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Properties

Label 16.0.46349544802...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{8}\cdot 3^{8}\cdot 5^{10}\cdot 41^{4}$
Root discriminant $16.95$
Ramified primes $2, 3, 5, 41$
Class number $1$
Class group Trivial
Galois group 16T799

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, -8, 36, -32, 72, -60, 165, -61, 231, -33, 165, -9, 62, -1, 12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 12*x^14 - x^13 + 62*x^12 - 9*x^11 + 165*x^10 - 33*x^9 + 231*x^8 - 61*x^7 + 165*x^6 - 60*x^5 + 72*x^4 - 32*x^3 + 36*x^2 - 8*x + 16)
 
gp: K = bnfinit(x^16 + 12*x^14 - x^13 + 62*x^12 - 9*x^11 + 165*x^10 - 33*x^9 + 231*x^8 - 61*x^7 + 165*x^6 - 60*x^5 + 72*x^4 - 32*x^3 + 36*x^2 - 8*x + 16, 1)
 

Normalized defining polynomial

\( x^{16} + 12 x^{14} - x^{13} + 62 x^{12} - 9 x^{11} + 165 x^{10} - 33 x^{9} + 231 x^{8} - 61 x^{7} + 165 x^{6} - 60 x^{5} + 72 x^{4} - 32 x^{3} + 36 x^{2} - 8 x + 16 \)

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:  \(46349544802500000000=2^{8}\cdot 3^{8}\cdot 5^{10}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.95$
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:  $2, 3, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{8} + \frac{1}{6} a^{7} + \frac{1}{6} a^{6} + \frac{1}{3} a^{5} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} + \frac{1}{6} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{9} - \frac{1}{2} a^{7} - \frac{1}{6} a^{6} - \frac{1}{6} a^{5} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3} + \frac{1}{6} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{12} a^{13} - \frac{1}{12} a^{10} - \frac{1}{12} a^{8} - \frac{1}{12} a^{7} + \frac{5}{12} a^{6} + \frac{5}{12} a^{5} + \frac{1}{4} a^{4} + \frac{1}{12} a^{3} - \frac{1}{6} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{12} a^{14} - \frac{1}{12} a^{11} - \frac{1}{12} a^{9} - \frac{1}{12} a^{8} + \frac{5}{12} a^{7} + \frac{5}{12} a^{6} + \frac{1}{4} a^{5} + \frac{1}{12} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3} a$, $\frac{1}{349224} a^{15} + \frac{2551}{87306} a^{14} - \frac{222}{14551} a^{13} - \frac{1459}{116408} a^{12} - \frac{10189}{174612} a^{11} + \frac{25771}{349224} a^{10} - \frac{18809}{116408} a^{9} - \frac{27173}{349224} a^{8} - \frac{105613}{349224} a^{7} - \frac{9351}{116408} a^{6} - \frac{21193}{116408} a^{5} + \frac{4084}{14551} a^{4} + \frac{11776}{43653} a^{3} - \frac{4803}{14551} a^{2} - \frac{4085}{29102} a + \frac{212}{14551}$

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:  \( -\frac{715}{87306} a^{15} + \frac{2933}{174612} a^{14} - \frac{6745}{58204} a^{13} + \frac{15641}{87306} a^{12} - \frac{40507}{58204} a^{11} + \frac{50221}{58204} a^{10} - \frac{124337}{58204} a^{9} + \frac{64091}{29102} a^{8} - \frac{47144}{14551} a^{7} + \frac{297637}{87306} a^{6} - \frac{172951}{87306} a^{5} + \frac{47866}{14551} a^{4} - \frac{106139}{174612} a^{3} + \frac{162677}{87306} a^{2} - \frac{30455}{29102} a + \frac{43568}{43653} \) (order $6$)
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:  \( 9917.65865529 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T799:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 62 conjugacy class representatives for t16n799 are not computed
Character table for t16n799 is not computed

Intermediate fields

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}) \), 4.4.9225.1, 4.0.1025.1, \(\Q(\sqrt{-3}, \sqrt{5})\), 8.0.85100625.1

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 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.4.4$x^{4} - 5$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.4.4.3$x^{4} + 2 x^{2} + 4 x + 4$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$41$41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.8.4.1$x^{8} + 57154 x^{4} - 68921 x^{2} + 816644929$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$