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Properties

Label 16.0.14505315669...641.12
Degree $16$
Signature $[0, 8]$
Discriminant $13^{12}\cdot 53^{8}$
Root discriminant $49.84$
Ramified primes $13, 53$
Class number $4$ (GRH)
Class group $[2, 2]$ (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![25281, 0, 76916, 0, 70814, 0, 25098, 0, 3099, 0, 54, 0, -25, 0, 5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 5*x^14 - 25*x^12 + 54*x^10 + 3099*x^8 + 25098*x^6 + 70814*x^4 + 76916*x^2 + 25281)
 
gp: K = bnfinit(x^16 + 5*x^14 - 25*x^12 + 54*x^10 + 3099*x^8 + 25098*x^6 + 70814*x^4 + 76916*x^2 + 25281, 1)
 

Normalized defining polynomial

\( x^{16} + 5 x^{14} - 25 x^{12} + 54 x^{10} + 3099 x^{8} + 25098 x^{6} + 70814 x^{4} + 76916 x^{2} + 25281 \)

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:  \(1450531566903202684958906641=13^{12}\cdot 53^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.84$
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:  $13, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{518} a^{12} + \frac{51}{518} a^{10} - \frac{55}{518} a^{8} + \frac{75}{259} a^{6} + \frac{3}{37} a^{4} - \frac{1}{2} a^{3} - \frac{181}{518} a^{2} + \frac{251}{518}$, $\frac{1}{518} a^{13} + \frac{51}{518} a^{11} - \frac{55}{518} a^{9} + \frac{75}{259} a^{7} + \frac{3}{37} a^{5} - \frac{1}{2} a^{4} - \frac{181}{518} a^{3} + \frac{251}{518} a$, $\frac{1}{8757044194514} a^{14} - \frac{368438924}{625503156751} a^{12} + \frac{799670748186}{4378522097257} a^{10} + \frac{1289043885431}{8757044194514} a^{8} - \frac{1862311153797}{4378522097257} a^{6} - \frac{1}{2} a^{5} - \frac{384911394103}{796094926774} a^{4} - \frac{1}{2} a^{3} + \frac{232489535518}{4378522097257} a^{2} - \frac{1602564279153}{8757044194514}$, $\frac{1}{1392370026927726} a^{15} + \frac{48137399701}{696185013463863} a^{13} - \frac{33435184813075}{198910003846818} a^{11} + \frac{52557057507767}{232061671154621} a^{9} + \frac{89485495168991}{232061671154621} a^{7} + \frac{332102557778}{21096515559511} a^{5} - \frac{1}{2} a^{4} - \frac{293741275981333}{1392370026927726} a^{3} - \frac{1}{2} a^{2} + \frac{285586183453405}{696185013463863} a$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 10178727.0774 \) (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

\(\Q(\sqrt{13}) \), 4.4.8957.1, 8.4.718600843493.1, 8.0.4252075997.1, 8.4.38085844705129.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$13$13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
53Data not computed