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Properties

Label 16.0.34381905092...0000.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{79}\cdot 5^{12}\cdot 13^{12}$
Root discriminant $701.49$
Ramified primes $2, 5, 13$
Class number $1451749392$ (GRH)
Class group $[3, 3, 12, 13442124]$ (GRH)
Galois group $C_{16}$ (as 16T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![102812781061250, 0, 313742585000000, 0, 25340747250000, 0, 779715300000, 0, 11781412500, 0, 96668000, 0, 439400, 0, 1040, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 1040*x^14 + 439400*x^12 + 96668000*x^10 + 11781412500*x^8 + 779715300000*x^6 + 25340747250000*x^4 + 313742585000000*x^2 + 102812781061250)
 
gp: K = bnfinit(x^16 + 1040*x^14 + 439400*x^12 + 96668000*x^10 + 11781412500*x^8 + 779715300000*x^6 + 25340747250000*x^4 + 313742585000000*x^2 + 102812781061250, 1)
 

Normalized defining polynomial

\( x^{16} + 1040 x^{14} + 439400 x^{12} + 96668000 x^{10} + 11781412500 x^{8} + 779715300000 x^{6} + 25340747250000 x^{4} + 313742585000000 x^{2} + 102812781061250 \)

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:  \(3438190509295256478027799611834368000000000000=2^{79}\cdot 5^{12}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $701.49$
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:  $2, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4160=2^{6}\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{4160}(1,·)$, $\chi_{4160}(707,·)$, $\chi_{4160}(649,·)$, $\chi_{4160}(203,·)$, $\chi_{4160}(1041,·)$, $\chi_{4160}(1747,·)$, $\chi_{4160}(1689,·)$, $\chi_{4160}(1243,·)$, $\chi_{4160}(2081,·)$, $\chi_{4160}(2787,·)$, $\chi_{4160}(2729,·)$, $\chi_{4160}(2283,·)$, $\chi_{4160}(3121,·)$, $\chi_{4160}(3827,·)$, $\chi_{4160}(3769,·)$, $\chi_{4160}(3323,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{65} a^{4}$, $\frac{1}{65} a^{5}$, $\frac{1}{65} a^{6}$, $\frac{1}{65} a^{7}$, $\frac{1}{24205025} a^{8} + \frac{8}{372385} a^{6} + \frac{20}{5729} a^{4} + \frac{1040}{5729} a^{2} + \frac{2721}{5729}$, $\frac{1}{41075927425} a^{9} - \frac{896014}{126387469} a^{7} + \frac{641908}{126387469} a^{5} - \frac{3304593}{9722113} a^{3} + \frac{4379677}{9722113} a$, $\frac{1}{41075927425} a^{10} + \frac{2}{126387469} a^{8} - \frac{91196}{126387469} a^{6} - \frac{3510166}{631937345} a^{4} + \frac{249742}{571889} a^{2} - \frac{1358}{5729}$, $\frac{1}{41075927425} a^{11} + \frac{4677733}{631937345} a^{7} + \frac{543129}{631937345} a^{5} + \frac{3644091}{9722113} a^{3} - \frac{515467}{9722113} a$, $\frac{1}{2669935282625} a^{12} + \frac{801}{41075927425} a^{8} - \frac{840469}{126387469} a^{6} - \frac{46884}{631937345} a^{4} + \frac{628993}{9722113} a^{2} + \frac{185}{5729}$, $\frac{1}{2669935282625} a^{13} - \frac{3125972}{631937345} a^{7} - \frac{4250592}{631937345} a^{5} + \frac{3193250}{9722113} a^{3} + \frac{1875461}{9722113} a$, $\frac{1}{2669935282625} a^{14} + \frac{418}{41075927425} a^{8} - \frac{2334679}{631937345} a^{6} - \frac{1235933}{631937345} a^{4} - \frac{1830787}{9722113} a^{2} - \frac{558}{5729}$, $\frac{1}{2669935282625} a^{15} + \frac{737777}{126387469} a^{7} - \frac{1172059}{631937345} a^{5} - \frac{1050959}{9722113} a^{3} - \frac{3894668}{9722113} a$

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

Class group and class number

$C_{3}\times C_{3}\times C_{12}\times C_{13442124}$, which has order $1451749392$ (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:  \( 23041805.370027676 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{16}$ (as 16T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 16
The 16 conjugacy class representatives for $C_{16}$
Character table for $C_{16}$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.8.38333925294080000.1

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 $16$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ $16$ R ${\href{/LocalNumberField/17.1.0.1}{1} }^{16}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ $16$ $16$

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
13Data not computed