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Properties

Label 16.16.4264489601...3184.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{32}\cdot 3^{8}\cdot 73^{6}$
Root discriminant $34.62$
Ramified primes $2, 3, 73$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2\wr C_2$ (as 16T46)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 152, 580, -520, -2818, 968, 5238, -1588, -4549, 1696, 1758, -848, -211, 148, -2, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 - 2*x^14 + 148*x^13 - 211*x^12 - 848*x^11 + 1758*x^10 + 1696*x^9 - 4549*x^8 - 1588*x^7 + 5238*x^6 + 968*x^5 - 2818*x^4 - 520*x^3 + 580*x^2 + 152*x + 4)
 
gp: K = bnfinit(x^16 - 8*x^15 - 2*x^14 + 148*x^13 - 211*x^12 - 848*x^11 + 1758*x^10 + 1696*x^9 - 4549*x^8 - 1588*x^7 + 5238*x^6 + 968*x^5 - 2818*x^4 - 520*x^3 + 580*x^2 + 152*x + 4, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} - 2 x^{14} + 148 x^{13} - 211 x^{12} - 848 x^{11} + 1758 x^{10} + 1696 x^{9} - 4549 x^{8} - 1588 x^{7} + 5238 x^{6} + 968 x^{5} - 2818 x^{4} - 520 x^{3} + 580 x^{2} + 152 x + 4 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4264489601111949714653184=2^{32}\cdot 3^{8}\cdot 73^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.62$
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:  $2, 3, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{6} a^{12} + \frac{1}{3} a^{10} - \frac{1}{2} a^{8} - \frac{1}{3} a^{6} - \frac{1}{2} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{6} a^{13} + \frac{1}{3} a^{11} - \frac{1}{2} a^{9} - \frac{1}{3} a^{7} - \frac{1}{2} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{6} a^{14} - \frac{1}{6} a^{10} - \frac{1}{3} a^{8} + \frac{1}{6} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3}$, $\frac{1}{6283175465382} a^{15} + \frac{318345955817}{6283175465382} a^{14} + \frac{166303979543}{6283175465382} a^{13} + \frac{61320817355}{2094391821794} a^{12} - \frac{2694424551}{6035711302} a^{11} - \frac{2361853108283}{6283175465382} a^{10} - \frac{1537989447353}{6283175465382} a^{9} - \frac{1457262654883}{6283175465382} a^{8} - \frac{351314415141}{2094391821794} a^{7} - \frac{745445522575}{6283175465382} a^{6} + \frac{120594375473}{6283175465382} a^{5} - \frac{405577429115}{6283175465382} a^{4} + \frac{1537305483371}{3141587732691} a^{3} - \frac{141154959380}{1047195910897} a^{2} - \frac{360619258776}{1047195910897} a + \frac{513553998613}{3141587732691}$

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:  $15$
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:  \( 22522502.0776 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\wr C_2$ (as 16T46):

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 $C_2^2\wr C_2$
Character table for $C_2^2\wr C_2$

Intermediate fields

\(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), 4.4.168192.1, 4.4.42048.2, 4.4.18688.1, 4.4.42048.1, 4.4.168192.2, \(\Q(\sqrt{2}, \sqrt{3})\), 4.4.10512.1, 8.8.28288548864.3, 8.8.28288548864.2, 8.8.28288548864.1

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 8 siblings: 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$73$73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.1.2$x^{2} + 365$$2$$1$$1$$C_2$$[\ ]_{2}$
73.2.1.2$x^{2} + 365$$2$$1$$1$$C_2$$[\ ]_{2}$
73.4.2.1$x^{4} + 1533 x^{2} + 644809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
73.4.2.1$x^{4} + 1533 x^{2} + 644809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$