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Properties

Label 16.0.66183076013...5024.7
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 17^{8}\cdot 47^{2}$
Root discriminant $26.69$
Ramified primes $2, 17, 47$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T984

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![32, 272, 1310, 3588, 5687, 5518, 3603, 1806, 675, 124, 20, 8, -11, -2, -1, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - x^14 - 2*x^13 - 11*x^12 + 8*x^11 + 20*x^10 + 124*x^9 + 675*x^8 + 1806*x^7 + 3603*x^6 + 5518*x^5 + 5687*x^4 + 3588*x^3 + 1310*x^2 + 272*x + 32)
 
gp: K = bnfinit(x^16 - 2*x^15 - x^14 - 2*x^13 - 11*x^12 + 8*x^11 + 20*x^10 + 124*x^9 + 675*x^8 + 1806*x^7 + 3603*x^6 + 5518*x^5 + 5687*x^4 + 3588*x^3 + 1310*x^2 + 272*x + 32, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - x^{14} - 2 x^{13} - 11 x^{12} + 8 x^{11} + 20 x^{10} + 124 x^{9} + 675 x^{8} + 1806 x^{7} + 3603 x^{6} + 5518 x^{5} + 5687 x^{4} + 3588 x^{3} + 1310 x^{2} + 272 x + 32 \)

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:  \(66183076013297341825024=2^{32}\cdot 17^{8}\cdot 47^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.69$
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:  $2, 17, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{4} a^{10} + \frac{1}{8} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{3}{8} a^{5} + \frac{1}{4} a^{4} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{12} + \frac{1}{8} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{7} + \frac{1}{8} a^{6} - \frac{1}{2} a^{5} - \frac{3}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{13} - \frac{1}{4} a^{10} - \frac{1}{8} a^{9} + \frac{1}{8} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{4} + \frac{1}{8} a^{3} - \frac{1}{4} a$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{10} + \frac{1}{4} a^{9} + \frac{1}{8} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{1139884843485104} a^{15} - \frac{50247551801089}{1139884843485104} a^{14} + \frac{35349382830985}{569942421742552} a^{13} + \frac{1279752336629}{569942421742552} a^{12} - \frac{46773918273753}{1139884843485104} a^{11} - \frac{260654394401227}{1139884843485104} a^{10} + \frac{527562177446805}{1139884843485104} a^{9} + \frac{53810073875813}{1139884843485104} a^{8} - \frac{29444641481258}{71242802717819} a^{7} - \frac{3580692829816}{71242802717819} a^{6} - \frac{62128300509765}{1139884843485104} a^{5} - \frac{193464634452161}{1139884843485104} a^{4} + \frac{23443168077039}{569942421742552} a^{3} + \frac{28973261402417}{569942421742552} a^{2} + \frac{37077868887035}{142485605435638} a - \frac{12842147756538}{71242802717819}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 765073.691654 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T984:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 38 conjugacy class representatives for t16n984
Character table for t16n984 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{34}) \), 4.0.1088.2 x2, 4.0.2312.1 x2, \(\Q(\sqrt{2}, \sqrt{17})\), 8.0.342102016.2

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.4.10.2$x^{4} + 2 x^{2} - 1$$4$$1$$10$$D_{4}$$[2, 3, 7/2]$
2.4.10.3$x^{4} + 6 x^{2} - 9$$4$$1$$10$$D_{4}$$[2, 3, 7/2]$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
$17$17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$47$$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$