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Properties

Label 16.0.47858458561...9296.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{8}\cdot 19^{8}$
Root discriminant $30.20$
Ramified primes $2, 3, 19$
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![390625, 0, 0, 0, -19375, 0, 0, 0, 336, 0, 0, 0, -31, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 31*x^12 + 336*x^8 - 19375*x^4 + 390625)
 
gp: K = bnfinit(x^16 - 31*x^12 + 336*x^8 - 19375*x^4 + 390625, 1)
 

Normalized defining polynomial

\( x^{16} - 31 x^{12} + 336 x^{8} - 19375 x^{4} + 390625 \)

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:  \(478584585616890104119296=2^{32}\cdot 3^{8}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.20$
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:  $2, 3, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(456=2^{3}\cdot 3\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{456}(1,·)$, $\chi_{456}(455,·)$, $\chi_{456}(265,·)$, $\chi_{456}(151,·)$, $\chi_{456}(77,·)$, $\chi_{456}(419,·)$, $\chi_{456}(341,·)$, $\chi_{456}(343,·)$, $\chi_{456}(37,·)$, $\chi_{456}(227,·)$, $\chi_{456}(229,·)$, $\chi_{456}(113,·)$, $\chi_{456}(305,·)$, $\chi_{456}(115,·)$, $\chi_{456}(379,·)$, $\chi_{456}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{9} a^{8} - \frac{2}{9} a^{4} + \frac{4}{9}$, $\frac{1}{45} a^{9} - \frac{11}{45} a^{5} - \frac{14}{45} a$, $\frac{1}{225} a^{10} - \frac{56}{225} a^{6} - \frac{14}{225} a^{2}$, $\frac{1}{1125} a^{11} - \frac{281}{1125} a^{7} + \frac{211}{1125} a^{3}$, $\frac{1}{1890000} a^{12} - \frac{121}{5625} a^{8} + \frac{2501}{5625} a^{4} + \frac{305}{3024}$, $\frac{1}{9450000} a^{13} - \frac{121}{28125} a^{9} - \frac{3124}{28125} a^{5} + \frac{3329}{15120} a$, $\frac{1}{47250000} a^{14} - \frac{121}{140625} a^{10} + \frac{25001}{140625} a^{6} + \frac{3329}{75600} a^{2}$, $\frac{1}{236250000} a^{15} - \frac{121}{703125} a^{11} + \frac{306251}{703125} a^{7} - \frac{147871}{378000} a^{3}$

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:  \( -\frac{341}{9450000} a^{13} + \frac{11}{28125} a^{9} + \frac{284}{28125} a^{5} + \frac{1375}{3024} a \) (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:  \( 372074.174823 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-38}) \), \(\Q(\sqrt{38}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{19}) \), \(\Q(\sqrt{-57}) \), \(\Q(\sqrt{57}) \), \(\Q(\sqrt{-114}) \), \(\Q(\sqrt{114}) \), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{6})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(i, \sqrt{38})\), \(\Q(i, \sqrt{19})\), \(\Q(i, \sqrt{57})\), \(\Q(i, \sqrt{114})\), \(\Q(\sqrt{2}, \sqrt{-19})\), \(\Q(\sqrt{2}, \sqrt{19})\), \(\Q(\sqrt{2}, \sqrt{-57})\), \(\Q(\sqrt{2}, \sqrt{57})\), \(\Q(\sqrt{-2}, \sqrt{19})\), \(\Q(\sqrt{-2}, \sqrt{-19})\), \(\Q(\sqrt{-2}, \sqrt{-57})\), \(\Q(\sqrt{-2}, \sqrt{57})\), \(\Q(\sqrt{6}, \sqrt{-38})\), \(\Q(\sqrt{6}, \sqrt{38})\), \(\Q(\sqrt{6}, \sqrt{-19})\), \(\Q(\sqrt{6}, \sqrt{19})\), \(\Q(\sqrt{-6}, \sqrt{-38})\), \(\Q(\sqrt{-6}, \sqrt{38})\), \(\Q(\sqrt{-6}, \sqrt{-19})\), \(\Q(\sqrt{-6}, \sqrt{19})\), \(\Q(\sqrt{3}, \sqrt{-38})\), \(\Q(\sqrt{3}, \sqrt{38})\), \(\Q(\sqrt{3}, \sqrt{-19})\), \(\Q(\sqrt{3}, \sqrt{19})\), \(\Q(\sqrt{-3}, \sqrt{-38})\), \(\Q(\sqrt{-3}, \sqrt{38})\), \(\Q(\sqrt{-3}, \sqrt{-19})\), \(\Q(\sqrt{-3}, \sqrt{19})\), \(\Q(\zeta_{24})\), 8.0.8540717056.1, 8.0.691798081536.8, 8.0.691798081536.2, 8.0.691798081536.3, 8.0.691798081536.9, 8.0.2702336256.1, 8.0.691798081536.5, 8.8.691798081536.1, 8.0.43237380096.5, 8.0.691798081536.7, 8.0.691798081536.6, 8.0.43237380096.3, 8.0.691798081536.4, 8.0.691798081536.1

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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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}$
$19$19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$