Properties

Label 8.0.50936769.1
Degree $8$
Signature $[0, 4]$
Discriminant $3^{4}\cdot 13^{2}\cdot 61^{2}$
Root discriminant $9.19$
Ramified primes $3, 13, 61$
Class number $1$
Class group Trivial
Galois group $C_2^2 \wr C_2$ (as 8T18)

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Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 - 7*x^6 + x^5 + 22*x^4 + 15*x^3 - 9*x^2 + 9)
 
gp: K = bnfinit(x^8 - x^7 - 7*x^6 + x^5 + 22*x^4 + 15*x^3 - 9*x^2 + 9, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 0, -9, 15, 22, 1, -7, -1, 1]);
 

Normalized defining polynomial

\( x^{8} - x^{7} - 7 x^{6} + x^{5} + 22 x^{4} + 15 x^{3} - 9 x^{2} + 9 \)

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

Invariants

Degree:  $8$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(50936769=3^{4}\cdot 13^{2}\cdot 61^{2}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $9.19$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $3, 13, 61$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $4$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{42} a^{6} + \frac{4}{21} a^{5} - \frac{5}{21} a^{4} - \frac{17}{42} a^{3} - \frac{11}{42} a^{2} + \frac{5}{14} a - \frac{5}{14}$, $\frac{1}{42} a^{7} + \frac{5}{21} a^{5} - \frac{1}{2} a^{4} - \frac{1}{42} a^{3} + \frac{19}{42} a^{2} - \frac{3}{14} a - \frac{1}{7}$

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

Class group and class number

Trivial group, which has order $1$

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $3$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -\frac{8}{21} a^{7} + \frac{17}{21} a^{6} + \frac{5}{3} a^{5} - \frac{44}{21} a^{4} - \frac{113}{21} a^{3} - \frac{1}{7} a^{2} + \frac{11}{7} a - \frac{13}{7} \) (order $6$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  \( \frac{5}{42} a^{7} - \frac{13}{42} a^{6} - \frac{2}{7} a^{5} + \frac{25}{42} a^{4} + \frac{8}{7} a^{3} - \frac{1}{3} a^{2} + \frac{2}{7} a + \frac{13}{14} \),  \( \frac{4}{7} a^{7} - \frac{25}{21} a^{6} - \frac{59}{21} a^{5} + \frac{82}{21} a^{4} + \frac{26}{3} a^{3} - \frac{43}{21} a^{2} - 3 a + \frac{31}{7} \),  \( \frac{5}{42} a^{7} - \frac{13}{42} a^{6} - \frac{2}{7} a^{5} + \frac{25}{42} a^{4} + \frac{8}{7} a^{3} - \frac{1}{3} a^{2} + \frac{9}{7} a - \frac{1}{14} \)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 7.78254046164 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Galois group

$C_2^2\wr C_2$ (as 8T18):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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

\(\Q(\sqrt{-3}) \), 4.0.549.1, 4.0.7137.1, 4.0.117.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$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}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$61$61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$