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Properties

Label 16.0.23043642001...0976.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{72}\cdot 47^{4}$
Root discriminant $59.25$
Ramified primes $2, 47$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T565)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4879681, 0, -830584, 0, 79524, 0, -3384, 0, 518, 0, -72, 0, 36, 0, -8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 + 36*x^12 - 72*x^10 + 518*x^8 - 3384*x^6 + 79524*x^4 - 830584*x^2 + 4879681)
 
gp: K = bnfinit(x^16 - 8*x^14 + 36*x^12 - 72*x^10 + 518*x^8 - 3384*x^6 + 79524*x^4 - 830584*x^2 + 4879681, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{14} + 36 x^{12} - 72 x^{10} + 518 x^{8} - 3384 x^{6} + 79524 x^{4} - 830584 x^{2} + 4879681 \)

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:  \(23043642001495833226013310976=2^{72}\cdot 47^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.25$
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:  $2, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4}$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{4} - \frac{3}{8}$, $\frac{1}{16} a^{9} - \frac{1}{16} a^{8} - \frac{1}{8} a^{5} + \frac{1}{8} a^{4} - \frac{3}{16} a + \frac{3}{16}$, $\frac{1}{752} a^{10} + \frac{39}{752} a^{8} - \frac{29}{376} a^{6} - \frac{83}{376} a^{4} - \frac{187}{752} a^{2} - \frac{3}{16}$, $\frac{1}{752} a^{11} - \frac{1}{94} a^{9} - \frac{1}{16} a^{8} - \frac{29}{376} a^{7} - \frac{9}{94} a^{5} + \frac{1}{8} a^{4} - \frac{187}{752} a^{3} + \frac{3}{16}$, $\frac{1}{8977376} a^{12} - \frac{106}{280543} a^{10} + \frac{316487}{8977376} a^{8} - \frac{94761}{1122172} a^{6} - \frac{858125}{8977376} a^{4} + \frac{10933}{23876} a^{2} + \frac{301}{4064}$, $\frac{1}{17954752} a^{13} - \frac{1}{17954752} a^{12} - \frac{53}{280543} a^{11} + \frac{53}{280543} a^{10} + \frac{316487}{17954752} a^{9} - \frac{316487}{17954752} a^{8} + \frac{92891}{1122172} a^{7} - \frac{92891}{1122172} a^{6} + \frac{1386219}{17954752} a^{5} + \frac{3102469}{17954752} a^{4} - \frac{3487}{23876} a^{3} + \frac{3487}{23876} a^{2} - \frac{2747}{8128} a - \frac{3349}{8128}$, $\frac{1}{26160073664} a^{14} + \frac{1073}{26160073664} a^{12} - \frac{4729245}{26160073664} a^{10} - \frac{1486213909}{26160073664} a^{8} + \frac{2299202307}{26160073664} a^{6} + \frac{95009229}{556597312} a^{4} + \frac{5391937}{11842496} a^{2} - \frac{30841}{251968}$, $\frac{1}{26160073664} a^{15} - \frac{6}{408751151} a^{13} - \frac{1}{17954752} a^{12} + \frac{212899}{26160073664} a^{11} + \frac{53}{280543} a^{10} - \frac{19520679}{1635004604} a^{9} + \frac{805685}{17954752} a^{8} + \frac{133727315}{26160073664} a^{7} - \frac{92891}{1122172} a^{6} - \frac{1096139}{34787332} a^{5} + \frac{858125}{17954752} a^{4} - \frac{4721007}{11842496} a^{3} + \frac{3487}{23876} a^{2} + \frac{221}{7874} a + \frac{3255}{8128}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( -1 \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 47062347.1598 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T565):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 28 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.0.2147483648.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
$47$$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
47.2.1.1$x^{2} - 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.1$x^{2} - 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.1$x^{2} - 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.1$x^{2} - 47$$2$$1$$1$$C_2$$[\ ]_{2}$