↑

Properties

Label 16.0.25051689769...4656.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{8}\cdot 17^{12}$
Root discriminant $29.00$
Ramified primes $2, 3, 17$
Class number $20$ (GRH)
Class group $[20]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -13, 0, 129, 0, -494, 0, 1430, 0, -494, 0, 129, 0, -13, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 13*x^14 + 129*x^12 - 494*x^10 + 1430*x^8 - 494*x^6 + 129*x^4 - 13*x^2 + 1)
 
gp: K = bnfinit(x^16 - 13*x^14 + 129*x^12 - 494*x^10 + 1430*x^8 - 494*x^6 + 129*x^4 - 13*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 13 x^{14} + 129 x^{12} - 494 x^{10} + 1430 x^{8} - 494 x^{6} + 129 x^{4} - 13 x^{2} + 1 \)

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:  \(250516897691366976454656=2^{16}\cdot 3^{8}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.00$
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:  $2, 3, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(204=2^{2}\cdot 3\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{204}(1,·)$, $\chi_{204}(67,·)$, $\chi_{204}(137,·)$, $\chi_{204}(203,·)$, $\chi_{204}(13,·)$, $\chi_{204}(149,·)$, $\chi_{204}(89,·)$, $\chi_{204}(157,·)$, $\chi_{204}(35,·)$, $\chi_{204}(101,·)$, $\chi_{204}(103,·)$, $\chi_{204}(169,·)$, $\chi_{204}(47,·)$, $\chi_{204}(115,·)$, $\chi_{204}(55,·)$, $\chi_{204}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{8} a^{6} - \frac{3}{8}$, $\frac{1}{8} a^{7} - \frac{3}{8} a$, $\frac{1}{8} a^{8} - \frac{3}{8} a^{2}$, $\frac{1}{8} a^{9} - \frac{3}{8} a^{3}$, $\frac{1}{8} a^{10} - \frac{3}{8} a^{4}$, $\frac{1}{8} a^{11} - \frac{3}{8} a^{5}$, $\frac{1}{1664} a^{12} - \frac{3}{64} a^{6} + \frac{545}{1664}$, $\frac{1}{1664} a^{13} - \frac{3}{64} a^{7} + \frac{545}{1664} a$, $\frac{1}{1664} a^{14} - \frac{3}{64} a^{8} + \frac{545}{1664} a^{2}$, $\frac{1}{1664} a^{15} - \frac{3}{64} a^{9} + \frac{545}{1664} a^{3}$

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

Class group and class number

$C_{20}$, which has order $20$ (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:  \( -\frac{733}{1664} a^{15} + \frac{4817}{832} a^{13} - \frac{461}{8} a^{11} + \frac{14431}{64} a^{9} - \frac{21071}{32} a^{7} + \frac{2383}{8} a^{5} - \frac{98925}{1664} a^{3} + \frac{4985}{832} a \) (order $12$)
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:  \( 165082.730127 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-17}) \), \(\Q(\sqrt{-51}) \), \(\Q(\sqrt{51}) \), \(\Q(\zeta_{12})\), \(\Q(i, \sqrt{17})\), \(\Q(i, \sqrt{51})\), \(\Q(\sqrt{-3}, \sqrt{17})\), \(\Q(\sqrt{-3}, \sqrt{-17})\), \(\Q(\sqrt{3}, \sqrt{17})\), \(\Q(\sqrt{3}, \sqrt{-17})\), 4.4.707472.1, 4.0.44217.1, 4.0.78608.1, 4.4.4913.1, 8.0.1731891456.1, 8.0.500516630784.2, 8.0.6179217664.1, 8.0.500516630784.3, 8.0.1955143089.1, 8.8.500516630784.1, 8.0.500516630784.1

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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{16}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$