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Properties

Label 16.8.79801761844...0625.1
Degree $16$
Signature $[8, 4]$
Discriminant $3^{16}\cdot 5^{8}\cdot 83^{4}$
Root discriminant $20.25$
Ramified primes $3, 5, 83$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^3:S_4.C_2$ (as 16T763)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-191, 280, 470, -935, -222, 1114, -198, -662, 238, 254, -117, -76, 42, 14, -10, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 10*x^14 + 14*x^13 + 42*x^12 - 76*x^11 - 117*x^10 + 254*x^9 + 238*x^8 - 662*x^7 - 198*x^6 + 1114*x^5 - 222*x^4 - 935*x^3 + 470*x^2 + 280*x - 191)
 
gp: K = bnfinit(x^16 - x^15 - 10*x^14 + 14*x^13 + 42*x^12 - 76*x^11 - 117*x^10 + 254*x^9 + 238*x^8 - 662*x^7 - 198*x^6 + 1114*x^5 - 222*x^4 - 935*x^3 + 470*x^2 + 280*x - 191, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 10 x^{14} + 14 x^{13} + 42 x^{12} - 76 x^{11} - 117 x^{10} + 254 x^{9} + 238 x^{8} - 662 x^{7} - 198 x^{6} + 1114 x^{5} - 222 x^{4} - 935 x^{3} + 470 x^{2} + 280 x - 191 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(798017618443531640625=3^{16}\cdot 5^{8}\cdot 83^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.25$
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:  $3, 5, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{15} a^{11} - \frac{2}{15} a^{10} + \frac{2}{5} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{15} a^{6} - \frac{1}{5} a^{5} - \frac{4}{15} a^{4} - \frac{1}{3} a^{3} - \frac{2}{15} a^{2} - \frac{2}{5} a + \frac{7}{15}$, $\frac{1}{15} a^{12} + \frac{2}{15} a^{10} + \frac{7}{15} a^{9} - \frac{1}{3} a^{8} - \frac{2}{5} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{2}{15} a^{4} + \frac{1}{5} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{15}$, $\frac{1}{15} a^{13} + \frac{1}{15} a^{10} - \frac{7}{15} a^{9} - \frac{1}{15} a^{8} + \frac{2}{15} a^{6} - \frac{2}{15} a^{5} + \frac{1}{15} a^{4} + \frac{4}{15} a^{2} + \frac{2}{5} a - \frac{4}{15}$, $\frac{1}{225} a^{14} + \frac{1}{75} a^{13} + \frac{1}{75} a^{12} - \frac{1}{225} a^{11} + \frac{26}{225} a^{10} + \frac{14}{75} a^{9} - \frac{13}{225} a^{8} - \frac{11}{225} a^{7} - \frac{14}{225} a^{6} + \frac{37}{75} a^{5} + \frac{7}{225} a^{4} + \frac{68}{225} a^{3} - \frac{7}{25} a^{2} + \frac{37}{75} a - \frac{49}{225}$, $\frac{1}{316125} a^{15} + \frac{581}{316125} a^{14} - \frac{677}{35125} a^{13} - \frac{5512}{316125} a^{12} - \frac{1514}{105375} a^{11} - \frac{424}{63225} a^{10} - \frac{46732}{316125} a^{9} + \frac{2011}{63225} a^{8} - \frac{49034}{105375} a^{7} + \frac{27749}{316125} a^{6} - \frac{21226}{63225} a^{5} + \frac{109654}{316125} a^{4} - \frac{66944}{316125} a^{3} - \frac{45056}{105375} a^{2} + \frac{132119}{316125} a + \frac{75163}{316125}$

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:  $11$
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:  \( 30064.8269648 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3:S_4.C_2$ (as 16T763):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 384
The 26 conjugacy class representatives for $C_2^3:S_4.C_2$
Character table for $C_2^3:S_4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.56025.1, 8.8.3138800625.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }$ R R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.12.14.3$x^{12} - 12 x^{11} + 3 x^{10} - 9 x^{7} - 6 x^{6} - 9 x^{3} + 9 x^{2} - 9$$6$$2$$14$$S_3 \times C_4$$[3/2]_{2}^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$83$83.4.0.1$x^{4} - x + 22$$1$$4$$0$$C_4$$[\ ]^{4}$
83.4.0.1$x^{4} - x + 22$$1$$4$$0$$C_4$$[\ ]^{4}$
83.8.4.1$x^{8} + 303116 x^{4} - 571787 x^{2} + 22969827364$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$