Properties

Label 16.0.59643097913...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{8}\cdot 13^{12}$
Root discriminant $30.62$
Ramified primes $2, 5, 13$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, 0, 108, 0, 1150, 0, -270, 0, 663, 0, -90, 0, 135, 0, -9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 9*x^14 + 135*x^12 - 90*x^10 + 663*x^8 - 270*x^6 + 1150*x^4 + 108*x^2 + 81)
 
gp: K = bnfinit(x^16 - 9*x^14 + 135*x^12 - 90*x^10 + 663*x^8 - 270*x^6 + 1150*x^4 + 108*x^2 + 81, 1)
 

Normalized defining polynomial

\( x^{16} - 9 x^{14} + 135 x^{12} - 90 x^{10} + 663 x^{8} - 270 x^{6} + 1150 x^{4} + 108 x^{2} + 81 \)

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:  \(596430979135513600000000=2^{16}\cdot 5^{8}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.62$
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:  \(260=2^{2}\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{260}(1,·)$, $\chi_{260}(131,·)$, $\chi_{260}(129,·)$, $\chi_{260}(79,·)$, $\chi_{260}(209,·)$, $\chi_{260}(259,·)$, $\chi_{260}(21,·)$, $\chi_{260}(151,·)$, $\chi_{260}(31,·)$, $\chi_{260}(161,·)$, $\chi_{260}(99,·)$, $\chi_{260}(229,·)$, $\chi_{260}(109,·)$, $\chi_{260}(239,·)$, $\chi_{260}(51,·)$, $\chi_{260}(181,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{9} a^{10} - \frac{1}{9} a^{6} - \frac{2}{9} a^{2}$, $\frac{1}{27} a^{11} - \frac{1}{9} a^{9} + \frac{2}{27} a^{7} - \frac{1}{9} a^{5} + \frac{10}{27} a^{3}$, $\frac{1}{27} a^{12} + \frac{2}{27} a^{8} + \frac{1}{9} a^{6} + \frac{10}{27} a^{4} + \frac{1}{9} a^{2}$, $\frac{1}{27} a^{13} + \frac{2}{27} a^{9} + \frac{1}{9} a^{7} + \frac{1}{27} a^{5} + \frac{1}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{3922918479} a^{14} - \frac{2456099}{3922918479} a^{12} - \frac{31374718}{3922918479} a^{10} + \frac{371689871}{3922918479} a^{8} + \frac{618090241}{3922918479} a^{6} + \frac{502427488}{3922918479} a^{4} + \frac{67067113}{1307639493} a^{2} + \frac{37341311}{145293277}$, $\frac{1}{3922918479} a^{15} - \frac{2456099}{3922918479} a^{13} - \frac{31374718}{3922918479} a^{11} + \frac{371689871}{3922918479} a^{9} + \frac{618090241}{3922918479} a^{7} + \frac{502427488}{3922918479} a^{5} + \frac{67067113}{1307639493} a^{3} + \frac{37341311}{145293277} a$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( \frac{4865}{932031} a^{15} - \frac{127732}{2796093} a^{13} + \frac{1934282}{2796093} a^{11} - \frac{794750}{2796093} a^{9} + \frac{8894494}{2796093} a^{7} - \frac{1376794}{2796093} a^{5} + \frac{13399556}{2796093} a^{3} + \frac{535048}{310677} a \) (order $4$)
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:  \( 213280.05844 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_4$ (as 16T2):

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 $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-65}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-13}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(i, \sqrt{65})\), \(\Q(i, \sqrt{13})\), \(\Q(i, \sqrt{5})\), \(\Q(\sqrt{-5}, \sqrt{13})\), \(\Q(\sqrt{5}, \sqrt{-13})\), \(\Q(\sqrt{5}, \sqrt{13})\), \(\Q(\sqrt{-5}, \sqrt{-13})\), 4.0.2197.1, 4.4.35152.1, 4.4.878800.1, 4.0.54925.1, 8.0.4569760000.1, 8.0.1235663104.1, 8.0.772289440000.3, 8.0.772289440000.2, 8.0.772289440000.1, 8.0.3016755625.1, 8.8.772289440000.1

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 ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$