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Properties

Label 16.8.17201207834...3824.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 3^{14}\cdot 11^{8}$
Root discriminant $24.53$
Ramified primes $2, 3, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_8:C_2^2$ (as 16T35)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, -6, 22, 29, -24, -152, 14, 276, 14, -152, -24, 29, 22, -6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 6*x^14 + 22*x^13 + 29*x^12 - 24*x^11 - 152*x^10 + 14*x^9 + 276*x^8 + 14*x^7 - 152*x^6 - 24*x^5 + 29*x^4 + 22*x^3 - 6*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 - 6*x^14 + 22*x^13 + 29*x^12 - 24*x^11 - 152*x^10 + 14*x^9 + 276*x^8 + 14*x^7 - 152*x^6 - 24*x^5 + 29*x^4 + 22*x^3 - 6*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 6 x^{14} + 22 x^{13} + 29 x^{12} - 24 x^{11} - 152 x^{10} + 14 x^{9} + 276 x^{8} + 14 x^{7} - 152 x^{6} - 24 x^{5} + 29 x^{4} + 22 x^{3} - 6 x^{2} - 4 x + 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(17201207834745791053824=2^{24}\cdot 3^{14}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{2}$, $\frac{1}{6} a^{12} + \frac{1}{3} a^{3} - \frac{1}{2}$, $\frac{1}{6} a^{13} + \frac{1}{3} a^{4} - \frac{1}{2} a$, $\frac{1}{11346} a^{14} + \frac{99}{1891} a^{13} - \frac{101}{3782} a^{12} - \frac{115}{5673} a^{11} - \frac{176}{1891} a^{10} + \frac{52}{1891} a^{9} - \frac{270}{1891} a^{8} + \frac{486}{1891} a^{7} - \frac{270}{1891} a^{6} + \frac{2047}{5673} a^{5} - \frac{176}{1891} a^{4} + \frac{592}{1891} a^{3} + \frac{3479}{11346} a^{2} + \frac{99}{1891} a + \frac{1261}{3782}$, $\frac{1}{34038} a^{15} - \frac{157}{3782} a^{13} - \frac{892}{17019} a^{12} - \frac{98}{5673} a^{11} + \frac{197}{1891} a^{10} + \frac{1076}{17019} a^{9} + \frac{131}{5673} a^{8} + \frac{2260}{5673} a^{7} + \frac{6655}{17019} a^{6} - \frac{808}{5673} a^{5} - \frac{2651}{5673} a^{4} + \frac{15293}{34038} a^{3} - \frac{2051}{5673} a^{2} + \frac{4673}{11346} a + \frac{7267}{17019}$

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:  $11$
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:  \( 206723.532105 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8:C_2^2$ (as 16T35):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_8:C_2^2$
Character table for $C_8:C_2^2$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{11}) \), 4.4.4752.1 x2, 4.4.13068.1 x2, \(\Q(\sqrt{3}, \sqrt{11})\), 8.4.32788343808.1 x2, 8.4.32788343808.2 x2, 8.8.2732361984.1

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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.12.13$x^{8} + 12 x^{4} + 16$$4$$2$$12$$D_4$$[2, 2]^{2}$
2.8.12.13$x^{8} + 12 x^{4} + 16$$4$$2$$12$$D_4$$[2, 2]^{2}$
3Data not computed
$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}$