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Properties

Label 16.0.22941779902...5625.3
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 5^{14}\cdot 29^{4}$
Root discriminant $21.63$
Ramified primes $3, 5, 29$
Class number $4$
Class group $[2, 2]$
Galois group $C_2\times C_2^3.C_4$ (as 16T99)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![58081, 0, -104554, 0, 84497, 0, -39487, 0, 11860, 0, -2378, 0, 317, 0, -26, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 26*x^14 + 317*x^12 - 2378*x^10 + 11860*x^8 - 39487*x^6 + 84497*x^4 - 104554*x^2 + 58081)
 
gp: K = bnfinit(x^16 - 26*x^14 + 317*x^12 - 2378*x^10 + 11860*x^8 - 39487*x^6 + 84497*x^4 - 104554*x^2 + 58081, 1)
 

Normalized defining polynomial

\( x^{16} - 26 x^{14} + 317 x^{12} - 2378 x^{10} + 11860 x^{8} - 39487 x^{6} + 84497 x^{4} - 104554 x^{2} + 58081 \)

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:  \(2294177990240478515625=3^{12}\cdot 5^{14}\cdot 29^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.63$
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:  $3, 5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{534363166262} a^{14} - \frac{32523013716}{267181583131} a^{12} + \frac{25719704343}{267181583131} a^{10} - \frac{23585277013}{267181583131} a^{8} + \frac{68253229688}{267181583131} a^{6} - \frac{1}{2} a^{5} + \frac{54180770231}{534363166262} a^{4} - \frac{1}{2} a^{3} - \frac{237696486309}{534363166262} a^{2} - \frac{1}{2} a + \frac{79104267842}{267181583131}$, $\frac{1}{128781523069142} a^{15} - \frac{7780788924515}{64390761534571} a^{13} - \frac{10903005499685}{128781523069142} a^{11} + \frac{3984138469952}{64390761534571} a^{9} - \frac{7611759451423}{128781523069142} a^{7} + \frac{19664936745244}{64390761534571} a^{5} - \frac{30162033796981}{128781523069142} a^{3} - \frac{1}{2} a^{2} + \frac{55999159410063}{128781523069142} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

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:  \( -\frac{12213679605}{128781523069142} a^{15} - \frac{254672493}{534363166262} a^{14} + \frac{303936587897}{128781523069142} a^{13} + \frac{3125063889}{267181583131} a^{12} - \frac{1779447337177}{64390761534571} a^{11} - \frac{69942013325}{534363166262} a^{10} + \frac{25409244577345}{128781523069142} a^{9} + \frac{470095686159}{534363166262} a^{8} - \frac{118703606104403}{128781523069142} a^{7} - \frac{2025347410099}{534363166262} a^{6} + \frac{355181804672125}{128781523069142} a^{5} + \frac{5379255217575}{534363166262} a^{4} - \frac{309172052929041}{64390761534571} a^{3} - \frac{3926099725406}{267181583131} a^{2} + \frac{470812933295179}{128781523069142} a + \frac{4479566019035}{534363166262} \) (order $30$)
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:  \( 28843.8333662 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^3.C_4$ (as 16T99):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $C_2\times C_2^3.C_4$
Character table for $C_2\times C_2^3.C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{5})\), \(\Q(\sqrt{-3}, \sqrt{5})\), 8.0.47897578125.1, 8.8.47897578125.1, \(\Q(\zeta_{15})\)

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
3Data not computed
5Data not computed
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$