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Properties

Label 16.0.16744268671...1744.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{50}\cdot 3^{12}\cdot 23^{4}$
Root discriminant $43.55$
Ramified primes $2, 3, 23$
Class number $80$ (GRH)
Class group $[2, 2, 2, 10]$ (GRH)
Galois group $C_2^2:Q_8$ (as 16T31)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14641, 0, 27104, 0, 32408, 0, 17744, 0, 8098, 0, 1696, 0, 248, 0, 16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 16*x^14 + 248*x^12 + 1696*x^10 + 8098*x^8 + 17744*x^6 + 32408*x^4 + 27104*x^2 + 14641)
 
gp: K = bnfinit(x^16 + 16*x^14 + 248*x^12 + 1696*x^10 + 8098*x^8 + 17744*x^6 + 32408*x^4 + 27104*x^2 + 14641, 1)
 

Normalized defining polynomial

\( x^{16} + 16 x^{14} + 248 x^{12} + 1696 x^{10} + 8098 x^{8} + 17744 x^{6} + 32408 x^{4} + 27104 x^{2} + 14641 \)

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:  \(167442686719647879731871744=2^{50}\cdot 3^{12}\cdot 23^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.55$
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:  $2, 3, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{40} a^{8} - \frac{3}{10} a^{6} - \frac{1}{5} a^{4} + \frac{3}{10} a^{2} - \frac{19}{40}$, $\frac{1}{40} a^{9} - \frac{3}{10} a^{7} - \frac{1}{5} a^{5} + \frac{3}{10} a^{3} - \frac{19}{40} a$, $\frac{1}{40} a^{10} + \frac{1}{5} a^{6} - \frac{1}{10} a^{4} + \frac{1}{8} a^{2} + \frac{3}{10}$, $\frac{1}{80} a^{11} - \frac{1}{80} a^{10} - \frac{1}{80} a^{9} - \frac{1}{80} a^{8} + \frac{1}{4} a^{7} + \frac{1}{20} a^{6} + \frac{1}{20} a^{5} + \frac{3}{20} a^{4} - \frac{7}{80} a^{3} - \frac{17}{80} a^{2} - \frac{9}{80} a - \frac{33}{80}$, $\frac{1}{80} a^{12} - \frac{1}{80} a^{8} + \frac{3}{10} a^{6} - \frac{3}{80} a^{4} - \frac{1}{5} a^{2} - \frac{29}{80}$, $\frac{1}{880} a^{13} + \frac{1}{176} a^{11} - \frac{1}{80} a^{10} + \frac{3}{440} a^{9} - \frac{1}{80} a^{8} - \frac{1}{44} a^{7} + \frac{1}{20} a^{6} + \frac{321}{880} a^{5} + \frac{3}{20} a^{4} - \frac{307}{880} a^{3} - \frac{17}{80} a^{2} + \frac{89}{440} a - \frac{33}{80}$, $\frac{1}{319000208560} a^{14} + \frac{82437921}{319000208560} a^{12} - \frac{2843387343}{319000208560} a^{10} - \frac{3234540417}{319000208560} a^{8} + \frac{2753540225}{63800041712} a^{6} + \frac{25369280069}{319000208560} a^{4} + \frac{22942212377}{63800041712} a^{2} + \frac{996557947}{2636365360}$, $\frac{1}{3509002294160} a^{15} + \frac{82437921}{3509002294160} a^{13} - \frac{683088995}{350900229416} a^{11} - \frac{1}{80} a^{10} - \frac{451377689}{219312643385} a^{9} - \frac{1}{80} a^{8} + \frac{348717920113}{3509002294160} a^{7} + \frac{1}{20} a^{6} - \frac{564781105767}{3509002294160} a^{5} + \frac{3}{20} a^{4} + \frac{820962280183}{1754501147080} a^{3} - \frac{17}{80} a^{2} - \frac{2066881}{659091340} a - \frac{33}{80}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{10}$, which has order $80$ (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:  \( 45687.5845647 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:Q_8$ (as 16T31):

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 $C_2^2:Q_8$
Character table for $C_2^2:Q_8$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), 4.0.158976.2, 4.0.39744.5, \(\Q(\sqrt{2}, \sqrt{3})\), 8.0.718886928384.34, 8.0.101093474304.27, 8.8.12230590464.1

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 16 sibling: 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
3Data not computed
$23$23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$