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Properties

Label 16.4.223291098234765625.1
Degree $16$
Signature $[4, 6]$
Discriminant $5^{8}\cdot 29^{6}\cdot 31^{2}$
Root discriminant $12.14$
Ramified primes $5, 29, 31$
Class number $1$
Class group Trivial
Galois group 16T876

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -4, -5, 4, 10, 1, -5, -5, -5, 1, 10, 4, -5, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 - 5*x^13 + 4*x^12 + 10*x^11 + x^10 - 5*x^9 - 5*x^8 - 5*x^7 + x^6 + 10*x^5 + 4*x^4 - 5*x^3 - 4*x^2 + 1)
 
gp: K = bnfinit(x^16 - 4*x^14 - 5*x^13 + 4*x^12 + 10*x^11 + x^10 - 5*x^9 - 5*x^8 - 5*x^7 + x^6 + 10*x^5 + 4*x^4 - 5*x^3 - 4*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{14} - 5 x^{13} + 4 x^{12} + 10 x^{11} + x^{10} - 5 x^{9} - 5 x^{8} - 5 x^{7} + x^{6} + 10 x^{5} + 4 x^{4} - 5 x^{3} - 4 x^{2} + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(223291098234765625=5^{8}\cdot 29^{6}\cdot 31^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12.14$
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:  $5, 29, 31$
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 \(a\))

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

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:  $9$
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:  \( 156.350498987 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T876:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 65 conjugacy class representatives for t16n876 are not computed
Character table for t16n876 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.4.15243125.1, 8.2.16294375.1, 8.6.472536875.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
$31$31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$