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Properties

Label 16.0.93492089436...3456.6
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 3^{12}$
Root discriminant $15.33$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
Galois group $C_2^2\wr C_2$ (as 16T39)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 8, -12, -104, 264, -48, -380, 568, -222, -280, 532, -456, 264, -112, 36, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 36*x^14 - 112*x^13 + 264*x^12 - 456*x^11 + 532*x^10 - 280*x^9 - 222*x^8 + 568*x^7 - 380*x^6 - 48*x^5 + 264*x^4 - 104*x^3 - 12*x^2 + 8*x + 1)
 
gp: K = bnfinit(x^16 - 8*x^15 + 36*x^14 - 112*x^13 + 264*x^12 - 456*x^11 + 532*x^10 - 280*x^9 - 222*x^8 + 568*x^7 - 380*x^6 - 48*x^5 + 264*x^4 - 104*x^3 - 12*x^2 + 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 36 x^{14} - 112 x^{13} + 264 x^{12} - 456 x^{11} + 532 x^{10} - 280 x^{9} - 222 x^{8} + 568 x^{7} - 380 x^{6} - 48 x^{5} + 264 x^{4} - 104 x^{3} - 12 x^{2} + 8 x + 1 \)

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:  \(9349208943630483456=2^{44}\cdot 3^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.33$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4} a^{8} + \frac{1}{4}$, $\frac{1}{4} a^{9} + \frac{1}{4} a$, $\frac{1}{4} a^{10} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{11} + \frac{1}{4} a^{3}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{8} + \frac{1}{8} a^{4} - \frac{1}{8}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{9} + \frac{1}{8} a^{5} - \frac{1}{8} a$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{10} + \frac{1}{8} a^{6} - \frac{1}{8} a^{2}$, $\frac{1}{355903592} a^{15} - \frac{1000069}{16177436} a^{14} + \frac{923933}{44487949} a^{13} + \frac{171425}{4044359} a^{12} + \frac{68311}{2941352} a^{11} - \frac{9442833}{88975898} a^{10} - \frac{1730485}{88975898} a^{9} + \frac{6910641}{88975898} a^{8} + \frac{8502161}{355903592} a^{7} + \frac{28183757}{177951796} a^{6} - \frac{19397283}{44487949} a^{5} + \frac{2224448}{44487949} a^{4} - \frac{44308529}{355903592} a^{3} - \frac{1558409}{88975898} a^{2} + \frac{44023797}{88975898} a - \frac{30359581}{88975898}$

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:  \( -\frac{1864801}{2404754} a^{15} + \frac{5467601}{874456} a^{14} - \frac{68136135}{2404754} a^{13} + \frac{38838505}{437228} a^{12} - \frac{4196977}{19874} a^{11} + \frac{3555014873}{9619016} a^{10} - \frac{2125995023}{4809508} a^{9} + \frac{614527425}{2404754} a^{8} + \frac{346856357}{2404754} a^{7} - \frac{4273615877}{9619016} a^{6} + \frac{791854059}{2404754} a^{5} + \frac{18046315}{4809508} a^{4} - \frac{476896053}{2404754} a^{3} + \frac{929392265}{9619016} a^{2} - \frac{11697495}{4809508} a - \frac{10554483}{2404754} \) (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:  \( 6164.45078274 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\wr C_2$ (as 16T39):

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 $C_2^2\wr C_2$
Character table for $C_2^2\wr C_2$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), 4.2.4608.2, 4.2.4608.1, 4.0.13824.1, 4.4.13824.1, 4.0.1728.1 x2, \(\Q(\zeta_{12})\), 4.2.6912.1 x2, 8.0.47775744.1, 8.0.21233664.1, 8.0.191102976.5, 8.0.3057647616.4, 8.4.3057647616.2, 8.4.3057647616.3, 8.0.3057647616.3

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 8 siblings: 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$