeta_{16})^+\), 4.0.10913792.5, 8.0.476443423277056.34, 8.0.60984758179463168.47, \(\Q(\zeta_{32})^+\)

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$73$73.8.4.1$x^{8} + 138554 x^{4} - 389017 x^{2} + 4799302729$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
73.8.4.1$x^{8} + 138554 x^{4} - 389017 x^{2} + 4799302729$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$