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Properties

Label 16.0.39836003145...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{62}\cdot 5^{12}\cdot 29^{12}$
Root discriminant $613.08$
Ramified primes $2, 5, 29$
Class number $3882901536$ (GRH)
Class group $[2, 18, 107858376]$ (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![2969901210240000, 0, 659978046720000, 0, 47949554448000, 0, 1329298056000, 0, 18086924450, 0, 133550800, 0, 545780, 0, 1160, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 1160*x^14 + 545780*x^12 + 133550800*x^10 + 18086924450*x^8 + 1329298056000*x^6 + 47949554448000*x^4 + 659978046720000*x^2 + 2969901210240000)
 
gp: K = bnfinit(x^16 + 1160*x^14 + 545780*x^12 + 133550800*x^10 + 18086924450*x^8 + 1329298056000*x^6 + 47949554448000*x^4 + 659978046720000*x^2 + 2969901210240000, 1)
 

Normalized defining polynomial

\( x^{16} + 1160 x^{14} + 545780 x^{12} + 133550800 x^{10} + 18086924450 x^{8} + 1329298056000 x^{6} + 47949554448000 x^{4} + 659978046720000 x^{2} + 2969901210240000 \)

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:  \(398360031450580799831535291203584000000000000=2^{62}\cdot 5^{12}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $613.08$
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:  $2, 5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4640=2^{5}\cdot 5\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{4640}(1,·)$, $\chi_{4640}(1027,·)$, $\chi_{4640}(1161,·)$, $\chi_{4640}(2187,·)$, $\chi_{4640}(2321,·)$, $\chi_{4640}(4483,·)$, $\chi_{4640}(3481,·)$, $\chi_{4640}(4507,·)$, $\chi_{4640}(289,·)$, $\chi_{4640}(3347,·)$, $\chi_{4640}(1449,·)$, $\chi_{4640}(1003,·)$, $\chi_{4640}(2609,·)$, $\chi_{4640}(2163,·)$, $\chi_{4640}(3769,·)$, $\chi_{4640}(3323,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{145} a^{4}$, $\frac{1}{145} a^{5}$, $\frac{1}{145} a^{6}$, $\frac{1}{145} a^{7}$, $\frac{1}{42050} a^{8}$, $\frac{1}{252300} a^{9} - \frac{1}{435} a^{7} - \frac{1}{435} a^{5} + \frac{1}{3} a^{3} + \frac{1}{6} a$, $\frac{1}{1513800} a^{10} + \frac{1}{189225} a^{8} + \frac{1}{522} a^{6} + \frac{2}{1305} a^{4} + \frac{1}{36} a^{2}$, $\frac{1}{9082800} a^{11} + \frac{1}{1135350} a^{9} + \frac{41}{15660} a^{7} - \frac{8}{3915} a^{5} - \frac{71}{216} a^{3}$, $\frac{1}{134334612000} a^{12} - \frac{61}{231611400} a^{10} + \frac{437}{231611400} a^{8} + \frac{608}{199665} a^{6} + \frac{6913}{3194640} a^{4} - \frac{143}{612} a^{2} + \frac{8}{17}$, $\frac{1}{806007672000} a^{13} - \frac{61}{1389668400} a^{11} + \frac{437}{1389668400} a^{9} - \frac{3523}{1197990} a^{7} - \frac{59183}{19167840} a^{5} - \frac{143}{3672} a^{3} + \frac{4}{51} a$, $\frac{1}{6512741265484809936000} a^{14} - \frac{5957863189}{1628185316371202484000} a^{12} - \frac{118404098089}{1604123464405125600} a^{10} + \frac{12146201574341}{1403608031354484900} a^{8} + \frac{438369333015601}{154880886218425920} a^{6} + \frac{8100734836489}{4302246839400720} a^{4} + \frac{37397637523}{824185218276} a^{2} - \frac{8403429490}{22894033841}$, $\frac{1}{39076447592908859616000} a^{15} - \frac{5957863189}{9769111898227214904000} a^{13} - \frac{118404098089}{9624740786430753600} a^{11} + \frac{12146201574341}{8421648188126909400} a^{9} - \frac{1697918752755791}{929285317310555520} a^{7} - \frac{743790793843}{890120035738080} a^{5} - \frac{2435158017305}{4945111309656} a^{3} - \frac{27095748586}{68682101523} a$

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

Class group and class number

$C_{2}\times C_{18}\times C_{107858376}$, which has order $3882901536$ (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:  \( 12166181.748695593 \) (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

\(\Q(\sqrt{145}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{290}) \), \(\Q(\sqrt{2}, \sqrt{145})\), 4.4.43059200.2, \(\Q(\zeta_{16})^+\), 8.8.1854094704640000.3, 8.0.19958958676508672000000.4, 8.0.19958958676508672000000.1

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 ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.8.31.2$x^{8} + 24 x^{6} + 4 x^{4} + 16 x^{2} + 34$$8$$1$$31$$C_8$$[3, 4, 5]$
2.8.31.2$x^{8} + 24 x^{6} + 4 x^{4} + 16 x^{2} + 34$$8$$1$$31$$C_8$$[3, 4, 5]$
5Data not computed
29Data not computed