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Properties

Label 16.16.4361204616...3616.2
Degree $16$
Signature $[16, 0]$
Discriminant $2^{72}\cdot 31^{4}$
Root discriminant $53.39$
Ramified primes $2, 31$
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![-254, 2208, 9328, -29696, -131584, -99136, 91216, 125184, 7196, -36752, -10320, 3232, 1392, -80, -64, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 64*x^14 - 80*x^13 + 1392*x^12 + 3232*x^11 - 10320*x^10 - 36752*x^9 + 7196*x^8 + 125184*x^7 + 91216*x^6 - 99136*x^5 - 131584*x^4 - 29696*x^3 + 9328*x^2 + 2208*x - 254)
 
gp: K = bnfinit(x^16 - 64*x^14 - 80*x^13 + 1392*x^12 + 3232*x^11 - 10320*x^10 - 36752*x^9 + 7196*x^8 + 125184*x^7 + 91216*x^6 - 99136*x^5 - 131584*x^4 - 29696*x^3 + 9328*x^2 + 2208*x - 254, 1)
 

Normalized defining polynomial

\( x^{16} - 64 x^{14} - 80 x^{13} + 1392 x^{12} + 3232 x^{11} - 10320 x^{10} - 36752 x^{9} + 7196 x^{8} + 125184 x^{7} + 91216 x^{6} - 99136 x^{5} - 131584 x^{4} - 29696 x^{3} + 9328 x^{2} + 2208 x - 254 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4361204616626257617397743616=2^{72}\cdot 31^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.39$
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:  $2, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{1292010967150390213093913969} a^{15} + \frac{341346263838114616466509544}{1292010967150390213093913969} a^{14} - \frac{380897857534960148859184673}{1292010967150390213093913969} a^{13} - \frac{397165950524216067670639437}{1292010967150390213093913969} a^{12} + \frac{4710399155162200828683347}{13319700692272064052514577} a^{11} - \frac{474221242055106155407866712}{1292010967150390213093913969} a^{10} - \frac{60089553851109619524852317}{1292010967150390213093913969} a^{9} + \frac{254029773028535660313384209}{1292010967150390213093913969} a^{8} + \frac{71913340667538777195166562}{1292010967150390213093913969} a^{7} + \frac{167122995601751259990110375}{1292010967150390213093913969} a^{6} + \frac{246886139112074148883241494}{1292010967150390213093913969} a^{5} + \frac{252965159795677153996041517}{1292010967150390213093913969} a^{4} + \frac{340029010546308318089153667}{1292010967150390213093913969} a^{3} + \frac{391962955268583771985330757}{1292010967150390213093913969} a^{2} - \frac{153342968462872454745651922}{1292010967150390213093913969} a + \frac{404841503588639518789905228}{1292010967150390213093913969}$

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:  $15$
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:  \( 673693032.387 \) (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

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), \(\Q(\zeta_{32})^+\)

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
31Data not computed