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Properties

Label 16.0.48794117818...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 29^{4}\cdot 41^{4}$
Root discriminant $19.63$
Ramified primes $5, 29, 41$
Class number $1$
Class group Trivial
Galois group $C_2^5.C_2$ (as 16T79)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6561, 0, 1458, 0, -405, 0, 108, 0, 109, 0, 12, 0, -5, 0, 2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 2*x^14 - 5*x^12 + 12*x^10 + 109*x^8 + 108*x^6 - 405*x^4 + 1458*x^2 + 6561)
 
gp: K = bnfinit(x^16 + 2*x^14 - 5*x^12 + 12*x^10 + 109*x^8 + 108*x^6 - 405*x^4 + 1458*x^2 + 6561, 1)
 

Normalized defining polynomial

\( x^{16} + 2 x^{14} - 5 x^{12} + 12 x^{10} + 109 x^{8} + 108 x^{6} - 405 x^{4} + 1458 x^{2} + 6561 \)

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:  \(487941178183837890625=5^{12}\cdot 29^{4}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.63$
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:  $5, 29, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{8} + \frac{1}{8} a^{6} + \frac{1}{8} a^{4} + \frac{1}{8} a^{2} + \frac{1}{8}$, $\frac{1}{24} a^{9} + \frac{5}{24} a^{7} + \frac{1}{24} a^{5} + \frac{3}{8} a^{3} - \frac{1}{2} a^{2} + \frac{1}{24} a$, $\frac{1}{72} a^{10} + \frac{1}{36} a^{8} - \frac{7}{36} a^{6} - \frac{1}{12} a^{4} + \frac{5}{36} a^{2} + \frac{1}{8}$, $\frac{1}{216} a^{11} + \frac{1}{108} a^{9} + \frac{11}{108} a^{7} - \frac{7}{36} a^{5} + \frac{41}{108} a^{3} - \frac{1}{2} a^{2} + \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{1296} a^{12} - \frac{1}{432} a^{11} + \frac{1}{648} a^{10} - \frac{1}{216} a^{9} - \frac{5}{1296} a^{8} + \frac{43}{216} a^{7} - \frac{23}{432} a^{6} - \frac{11}{72} a^{5} - \frac{53}{1296} a^{4} - \frac{41}{216} a^{3} + \frac{11}{24} a^{2} - \frac{3}{16} a - \frac{7}{16}$, $\frac{1}{3888} a^{13} - \frac{7}{3888} a^{11} - \frac{23}{3888} a^{9} - \frac{1}{16} a^{8} + \frac{19}{1296} a^{7} - \frac{1}{16} a^{6} + \frac{1}{3888} a^{5} - \frac{1}{16} a^{4} + \frac{25}{54} a^{3} - \frac{1}{16} a^{2} + \frac{1}{3} a + \frac{7}{16}$, $\frac{1}{221616} a^{14} - \frac{13}{55404} a^{12} - \frac{1}{432} a^{11} - \frac{275}{221616} a^{10} + \frac{7}{432} a^{9} + \frac{1451}{36936} a^{8} + \frac{23}{432} a^{7} + \frac{1229}{110808} a^{6} - \frac{19}{144} a^{5} - \frac{637}{8208} a^{4} + \frac{215}{432} a^{3} + \frac{667}{1368} a^{2} + \frac{1}{3} a - \frac{3}{304}$, $\frac{1}{664848} a^{15} - \frac{13}{166212} a^{13} + \frac{79}{41553} a^{11} - \frac{1}{144} a^{10} + \frac{491}{27702} a^{9} - \frac{1}{72} a^{8} - \frac{16237}{83106} a^{7} + \frac{7}{72} a^{6} - \frac{979}{24624} a^{5} + \frac{1}{24} a^{4} - \frac{329}{684} a^{3} - \frac{5}{72} a^{2} + \frac{20}{57} a + \frac{7}{16}$

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:  \( \frac{17}{12312} a^{14} - \frac{5}{6156} a^{12} - \frac{5}{3078} a^{10} + \frac{163}{12312} a^{8} + \frac{803}{12312} a^{6} - \frac{319}{6156} a^{4} - \frac{547}{1368} a^{2} + \frac{415}{152} \) (order $10$)
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:  \( 16899.0446055 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2$ (as 16T79):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $C_2^5.C_2$
Character table for $C_2^5.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.29725.1, 4.0.3625.1, 4.0.1025.1, \(\Q(\zeta_{5})\), 4.4.5125.1, 4.4.725.1, 4.4.148625.1, 8.8.22089390625.2, 8.0.22089390625.8, 8.0.26265625.1, 8.0.883575625.1, 8.0.13140625.1, 8.0.22089390625.4, 8.0.22089390625.6

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$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}$
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$