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Properties

Label 16.0.45905397709...6736.6
Degree $16$
Signature $[0, 8]$
Discriminant $2^{64}\cdot 3^{8}\cdot 41^{14}$
Root discriminant $714.28$
Ramified primes $2, 3, 41$
Class number $3295289344$ (GRH)
Class group $[2, 2, 2, 2, 8, 8, 3218056]$ (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![16775171361, 0, 89879331456, 0, 36798327216, 0, 5736432672, 0, 405682372, 0, 12664736, 0, 144648, 0, 656, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 656*x^14 + 144648*x^12 + 12664736*x^10 + 405682372*x^8 + 5736432672*x^6 + 36798327216*x^4 + 89879331456*x^2 + 16775171361)
 
gp: K = bnfinit(x^16 + 656*x^14 + 144648*x^12 + 12664736*x^10 + 405682372*x^8 + 5736432672*x^6 + 36798327216*x^4 + 89879331456*x^2 + 16775171361, 1)
 

Normalized defining polynomial

\( x^{16} + 656 x^{14} + 144648 x^{12} + 12664736 x^{10} + 405682372 x^{8} + 5736432672 x^{6} + 36798327216 x^{4} + 89879331456 x^{2} + 16775171361 \)

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:  \(4590539770923916035789355095649376383046516736=2^{64}\cdot 3^{8}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $714.28$
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:  $2, 3, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3936=2^{5}\cdot 3\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{3936}(1,·)$, $\chi_{3936}(1667,·)$, $\chi_{3936}(2053,·)$, $\chi_{3936}(73,·)$, $\chi_{3936}(3851,·)$, $\chi_{3936}(3289,·)$, $\chi_{3936}(3611,·)$, $\chi_{3936}(2077,·)$, $\chi_{3936}(1895,·)$, $\chi_{3936}(301,·)$, $\chi_{3936}(1967,·)$, $\chi_{3936}(1393,·)$, $\chi_{3936}(3827,·)$, $\chi_{3936}(2293,·)$, $\chi_{3936}(2615,·)$, $\chi_{3936}(575,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{9} a^{6} + \frac{4}{9} a^{4} + \frac{1}{9} a^{2}$, $\frac{1}{9} a^{7} + \frac{1}{9} a^{5} - \frac{2}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{369} a^{8} - \frac{1}{3} a^{4} + \frac{1}{9} a^{2}$, $\frac{1}{14391} a^{9} + \frac{2}{39} a^{7} + \frac{14}{117} a^{5} + \frac{145}{351} a^{3} - \frac{2}{13} a$, $\frac{1}{14391} a^{10} - \frac{1}{4797} a^{8} + \frac{1}{117} a^{6} + \frac{106}{351} a^{4} - \frac{44}{117} a^{2}$, $\frac{1}{14391} a^{11} + \frac{2}{39} a^{7} - \frac{41}{351} a^{5} + \frac{49}{117} a^{3} + \frac{8}{39} a$, $\frac{1}{388557} a^{12} - \frac{10}{388557} a^{10} - \frac{4}{4797} a^{8} + \frac{124}{9477} a^{6} + \frac{3494}{9477} a^{4} - \frac{140}{1053} a^{2} - \frac{1}{3}$, $\frac{1}{1165671} a^{13} - \frac{10}{1165671} a^{11} - \frac{1415}{28431} a^{7} - \frac{2905}{28431} a^{5} - \frac{1004}{3159} a^{3} - \frac{46}{117} a$, $\frac{1}{59694762354779165000445981} a^{14} + \frac{678411517623237313}{1455969813531199146352341} a^{12} + \frac{163467327075372115106}{6632751372753240555605109} a^{10} - \frac{644259033430849314398}{1455969813531199146352341} a^{8} - \frac{6302437149301274588866}{1455969813531199146352341} a^{6} - \frac{25372438669293594813125}{53924807908562931346383} a^{4} - \frac{3762997794941810242838}{17974935969520977115461} a^{2} + \frac{985165795760315357}{3939280291369927047}$, $\frac{1}{537252861193012485004013829} a^{15} + \frac{181446803585979884666}{537252861193012485004013829} a^{13} - \frac{1850817995245117248260}{59694762354779165000445981} a^{11} - \frac{13970433930227222348845}{537252861193012485004013829} a^{9} - \frac{460131162396865089965548}{13103728321780792317171069} a^{7} + \frac{43903362816147696186272}{1455969813531199146352341} a^{5} - \frac{72675454118736857360707}{161774423725688794039149} a^{3} + \frac{74522546576346290044}{460895794090281464499} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{8}\times C_{8}\times C_{3218056}$, which has order $3295289344$ (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:  \( 99340914.09265693 \) (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

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{82}) \), \(\Q(\sqrt{246}) \), \(\Q(\sqrt{3}, \sqrt{82})\), 4.4.141150208.1, 4.4.1270351872.1, 8.8.6455175514775617536.4, 8.0.418231618537560997888.74, 8.0.33876761101542440828928.3

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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
41Data not computed