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Properties

Label 16.0.440301256704000000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{8}\cdot 5^{6}$
Root discriminant $12.67$
Ramified primes $2, 3, 5$
Class number $1$
Class group Trivial
Galois group $D_8:C_2$ (as 16T44)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 14, -40, 62, -60, 78, -164, 295, -384, 370, -276, 168, -84, 32, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 168*x^12 - 276*x^11 + 370*x^10 - 384*x^9 + 295*x^8 - 164*x^7 + 78*x^6 - 60*x^5 + 62*x^4 - 40*x^3 + 14*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 168*x^12 - 276*x^11 + 370*x^10 - 384*x^9 + 295*x^8 - 164*x^7 + 78*x^6 - 60*x^5 + 62*x^4 - 40*x^3 + 14*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 32 x^{14} - 84 x^{13} + 168 x^{12} - 276 x^{11} + 370 x^{10} - 384 x^{9} + 295 x^{8} - 164 x^{7} + 78 x^{6} - 60 x^{5} + 62 x^{4} - 40 x^{3} + 14 x^{2} - 4 x + 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(440301256704000000=2^{32}\cdot 3^{8}\cdot 5^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12.67$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{55} a^{14} + \frac{21}{55} a^{13} - \frac{18}{55} a^{12} + \frac{4}{11} a^{11} + \frac{3}{11} a^{10} + \frac{14}{55} a^{9} + \frac{21}{55} a^{8} + \frac{19}{55} a^{7} - \frac{13}{55} a^{6} - \frac{27}{55} a^{5} - \frac{3}{55} a^{4} - \frac{9}{55} a^{3} + \frac{18}{55} a^{2} - \frac{2}{5} a - \frac{1}{55}$, $\frac{1}{894245} a^{15} - \frac{351}{178849} a^{14} - \frac{332114}{894245} a^{13} + \frac{1538}{81295} a^{12} + \frac{62441}{178849} a^{11} + \frac{67039}{894245} a^{10} + \frac{29332}{894245} a^{9} + \frac{443973}{894245} a^{8} - \frac{131272}{894245} a^{7} + \frac{226451}{894245} a^{6} + \frac{379}{81295} a^{5} + \frac{407204}{894245} a^{4} - \frac{102853}{894245} a^{3} + \frac{23951}{178849} a^{2} - \frac{316339}{894245} a - \frac{356879}{894245}$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{5652364}{894245} a^{15} + \frac{41727111}{894245} a^{14} - \frac{155011478}{894245} a^{13} + \frac{75688198}{178849} a^{12} - \frac{12975562}{16259} a^{11} + \frac{101264864}{81295} a^{10} - \frac{1392977744}{894245} a^{9} + \frac{1293447214}{894245} a^{8} - \frac{847056648}{894245} a^{7} + \frac{382170398}{894245} a^{6} - \frac{187970568}{894245} a^{5} + \frac{213637051}{894245} a^{4} - \frac{19510592}{81295} a^{3} + \frac{91228733}{894245} a^{2} - \frac{20479676}{894245} a + \frac{1650439}{178849} \) (order $12$)
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:  \( 925.520714294 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_8:C_2$ (as 16T44):

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 $D_8:C_2$
Character table for $D_8:C_2$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), 4.0.2880.1, 4.0.320.1, \(\Q(\zeta_{12})\), 8.0.8294400.2

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 R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$