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Properties

Label 16.0.20555051435...3616.5
Degree $16$
Signature $[0, 8]$
Discriminant $2^{64}\cdot 3^{8}\cdot 19^{8}$
Root discriminant $120.80$
Ramified primes $2, 3, 19$
Class number $1079104$ (GRH)
Class group $[2, 539552]$ (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![15567303361, 0, 6746464256, 0, 2529924096, 0, 361417728, 0, 25354560, 0, 965888, 0, 20384, 0, 224, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 224*x^14 + 20384*x^12 + 965888*x^10 + 25354560*x^8 + 361417728*x^6 + 2529924096*x^4 + 6746464256*x^2 + 15567303361)
 
gp: K = bnfinit(x^16 + 224*x^14 + 20384*x^12 + 965888*x^10 + 25354560*x^8 + 361417728*x^6 + 2529924096*x^4 + 6746464256*x^2 + 15567303361, 1)
 

Normalized defining polynomial

\( x^{16} + 224 x^{14} + 20384 x^{12} + 965888 x^{10} + 25354560 x^{8} + 361417728 x^{6} + 2529924096 x^{4} + 6746464256 x^{2} + 15567303361 \)

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:  \(2055505143594254982418411202543616=2^{64}\cdot 3^{8}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $120.80$
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:  $2, 3, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1824=2^{5}\cdot 3\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{1824}(1,·)$, $\chi_{1824}(455,·)$, $\chi_{1824}(457,·)$, $\chi_{1824}(911,·)$, $\chi_{1824}(913,·)$, $\chi_{1824}(1367,·)$, $\chi_{1824}(1369,·)$, $\chi_{1824}(1823,·)$, $\chi_{1824}(227,·)$, $\chi_{1824}(229,·)$, $\chi_{1824}(683,·)$, $\chi_{1824}(685,·)$, $\chi_{1824}(1139,·)$, $\chi_{1824}(1141,·)$, $\chi_{1824}(1595,·)$, $\chi_{1824}(1597,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{13021} a^{8} + \frac{112}{13021} a^{6} + \frac{3920}{13021} a^{4} + \frac{4841}{13021} a^{2} - \frac{1294}{13021}$, $\frac{1}{1624617149} a^{9} + \frac{113165623}{1624617149} a^{7} - \frac{282096045}{1624617149} a^{5} + \frac{224330629}{1624617149} a^{3} - \frac{54533242}{1624617149} a$, $\frac{1}{1624617149} a^{10} + \frac{140}{1624617149} a^{8} + \frac{40307051}{1624617149} a^{6} + \frac{136118946}{1624617149} a^{4} - \frac{392657232}{1624617149} a^{2} + \frac{1768}{13021}$, $\frac{1}{1624617149} a^{11} + \frac{443291321}{1624617149} a^{7} + \frac{638753670}{1624617149} a^{5} + \frac{693397688}{1624617149} a^{3} - \frac{267840273}{1624617149} a$, $\frac{1}{1624617149} a^{12} - \frac{12936}{1624617149} a^{8} - \frac{272808644}{1624617149} a^{6} - \frac{343557471}{1624617149} a^{4} - \frac{184494581}{1624617149} a^{2} + \frac{1169}{13021}$, $\frac{1}{1624617149} a^{13} - \frac{142360765}{1624617149} a^{7} - \frac{22340653}{56021281} a^{5} + \frac{190294049}{1624617149} a^{3} - \frac{212320885}{1624617149} a$, $\frac{1}{1624617149} a^{14} + \frac{664}{1624617149} a^{8} + \frac{675046770}{1624617149} a^{6} - \frac{621203527}{1624617149} a^{4} + \frac{121685728}{1624617149} a^{2} - \frac{5081}{13021}$, $\frac{1}{1624617149} a^{15} + \frac{265461952}{1624617149} a^{7} - \frac{140401782}{1624617149} a^{5} + \frac{630925780}{1624617149} a^{3} - \frac{165455879}{1624617149} a$

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

Class group and class number

$C_{2}\times C_{539552}$, which has order $1079104$ (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

\(\Q(\sqrt{-114}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-57}) \), \(\Q(\sqrt{2}, \sqrt{-57})\), \(\Q(\zeta_{16})^+\), 4.0.6653952.2, 8.0.177100308873216.55, 8.0.22668839535771648.22, \(\Q(\zeta_{32})^+\)

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

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
3Data not computed
19Data not computed