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Properties

Label 16.0.22986704741...9376.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{8}\cdot 13^{8}$
Root discriminant $24.98$
Ramified primes $2, 3, 13$
Class number $12$ (GRH)
Class group $[12]$ (GRH)
Galois group $C_2^4$ (as 16T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6561, 0, 0, 0, -2511, 0, 0, 0, 880, 0, 0, 0, -31, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 31*x^12 + 880*x^8 - 2511*x^4 + 6561)
 
gp: K = bnfinit(x^16 - 31*x^12 + 880*x^8 - 2511*x^4 + 6561, 1)
 

Normalized defining polynomial

\( x^{16} - 31 x^{12} + 880 x^{8} - 2511 x^{4} + 6561 \)

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:  \(22986704741655040229376=2^{32}\cdot 3^{8}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.98$
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:  $2, 3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(312=2^{3}\cdot 3\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{312}(1,·)$, $\chi_{312}(259,·)$, $\chi_{312}(77,·)$, $\chi_{312}(79,·)$, $\chi_{312}(209,·)$, $\chi_{312}(131,·)$, $\chi_{312}(25,·)$, $\chi_{312}(155,·)$, $\chi_{312}(157,·)$, $\chi_{312}(287,·)$, $\chi_{312}(103,·)$, $\chi_{312}(233,·)$, $\chi_{312}(235,·)$, $\chi_{312}(181,·)$, $\chi_{312}(311,·)$, $\chi_{312}(53,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{7} a^{8} + \frac{2}{7} a^{4} - \frac{3}{7}$, $\frac{1}{21} a^{9} + \frac{2}{21} a^{5} + \frac{4}{21} a$, $\frac{1}{63} a^{10} + \frac{23}{63} a^{6} + \frac{25}{63} a^{2}$, $\frac{1}{189} a^{11} + \frac{23}{189} a^{7} - \frac{38}{189} a^{3}$, $\frac{1}{498960} a^{12} - \frac{34}{567} a^{8} + \frac{163}{567} a^{4} - \frac{2671}{6160}$, $\frac{1}{1496880} a^{13} - \frac{34}{1701} a^{9} + \frac{730}{1701} a^{5} - \frac{2671}{18480} a$, $\frac{1}{4490640} a^{14} - \frac{34}{5103} a^{10} + \frac{730}{5103} a^{6} + \frac{15809}{55440} a^{2}$, $\frac{1}{13471920} a^{15} - \frac{34}{15309} a^{11} + \frac{5833}{15309} a^{7} - \frac{39631}{166320} a^{3}$

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

Class group and class number

$C_{12}$, which has order $12$ (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:  \( -\frac{589}{1496880} a^{13} + \frac{19}{1701} a^{9} - \frac{508}{1701} a^{5} + \frac{513}{6160} a \) (order $24$)
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:  \( 68660.2329617 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4$ (as 16T3):

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 $C_2^4$
Character table for $C_2^4$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-13}) \), \(\Q(\sqrt{26}) \), \(\Q(\sqrt{-26}) \), \(\Q(\sqrt{78}) \), \(\Q(\sqrt{-78}) \), \(\Q(\sqrt{39}) \), \(\Q(\sqrt{-39}) \), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{6})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(i, \sqrt{13})\), \(\Q(i, \sqrt{26})\), \(\Q(i, \sqrt{78})\), \(\Q(i, \sqrt{39})\), \(\Q(\sqrt{2}, \sqrt{13})\), \(\Q(\sqrt{2}, \sqrt{-13})\), \(\Q(\sqrt{2}, \sqrt{39})\), \(\Q(\sqrt{2}, \sqrt{-39})\), \(\Q(\sqrt{-2}, \sqrt{13})\), \(\Q(\sqrt{-2}, \sqrt{-13})\), \(\Q(\sqrt{-2}, \sqrt{-39})\), \(\Q(\sqrt{-2}, \sqrt{39})\), \(\Q(\sqrt{6}, \sqrt{13})\), \(\Q(\sqrt{6}, \sqrt{-13})\), \(\Q(\sqrt{6}, \sqrt{26})\), \(\Q(\sqrt{6}, \sqrt{-26})\), \(\Q(\sqrt{-6}, \sqrt{13})\), \(\Q(\sqrt{-6}, \sqrt{-13})\), \(\Q(\sqrt{-6}, \sqrt{26})\), \(\Q(\sqrt{-6}, \sqrt{-26})\), \(\Q(\sqrt{3}, \sqrt{13})\), \(\Q(\sqrt{3}, \sqrt{-13})\), \(\Q(\sqrt{3}, \sqrt{26})\), \(\Q(\sqrt{3}, \sqrt{-26})\), \(\Q(\sqrt{-3}, \sqrt{13})\), \(\Q(\sqrt{-3}, \sqrt{-13})\), \(\Q(\sqrt{-3}, \sqrt{26})\), \(\Q(\sqrt{-3}, \sqrt{-26})\), \(\Q(\zeta_{24})\), 8.0.1871773696.1, 8.0.151613669376.8, 8.0.151613669376.3, 8.0.151613669376.2, 8.0.592240896.1, 8.0.151613669376.9, 8.8.151613669376.1, 8.0.151613669376.5, 8.0.9475854336.2, 8.0.151613669376.6, 8.0.9475854336.1, 8.0.151613669376.7, 8.0.151613669376.4, 8.0.151613669376.1

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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$