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Properties

Label 16.4.46259590302...3504.3
Degree $16$
Signature $[4, 6]$
Discriminant $2^{24}\cdot 3^{14}\cdot 7^{8}$
Root discriminant $19.57$
Ramified primes $2, 3, 7$
Class number $1$
Class group Trivial
Galois group $D_8:C_2$ (as 16T44)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, -210, 623, -690, -203, 1080, -598, -450, 604, -96, -172, 120, -38, 0, 11, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 11*x^14 - 38*x^12 + 120*x^11 - 172*x^10 - 96*x^9 + 604*x^8 - 450*x^7 - 598*x^6 + 1080*x^5 - 203*x^4 - 690*x^3 + 623*x^2 - 210*x + 25)
 
gp: K = bnfinit(x^16 - 6*x^15 + 11*x^14 - 38*x^12 + 120*x^11 - 172*x^10 - 96*x^9 + 604*x^8 - 450*x^7 - 598*x^6 + 1080*x^5 - 203*x^4 - 690*x^3 + 623*x^2 - 210*x + 25, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 11 x^{14} - 38 x^{12} + 120 x^{11} - 172 x^{10} - 96 x^{9} + 604 x^{8} - 450 x^{7} - 598 x^{6} + 1080 x^{5} - 203 x^{4} - 690 x^{3} + 623 x^{2} - 210 x + 25 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(462595903021859733504=2^{24}\cdot 3^{14}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.57$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{9} - \frac{2}{5} a^{7} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5} a$, $\frac{1}{5} a^{11} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{5} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{8} + \frac{2}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{9} + \frac{2}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{25} a^{14} - \frac{2}{25} a^{13} - \frac{1}{25} a^{12} - \frac{1}{25} a^{11} + \frac{2}{25} a^{10} + \frac{12}{25} a^{9} - \frac{12}{25} a^{8} - \frac{7}{25} a^{7} - \frac{6}{25} a^{6} - \frac{6}{25} a^{5} - \frac{8}{25} a^{4} + \frac{2}{25} a^{3} + \frac{7}{25} a^{2} - \frac{2}{5} a$, $\frac{1}{7128474125} a^{15} - \frac{18154159}{7128474125} a^{14} - \frac{215538437}{7128474125} a^{13} - \frac{57476564}{7128474125} a^{12} - \frac{617445571}{7128474125} a^{11} - \frac{563730742}{7128474125} a^{10} + \frac{1810027279}{7128474125} a^{9} - \frac{2308986508}{7128474125} a^{8} - \frac{368603422}{7128474125} a^{7} - \frac{2507848459}{7128474125} a^{6} + \frac{2668745679}{7128474125} a^{5} + \frac{2436780843}{7128474125} a^{4} - \frac{1420862382}{7128474125} a^{3} + \frac{1221469906}{7128474125} a^{2} + \frac{495871556}{1425694825} a - \frac{139891148}{285138965}$

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:  $9$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 19910.9117506 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_8:C_2$ (as 16T44):

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 $D_8:C_2$
Character table for $D_8:C_2$

Intermediate fields

\(\Q(\sqrt{21}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{6}) \), 4.2.84672.2, 4.2.1323.1, \(\Q(\sqrt{6}, \sqrt{14})\), 8.4.7169347584.4

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 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 R R ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$