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Properties

Label 8.0.191102976.7
Degree $8$
Signature $[0, 4]$
Discriminant $2^{18}\cdot 3^{6}$
Root discriminant $10.84$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
Galois group $S_4\times C_2$ (as 8T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -8, 14, -16, 19, -16, 10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 4*x^7 + 10*x^6 - 16*x^5 + 19*x^4 - 16*x^3 + 14*x^2 - 8*x + 2)
 
gp: K = bnfinit(x^8 - 4*x^7 + 10*x^6 - 16*x^5 + 19*x^4 - 16*x^3 + 14*x^2 - 8*x + 2, 1)
 

Normalized defining polynomial

\( x^{8} - 4 x^{7} + 10 x^{6} - 16 x^{5} + 19 x^{4} - 16 x^{3} + 14 x^{2} - 8 x + 2 \)

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

Invariants

Degree:  $8$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(191102976=2^{18}\cdot 3^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $10.84$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{1}{3} a^{5} - \frac{4}{9} a^{4} - \frac{1}{9} a^{3} - \frac{1}{3} a^{2} - \frac{1}{9} a - \frac{4}{9}$

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

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $3$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{8}{9} a^{7} + \frac{28}{9} a^{6} - \frac{22}{3} a^{5} + \frac{95}{9} a^{4} - \frac{100}{9} a^{3} + \frac{23}{3} a^{2} - \frac{64}{9} a + \frac{23}{9} \) (order $4$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  \( \frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{1}{3} a^{5} + \frac{14}{9} a^{4} - \frac{19}{9} a^{3} + \frac{8}{3} a^{2} - \frac{10}{9} a + \frac{23}{9} \),  \( \frac{1}{9} a^{7} - \frac{8}{9} a^{6} + \frac{8}{3} a^{5} - \frac{49}{9} a^{4} + \frac{62}{9} a^{3} - \frac{19}{3} a^{2} + \frac{35}{9} a - \frac{31}{9} \),  \( \frac{13}{9} a^{7} - \frac{50}{9} a^{6} + \frac{41}{3} a^{5} - \frac{187}{9} a^{4} + \frac{212}{9} a^{3} - \frac{52}{3} a^{2} + \frac{131}{9} a - \frac{61}{9} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 42.4587327582 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_4$ (as 8T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 48
The 10 conjugacy class representatives for $S_4\times C_2$
Character table for $S_4\times C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), 4.2.13824.1

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

Sibling fields

Degree 6 siblings: data not computed
Degree 8 sibling: data not computed
Degree 12 siblings: data not computed
Degree 16 sibling: data not computed
Degree 24 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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.4.9.4$x^{4} + 2 x^{2} + 10$$4$$1$$9$$D_{4}$$[2, 3, 7/2]$
2.4.9.4$x^{4} + 2 x^{2} + 10$$4$$1$$9$$D_{4}$$[2, 3, 7/2]$
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.6.6.4$x^{6} + 3 x^{4} + 6 x^{3} + 9 x^{2} + 63 x + 9$$3$$2$$6$$D_{6}$$[3/2]_{2}^{2}$