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Properties

Label 16.0.21610371432...8464.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 3^{11}\cdot 37^{5}$
Root discriminant $44.25$
Ramified primes $2, 3, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1642

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1776, 0, -3360, 0, 39768, 0, -27360, 0, 6400, 0, -312, 0, -56, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 56*x^12 - 312*x^10 + 6400*x^8 - 27360*x^6 + 39768*x^4 - 3360*x^2 + 1776)
 
gp: K = bnfinit(x^16 - 56*x^12 - 312*x^10 + 6400*x^8 - 27360*x^6 + 39768*x^4 - 3360*x^2 + 1776, 1)
 

Normalized defining polynomial

\( x^{16} - 56 x^{12} - 312 x^{10} + 6400 x^{8} - 27360 x^{6} + 39768 x^{4} - 3360 x^{2} + 1776 \)

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:  \(216103714323709222887358464=2^{44}\cdot 3^{11}\cdot 37^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.25$
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:  $2, 3, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{4} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{24} a^{12} + \frac{1}{12} a^{8}$, $\frac{1}{24} a^{13} + \frac{1}{12} a^{9}$, $\frac{1}{61823259880824} a^{14} - \frac{132636980825}{61823259880824} a^{12} + \frac{95043291827}{1188908843862} a^{10} + \frac{268958775739}{2377817687724} a^{8} + \frac{892725441331}{5151938323402} a^{6} - \frac{57101214667}{396302947954} a^{4} + \frac{1036607093905}{2575969161701} a^{2} - \frac{934039292431}{2575969161701}$, $\frac{1}{61823259880824} a^{15} - \frac{132636980825}{61823259880824} a^{13} + \frac{95043291827}{1188908843862} a^{11} + \frac{268958775739}{2377817687724} a^{9} + \frac{892725441331}{5151938323402} a^{7} - \frac{57101214667}{396302947954} a^{5} + \frac{1036607093905}{2575969161701} a^{3} - \frac{934039292431}{2575969161701} a$

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:  \( \frac{9781}{57599382} a^{14} + \frac{9369}{38399588} a^{12} - \frac{501289}{57599382} a^{10} - \frac{2481251}{38399588} a^{8} + \frac{9276586}{9599897} a^{6} - \frac{66707695}{19199794} a^{4} + \frac{41007670}{9599897} a^{2} + \frac{2194930}{9599897} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 25496709.4741 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1642:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 73 conjugacy class representatives for t16n1642 are not computed
Character table for t16n1642 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.333.1, 8.0.1362604032.3

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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.7.1$x^{8} + 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
37Data not computed