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Properties

Label 16.0.12835393105...9664.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{38}\cdot 3^{4}\cdot 7^{8}$
Root discriminant $18.06$
Ramified primes $2, 3, 7$
Class number $1$
Class group Trivial
Galois group $C_2^4.C_2^3$ (as 16T329)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![324, 0, -864, 0, 1170, 0, -864, 0, 385, 0, -100, 0, 16, 0, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 + 16*x^12 - 100*x^10 + 385*x^8 - 864*x^6 + 1170*x^4 - 864*x^2 + 324)
 
gp: K = bnfinit(x^16 - 4*x^14 + 16*x^12 - 100*x^10 + 385*x^8 - 864*x^6 + 1170*x^4 - 864*x^2 + 324, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{14} + 16 x^{12} - 100 x^{10} + 385 x^{8} - 864 x^{6} + 1170 x^{4} - 864 x^{2} + 324 \)

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:  \(128353931059122929664=2^{38}\cdot 3^{4}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.06$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{8} + \frac{1}{6} a^{6} - \frac{1}{6} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{12} a^{11} - \frac{1}{12} a^{10} - \frac{1}{12} a^{9} + \frac{1}{12} a^{8} - \frac{1}{6} a^{7} + \frac{1}{6} a^{6} + \frac{1}{6} a^{5} - \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{792} a^{12} + \frac{13}{396} a^{10} - \frac{149}{792} a^{8} + \frac{41}{198} a^{6} - \frac{7}{396} a^{4} - \frac{10}{33} a^{2} - \frac{9}{22}$, $\frac{1}{792} a^{13} + \frac{13}{396} a^{11} - \frac{149}{792} a^{9} + \frac{41}{198} a^{7} - \frac{7}{396} a^{5} - \frac{10}{33} a^{3} - \frac{9}{22} a$, $\frac{1}{109296} a^{14} + \frac{1}{9936} a^{12} - \frac{661}{9936} a^{10} + \frac{24971}{109296} a^{8} + \frac{4901}{54648} a^{6} + \frac{2833}{18216} a^{4} - \frac{613}{3036} a^{2} + \frac{331}{1012}$, $\frac{1}{218592} a^{15} - \frac{1}{218592} a^{14} - \frac{127}{218592} a^{13} + \frac{127}{218592} a^{12} + \frac{7357}{218592} a^{11} - \frac{7357}{218592} a^{10} - \frac{27331}{218592} a^{9} + \frac{27331}{218592} a^{8} + \frac{2693}{109296} a^{7} - \frac{2693}{109296} a^{6} - \frac{8989}{36432} a^{5} + \frac{8989}{36432} a^{4} - \frac{235}{2024} a^{3} + \frac{235}{2024} a^{2} + \frac{745}{2024} a - \frac{745}{2024}$

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

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3195.01813455 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T329):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 29 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{-14}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{2}, \sqrt{-7})\), 4.0.392.1 x2, 4.2.448.1 x2, 8.0.9834496.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.22.12$x^{8} + 10 x^{4} + 4$$4$$2$$22$$Q_8:C_2$$[2, 3, 4]^{2}$
2.8.16.13$x^{8} + 6 x^{6} + 4 x^{5} + 2 x^{4} + 4$$4$$2$$16$$D_4\times C_2$$[2, 2, 3]^{2}$
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$