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Properties

Label 16.8.14958734309...5296.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{48}\cdot 3^{12}$
Root discriminant $18.24$
Ramified primes $2, 3$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2:D_4$ (as 16T43)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, -8, 16, 64, 56, 40, -56, -112, 10, 0, -20, 24, 28, -8, -8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 - 8*x^13 + 28*x^12 + 24*x^11 - 20*x^10 + 10*x^8 - 112*x^7 - 56*x^6 + 40*x^5 + 56*x^4 + 64*x^3 + 16*x^2 - 8*x - 2)
 
gp: K = bnfinit(x^16 - 8*x^14 - 8*x^13 + 28*x^12 + 24*x^11 - 20*x^10 + 10*x^8 - 112*x^7 - 56*x^6 + 40*x^5 + 56*x^4 + 64*x^3 + 16*x^2 - 8*x - 2, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{14} - 8 x^{13} + 28 x^{12} + 24 x^{11} - 20 x^{10} + 10 x^{8} - 112 x^{7} - 56 x^{6} + 40 x^{5} + 56 x^{4} + 64 x^{3} + 16 x^{2} - 8 x - 2 \)

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:  \(149587343098087735296=2^{48}\cdot 3^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.24$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13} a^{12} - \frac{5}{13} a^{11} + \frac{4}{13} a^{10} - \frac{6}{13} a^{9} - \frac{1}{13} a^{8} + \frac{5}{13} a^{7} + \frac{1}{13} a^{6} + \frac{5}{13} a^{5} + \frac{5}{13} a^{4} - \frac{3}{13} a^{3} + \frac{3}{13} a^{2} - \frac{2}{13}$, $\frac{1}{65} a^{13} - \frac{2}{65} a^{12} + \frac{2}{65} a^{11} - \frac{7}{65} a^{10} + \frac{4}{13} a^{9} - \frac{24}{65} a^{8} + \frac{29}{65} a^{7} + \frac{8}{65} a^{6} + \frac{4}{13} a^{5} - \frac{1}{65} a^{4} + \frac{7}{65} a^{3} - \frac{6}{13} a^{2} + \frac{11}{65} a - \frac{19}{65}$, $\frac{1}{65} a^{14} - \frac{2}{65} a^{12} - \frac{3}{65} a^{11} + \frac{6}{65} a^{10} + \frac{16}{65} a^{9} - \frac{19}{65} a^{8} + \frac{1}{65} a^{7} - \frac{29}{65} a^{6} - \frac{2}{5} a^{5} + \frac{1}{13} a^{4} - \frac{16}{65} a^{3} + \frac{16}{65} a^{2} + \frac{3}{65} a + \frac{27}{65}$, $\frac{1}{1515722195} a^{15} - \frac{3324158}{1515722195} a^{14} - \frac{8131048}{1515722195} a^{13} + \frac{12686}{1126931} a^{12} + \frac{479495218}{1515722195} a^{11} - \frac{75598895}{303144439} a^{10} + \frac{668613533}{1515722195} a^{9} + \frac{47359357}{1515722195} a^{8} + \frac{267149249}{1515722195} a^{7} - \frac{34055214}{116594015} a^{6} + \frac{35744041}{116594015} a^{5} + \frac{94904648}{303144439} a^{4} + \frac{107461437}{1515722195} a^{3} - \frac{58188614}{303144439} a^{2} - \frac{161431493}{1515722195} a - \frac{305042432}{1515722195}$

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:  \( 20519.3606285 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:D_4$ (as 16T43):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_2^2:D_4$
Character table for $C_2^2:D_4$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), 4.2.9216.1, 4.2.1024.1, 4.4.27648.1 x2, 4.4.13824.1 x2, \(\Q(\sqrt{2}, \sqrt{3})\), 8.8.3057647616.1, 8.4.339738624.2, 8.4.764411904.3

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

Sibling fields

Galois closure: data not computed
Degree 16 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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
3Data not computed