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Properties

Label 16.0.21236059055...3504.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{10}\cdot 11^{8}$
Root discriminant $18.64$
Ramified primes $2, 3, 11$
Class number $2$
Class group $[2]$
Galois group $C_2\wr C_2^2$ (as 16T149)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12, 0, -8, -8, -60, -28, 112, 128, -71, -130, 4, 56, 19, -16, 0, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 16*x^13 + 19*x^12 + 56*x^11 + 4*x^10 - 130*x^9 - 71*x^8 + 128*x^7 + 112*x^6 - 28*x^5 - 60*x^4 - 8*x^3 - 8*x^2 + 12)
 
gp: K = bnfinit(x^16 - 2*x^15 - 16*x^13 + 19*x^12 + 56*x^11 + 4*x^10 - 130*x^9 - 71*x^8 + 128*x^7 + 112*x^6 - 28*x^5 - 60*x^4 - 8*x^3 - 8*x^2 + 12, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 16 x^{13} + 19 x^{12} + 56 x^{11} + 4 x^{10} - 130 x^{9} - 71 x^{8} + 128 x^{7} + 112 x^{6} - 28 x^{5} - 60 x^{4} - 8 x^{3} - 8 x^{2} + 12 \)

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:  \(212360590552417173504=2^{24}\cdot 3^{10}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.64$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{4} a^{12} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{12} a^{13} - \frac{1}{12} a^{12} + \frac{5}{12} a^{9} + \frac{1}{12} a^{8} - \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{6} a^{3} + \frac{1}{3} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{12} a^{14} - \frac{1}{12} a^{12} + \frac{5}{12} a^{10} - \frac{1}{2} a^{9} - \frac{1}{12} a^{8} - \frac{1}{12} a^{6} + \frac{5}{12} a^{4} - \frac{1}{2} a^{3} - \frac{1}{6} a^{2} - \frac{1}{2}$, $\frac{1}{163372340844} a^{15} - \frac{1075067917}{163372340844} a^{14} + \frac{419286875}{40843085211} a^{13} - \frac{1963517162}{40843085211} a^{12} - \frac{31354609831}{163372340844} a^{11} - \frac{49498640963}{163372340844} a^{10} - \frac{39632506265}{81686170422} a^{9} + \frac{7143200699}{81686170422} a^{8} - \frac{35125095871}{163372340844} a^{7} - \frac{29763584789}{163372340844} a^{6} + \frac{33576253525}{81686170422} a^{5} + \frac{3667207342}{40843085211} a^{4} + \frac{2386955219}{81686170422} a^{3} + \frac{25110902713}{81686170422} a^{2} + \frac{339573212}{13614361737} a - \frac{1749816352}{13614361737}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( \frac{1187}{54238} a^{15} - \frac{30167}{162714} a^{14} + \frac{103261}{325428} a^{13} - \frac{53993}{108476} a^{12} + \frac{155181}{54238} a^{11} - \frac{341197}{162714} a^{10} - \frac{1885567}{325428} a^{9} - \frac{1126177}{325428} a^{8} + \frac{1123012}{81357} a^{7} + \frac{205916}{27119} a^{6} - \frac{949453}{108476} a^{5} - \frac{2878103}{325428} a^{4} + \frac{57499}{162714} a^{3} + \frac{35330}{27119} a^{2} - \frac{34551}{54238} a + \frac{149345}{54238} \) (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:  \( 27883.3172657 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_2^2$ (as 16T149):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 16 conjugacy class representatives for $C_2\wr C_2^2$
Character table for $C_2\wr C_2^2$

Intermediate fields

\(\Q(\sqrt{-11}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-3}) \), 4.0.5808.1, \(\Q(\sqrt{-3}, \sqrt{-11})\), 4.0.5808.2, 8.0.303595776.3, 8.0.4857532416.4 x2, 8.0.910787328.1 x2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 siblings: data not computed
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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.12.15$x^{8} + 2 x^{7} + 2 x^{4} + 12$$4$$2$$12$$C_2^2:C_4$$[2, 2]^{4}$
2.8.12.15$x^{8} + 2 x^{7} + 2 x^{4} + 12$$4$$2$$12$$C_2^2:C_4$$[2, 2]^{4}$
$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}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$11$11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$