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Properties

Label 16.0.93165153442...9584.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 3^{14}\cdot 11^{6}$
Root discriminant $36.36$
Ramified primes $2, 3, 11$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^2.C_2^5.C_2$ (as 16T493)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4356, 0, 3960, 0, 3744, 0, 3744, 0, 2073, 0, 570, 0, 72, 0, 6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 6*x^14 + 72*x^12 + 570*x^10 + 2073*x^8 + 3744*x^6 + 3744*x^4 + 3960*x^2 + 4356)
 
gp: K = bnfinit(x^16 + 6*x^14 + 72*x^12 + 570*x^10 + 2073*x^8 + 3744*x^6 + 3744*x^4 + 3960*x^2 + 4356, 1)
 

Normalized defining polynomial

\( x^{16} + 6 x^{14} + 72 x^{12} + 570 x^{10} + 2073 x^{8} + 3744 x^{6} + 3744 x^{4} + 3960 x^{2} + 4356 \)

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:  \(9316515344280166632259584=2^{40}\cdot 3^{14}\cdot 11^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.36$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8}$, $\frac{1}{3} a^{9}$, $\frac{1}{39} a^{10} - \frac{1}{39} a^{8} + \frac{5}{13} a^{6} - \frac{3}{13} a^{4} - \frac{5}{13} a^{2} - \frac{6}{13}$, $\frac{1}{39} a^{11} - \frac{1}{39} a^{9} + \frac{5}{13} a^{7} - \frac{3}{13} a^{5} - \frac{5}{13} a^{3} - \frac{6}{13} a$, $\frac{1}{2028} a^{12} - \frac{4}{507} a^{10} + \frac{67}{1014} a^{8} + \frac{3}{26} a^{6} - \frac{233}{676} a^{4} - \frac{11}{338} a^{2} - \frac{163}{338}$, $\frac{1}{2028} a^{13} - \frac{4}{507} a^{11} + \frac{67}{1014} a^{9} + \frac{3}{26} a^{7} - \frac{233}{676} a^{5} - \frac{11}{338} a^{3} - \frac{163}{338} a$, $\frac{1}{58290804} a^{14} - \frac{346}{4857567} a^{12} + \frac{12629}{9715134} a^{10} - \frac{199027}{9715134} a^{8} + \frac{48959}{19430268} a^{6} - \frac{1544599}{3238378} a^{4} + \frac{1112935}{3238378} a^{2} + \frac{70604}{147199}$, $\frac{1}{116581608} a^{15} - \frac{173}{4857567} a^{13} - \frac{236477}{19430268} a^{11} - \frac{3188299}{19430268} a^{9} - \frac{7424221}{38860536} a^{7} - \frac{797281}{6476756} a^{5} - \frac{879913}{6476756} a^{3} + \frac{69271}{147199} a$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 817007.205019 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.C_2^5.C_2$ (as 16T493):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^2.C_2^5.C_2$
Character table for $C_2^2.C_2^5.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 4.2.6336.1, 4.0.4752.1, 4.2.1728.1, 8.0.361304064.2

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 R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$2.8.20.104$x^{8} + 2 x^{6} + 4 x^{5} + 12 x^{2} + 6$$8$$1$$20$$C_4\wr C_2$$[2, 2, 3, 7/2]^{2}$
2.8.20.101$x^{8} + 2 x^{6} + 4 x^{5} + 8 x^{3} + 14$$8$$1$$20$$C_4\wr C_2$$[2, 2, 3, 7/2]^{2}$
3Data not computed
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$