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Properties

Label 16.0.11945283218...0096.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{20}\cdot 3^{12}\cdot 11^{8}$
Root discriminant $17.98$
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![81, 0, -297, 0, 297, 0, 66, 0, -158, 0, -22, 0, 33, 0, 11, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 11*x^14 + 33*x^12 - 22*x^10 - 158*x^8 + 66*x^6 + 297*x^4 - 297*x^2 + 81)
 
gp: K = bnfinit(x^16 + 11*x^14 + 33*x^12 - 22*x^10 - 158*x^8 + 66*x^6 + 297*x^4 - 297*x^2 + 81, 1)
 

Normalized defining polynomial

\( x^{16} + 11 x^{14} + 33 x^{12} - 22 x^{10} - 158 x^{8} + 66 x^{6} + 297 x^{4} - 297 x^{2} + 81 \)

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:  \(119452832185734660096=2^{20}\cdot 3^{12}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.98$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{12} a^{10} - \frac{1}{12} a^{8} - \frac{1}{4} a^{6} + \frac{5}{12} a^{4} - \frac{5}{12} a^{2} - \frac{1}{4}$, $\frac{1}{12} a^{11} - \frac{1}{12} a^{9} - \frac{1}{4} a^{7} + \frac{5}{12} a^{5} - \frac{5}{12} a^{3} - \frac{1}{4} a$, $\frac{1}{36} a^{12} - \frac{1}{36} a^{10} - \frac{1}{4} a^{8} + \frac{5}{36} a^{6} + \frac{7}{36} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{108} a^{13} - \frac{1}{27} a^{11} - \frac{2}{9} a^{9} - \frac{1}{27} a^{7} - \frac{11}{27} a^{5} - \frac{4}{9} a^{3} - \frac{5}{12} a$, $\frac{1}{20952} a^{14} - \frac{1}{216} a^{13} + \frac{179}{20952} a^{12} + \frac{1}{54} a^{11} - \frac{25}{1164} a^{10} + \frac{1}{9} a^{9} + \frac{2347}{10476} a^{8} + \frac{1}{54} a^{7} + \frac{1367}{10476} a^{6} + \frac{11}{54} a^{5} - \frac{281}{1164} a^{4} + \frac{2}{9} a^{3} - \frac{875}{2328} a^{2} + \frac{5}{24} a + \frac{265}{776}$, $\frac{1}{62856} a^{15} - \frac{14}{7857} a^{13} - \frac{1}{72} a^{12} + \frac{205}{5238} a^{11} - \frac{1}{36} a^{10} + \frac{2483}{15714} a^{9} - \frac{1}{12} a^{8} + \frac{1142}{7857} a^{7} - \frac{7}{36} a^{6} - \frac{2119}{5238} a^{5} + \frac{7}{36} a^{4} - \frac{293}{6984} a^{3} - \frac{5}{12} a^{2} + \frac{7}{97} a + \frac{3}{8}$

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:  \( -\frac{373}{10476} a^{14} - \frac{2101}{5238} a^{12} - \frac{4481}{3492} a^{10} + \frac{4741}{10476} a^{8} + \frac{63029}{10476} a^{6} + \frac{803}{3492} a^{4} - \frac{6223}{582} a^{2} + \frac{2229}{388} \) (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:  \( 22523.3662566 \)
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

\(\Q(\sqrt{33}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-3}) \), 4.0.13068.1, 4.4.13068.1, \(\Q(\sqrt{-3}, \sqrt{-11})\), 8.0.170772624.1, 8.4.10929447936.1 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.8.12.13$x^{8} + 12 x^{4} + 16$$4$$2$$12$$D_4$$[2, 2]^{2}$
$3$3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$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}$