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Properties

Label 16.0.69499899149...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 59^{12}$
Root discriminant $47.60$
Ramified primes $5, 59$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_2^2.SD_{16}$ (as 16T163)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1172889, 0, 473993, 0, 165759, 0, -40997, 0, 1730, 0, -487, 0, 21, 0, -2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^14 + 21*x^12 - 487*x^10 + 1730*x^8 - 40997*x^6 + 165759*x^4 + 473993*x^2 + 1172889)
 
gp: K = bnfinit(x^16 - 2*x^14 + 21*x^12 - 487*x^10 + 1730*x^8 - 40997*x^6 + 165759*x^4 + 473993*x^2 + 1172889, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{14} + 21 x^{12} - 487 x^{10} + 1730 x^{8} - 40997 x^{6} + 165759 x^{4} + 473993 x^{2} + 1172889 \)

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:  \(694998991499817467531640625=5^{8}\cdot 59^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.60$
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:  $5, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{10} a^{10} - \frac{1}{10} a^{8} - \frac{1}{5} a^{6} - \frac{1}{2} a^{5} + \frac{1}{10} a^{4} - \frac{3}{10} a^{2} - \frac{1}{2} a - \frac{3}{10}$, $\frac{1}{10} a^{11} - \frac{1}{10} a^{9} - \frac{1}{5} a^{7} - \frac{2}{5} a^{5} - \frac{1}{2} a^{4} + \frac{1}{5} a^{3} - \frac{1}{2} a^{2} - \frac{3}{10} a - \frac{1}{2}$, $\frac{1}{100} a^{12} - \frac{3}{100} a^{10} + \frac{1}{20} a^{8} + \frac{3}{20} a^{6} - \frac{1}{2} a^{5} - \frac{3}{10} a^{4} - \frac{1}{2} a^{3} - \frac{1}{50} a^{2} - \frac{1}{2} a + \frac{31}{100}$, $\frac{1}{5700} a^{13} + \frac{69}{1900} a^{11} - \frac{67}{380} a^{9} + \frac{199}{1140} a^{7} + \frac{13}{95} a^{5} - \frac{1}{2} a^{4} + \frac{367}{1425} a^{3} - \frac{1099}{5700} a$, $\frac{1}{711095406451953000} a^{14} - \frac{1}{11400} a^{13} + \frac{18286058620633}{4740636043013020} a^{12} + \frac{121}{3800} a^{11} - \frac{1097318798905771}{29628975268831375} a^{10} - \frac{161}{760} a^{9} - \frac{114717217901051}{35554770322597650} a^{8} + \frac{143}{2280} a^{7} + \frac{3179547738311221}{47406360430130200} a^{6} - \frac{7}{380} a^{5} - \frac{25889239471384621}{355547703225976500} a^{4} - \frac{41}{1425} a^{3} + \frac{5050195805581}{5688763251615624} a^{2} + \frac{2239}{11400} a - \frac{13854334715529}{656597789891000}$, $\frac{1}{13510812722587107000} a^{15} - \frac{54171215944321}{900720848172473800} a^{13} - \frac{1}{200} a^{12} - \frac{194848515079507203}{4503604240862369000} a^{11} + \frac{3}{200} a^{10} - \frac{565779717000906839}{2702162544517421400} a^{9} + \frac{9}{40} a^{8} + \frac{102920869288558913}{450360424086236900} a^{7} + \frac{7}{40} a^{6} + \frac{335742354487156226}{1688851590323388375} a^{5} - \frac{7}{20} a^{4} - \frac{901542720485936879}{2702162544517421400} a^{3} - \frac{6}{25} a^{2} + \frac{490306176960127}{1559419750991125} a - \frac{81}{200}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $7$
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:  \( 6722973.23486 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.SD_{16}$ (as 16T163):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 19 conjugacy class representatives for $C_2^2.SD_{16}$
Character table for $C_2^2.SD_{16}$

Intermediate fields

\(\Q(\sqrt{-59}) \), 4.0.17405.1, 8.0.5272566705125.1, 8.0.1514670125.1, 8.0.26362833525625.1

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

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ R

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
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$59$59.8.6.1$x^{8} - 59 x^{4} + 55696$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
59.8.6.1$x^{8} - 59 x^{4} + 55696$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$