Properties

Label 8.4.3186376704.2
Degree $8$
Signature $[4, 2]$
Discriminant $2^{14}\cdot 3^{4}\cdot 7^{4}$
Root discriminant $15.41$
Ramified primes $2, 3, 7$
Class number $1$
Class group Trivial
Galois group $(A_4\wr C_2):C_2$ (as 8T45)

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Show commands for: SageMath / Pari/GP / Magma

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

Normalized defining polynomial

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

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

Invariants

Degree:  $8$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[4, 2]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(3186376704=2^{14}\cdot 3^{4}\cdot 7^{4}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $15.41$
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:  $2, 3, 7$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{11} a^{7} + \frac{3}{11} a^{6} + \frac{4}{11} a^{4} + \frac{2}{11} a^{3} + \frac{3}{11} a^{2} - \frac{3}{11} a - \frac{1}{11}$

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:  $5$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 146.850411209 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Galois group

$(A_4\wr C_2):C_2$ (as 8T45):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 576
The 16 conjugacy class representatives for $(A_4\wr C_2):C_2$
Character table for $(A_4\wr C_2):C_2$

Intermediate fields

\(\Q(\sqrt{7}) \)

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

Sibling fields

Degree 12 siblings: data not computed
Degree 16 siblings: data not computed
Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 siblings: data not computed
Degree 36 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 ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$2$2.8.14.2$x^{8} + 2 x^{7} + 2$$8$$1$$14$$A_4\times C_2$$[2, 2, 2]^{3}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.3.4.4$x^{3} + 3 x^{2} + 3$$3$$1$$4$$S_3$$[2]^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.4.2t1.a.a$1$ $ 2^{2}$ $x^{2} + 1$ $C_2$ (as 2T1) $1$ $-1$
1.7.2t1.a.a$1$ $ 7 $ $x^{2} - x + 2$ $C_2$ (as 2T1) $1$ $-1$
* 1.28.2t1.a.a$1$ $ 2^{2} \cdot 7 $ $x^{2} - 7$ $C_2$ (as 2T1) $1$ $1$
2.567.3t2.b.a$2$ $ 3^{4} \cdot 7 $ $x^{3} - 3 x - 5$ $S_3$ (as 3T2) $1$ $0$
2.15876.6t3.c.a$2$ $ 2^{2} \cdot 3^{4} \cdot 7^{2}$ $x^{6} - 12 x^{4} - 14 x^{3} - 27 x^{2} - 126 x - 126$ $D_{6}$ (as 6T3) $1$ $0$
2.324.3t2.b.a$2$ $ 2^{2} \cdot 3^{4}$ $x^{3} - 3 x - 4$ $S_3$ (as 3T2) $1$ $0$
2.9072.6t3.b.a$2$ $ 2^{4} \cdot 3^{4} \cdot 7 $ $x^{6} - 6 x^{4} + 9 x^{2} - 7$ $D_{6}$ (as 6T3) $1$ $0$
4.63504.6t9.a.a$4$ $ 2^{4} \cdot 3^{4} \cdot 7^{2}$ $x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} - x^{2} - 2 x - 3$ $S_3^2$ (as 6T9) $1$ $0$
* 6.113799168.8t45.a.a$6$ $ 2^{12} \cdot 3^{4} \cdot 7^{3}$ $x^{8} - 2 x^{7} - 4 x^{6} + 4 x^{5} + 4 x^{4} + 4 x^{3} + 4 x^{2} - 8 x - 6$ $(A_4\wr C_2):C_2$ (as 8T45) $1$ $2$
6.113799168.12t161.a.a$6$ $ 2^{12} \cdot 3^{4} \cdot 7^{3}$ $x^{8} - 2 x^{7} - 4 x^{6} + 4 x^{5} + 4 x^{4} + 4 x^{3} + 4 x^{2} - 8 x - 6$ $(A_4\wr C_2):C_2$ (as 8T45) $1$ $-2$
9.256096265048064.18t185.a.a$9$ $ 2^{12} \cdot 3^{12} \cdot 7^{6}$ $x^{8} - 2 x^{7} - 4 x^{6} + 4 x^{5} + 4 x^{4} + 4 x^{3} + 4 x^{2} - 8 x - 6$ $(A_4\wr C_2):C_2$ (as 8T45) $1$ $-1$
9.746636341248.12t165.a.a$9$ $ 2^{12} \cdot 3^{12} \cdot 7^{3}$ $x^{8} - 2 x^{7} - 4 x^{6} + 4 x^{5} + 4 x^{4} + 4 x^{3} + 4 x^{2} - 8 x - 6$ $(A_4\wr C_2):C_2$ (as 8T45) $1$ $1$
9.47784725839872.18t185.a.a$9$ $ 2^{18} \cdot 3^{12} \cdot 7^{3}$ $x^{8} - 2 x^{7} - 4 x^{6} + 4 x^{5} + 4 x^{4} + 4 x^{3} + 4 x^{2} - 8 x - 6$ $(A_4\wr C_2):C_2$ (as 8T45) $1$ $-1$
9.16390160963076096.18t179.a.a$9$ $ 2^{18} \cdot 3^{12} \cdot 7^{6}$ $x^{8} - 2 x^{7} - 4 x^{6} + 4 x^{5} + 4 x^{4} + 4 x^{3} + 4 x^{2} - 8 x - 6$ $(A_4\wr C_2):C_2$ (as 8T45) $1$ $1$
12.6882294149039505014784.24t1503.a.a$12$ $ 2^{24} \cdot 3^{20} \cdot 7^{6}$ $x^{8} - 2 x^{7} - 4 x^{6} + 4 x^{5} + 4 x^{4} + 4 x^{3} + 4 x^{2} - 8 x - 6$ $(A_4\wr C_2):C_2$ (as 8T45) $1$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.