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Properties

Label 8.4.554696704.2
Degree $8$
Signature $[4, 2]$
Discriminant $2^{20}\cdot 23^{2}$
Root discriminant $12.39$
Ramified primes $2, 23$
Class number $1$
Class group Trivial
Galois group $(A_4\wr C_2):C_2$ (as 8T45)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -4, 4, 12, -8, -4, 8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 4*x^7 + 8*x^6 - 4*x^5 - 8*x^4 + 12*x^3 + 4*x^2 - 4*x - 1)
 
gp: K = bnfinit(x^8 - 4*x^7 + 8*x^6 - 4*x^5 - 8*x^4 + 12*x^3 + 4*x^2 - 4*x - 1, 1)
 

Normalized defining polynomial

\( x^{8} - 4 x^{7} + 8 x^{6} - 4 x^{5} - 8 x^{4} + 12 x^{3} + 4 x^{2} - 4 x - 1 \)

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

Invariants

Degree:  $8$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(554696704=2^{20}\cdot 23^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12.39$
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:  $2, 23$
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 \(a\))

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

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:  $5$
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:  \( a \),  \( \frac{3}{2} a^{7} - \frac{13}{2} a^{6} + \frac{29}{2} a^{5} - 12 a^{4} - \frac{11}{2} a^{3} + \frac{37}{2} a^{2} - \frac{3}{2} a - 2 \),  \( \frac{1}{2} a^{7} - 2 a^{6} + 4 a^{5} - 2 a^{4} - \frac{9}{2} a^{3} + 7 a^{2} + a - 3 \),  \( \frac{1}{2} a^{6} - 2 a^{5} + 4 a^{4} - 2 a^{3} - \frac{7}{2} a^{2} + 5 a + 2 \),  \( a^{7} - \frac{9}{2} a^{6} + \frac{21}{2} a^{5} - 10 a^{4} - 2 a^{3} + \frac{25}{2} a^{2} - \frac{5}{2} a - 2 \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 22.4716617767 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

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

\(\Q(\sqrt{2}) \)

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 ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ R ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.8.20.3$x^{8} + 4 x^{7} + 6 x^{4} + 4$$4$$2$$20$$D_4\times C_2$$[2, 3, 7/2]^{2}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.3.2.1$x^{3} - 23$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.2e3.2t1.2c1$1$ $ 2^{3}$ $x^{2} + 2$ $C_2$ (as 2T1) $1$ $-1$
* 1.2e3.2t1.1c1$1$ $ 2^{3}$ $x^{2} - 2$ $C_2$ (as 2T1) $1$ $1$
1.2e2.2t1.1c1$1$ $ 2^{2}$ $x^{2} + 1$ $C_2$ (as 2T1) $1$ $-1$
2.2e3_23e2.3t2.1c1$2$ $ 2^{3} \cdot 23^{2}$ $x^{3} - x^{2} - 15 x - 29$ $S_3$ (as 3T2) $1$ $0$
2.2e5_23e2.6t3.2c1$2$ $ 2^{5} \cdot 23^{2}$ $x^{6} - 16 x^{4} + 93 x^{2} - 242$ $D_{6}$ (as 6T3) $1$ $0$
2.2e2_23e2.3t2.1c1$2$ $ 2^{2} \cdot 23^{2}$ $x^{3} - x^{2} + 8 x - 6$ $S_3$ (as 3T2) $1$ $0$
2.2e6_23e2.6t3.1c1$2$ $ 2^{6} \cdot 23^{2}$ $x^{6} - 2 x^{5} + 11 x^{4} - 20 x^{3} + 84 x^{2} - 144 x - 136$ $D_{6}$ (as 6T3) $1$ $0$
4.2e8_23e2.6t9.1c1$4$ $ 2^{8} \cdot 23^{2}$ $x^{6} - 2 x^{5} - x^{4} + 4 x^{3} - 8 x + 2$ $S_3^2$ (as 6T9) $1$ $0$
6.2e19_23e2.12t161.2c1$6$ $ 2^{19} \cdot 23^{2}$ $x^{8} - 4 x^{7} + 8 x^{6} - 4 x^{5} - 8 x^{4} + 12 x^{3} + 4 x^{2} - 4 x - 1$ $(A_4\wr C_2):C_2$ (as 8T45) $1$ $-2$
* 6.2e17_23e2.8t45.1c1$6$ $ 2^{17} \cdot 23^{2}$ $x^{8} - 4 x^{7} + 8 x^{6} - 4 x^{5} - 8 x^{4} + 12 x^{3} + 4 x^{2} - 4 x - 1$ $(A_4\wr C_2):C_2$ (as 8T45) $1$ $2$
9.2e28_23e6.18t185.1c1$9$ $ 2^{28} \cdot 23^{6}$ $x^{8} - 4 x^{7} + 8 x^{6} - 4 x^{5} - 8 x^{4} + 12 x^{3} + 4 x^{2} - 4 x - 1$ $(A_4\wr C_2):C_2$ (as 8T45) $1$ $-1$
9.2e21_23e6.18t185.1c1$9$ $ 2^{21} \cdot 23^{6}$ $x^{8} - 4 x^{7} + 8 x^{6} - 4 x^{5} - 8 x^{4} + 12 x^{3} + 4 x^{2} - 4 x - 1$ $(A_4\wr C_2):C_2$ (as 8T45) $1$ $-1$
9.2e21_23e6.12t165.1c1$9$ $ 2^{21} \cdot 23^{6}$ $x^{8} - 4 x^{7} + 8 x^{6} - 4 x^{5} - 8 x^{4} + 12 x^{3} + 4 x^{2} - 4 x - 1$ $(A_4\wr C_2):C_2$ (as 8T45) $1$ $1$
9.2e26_23e6.18t179.2c1$9$ $ 2^{26} \cdot 23^{6}$ $x^{8} - 4 x^{7} + 8 x^{6} - 4 x^{5} - 8 x^{4} + 12 x^{3} + 4 x^{2} - 4 x - 1$ $(A_4\wr C_2):C_2$ (as 8T45) $1$ $1$
12.2e36_23e10.24t1504.1c1$12$ $ 2^{36} \cdot 23^{10}$ $x^{8} - 4 x^{7} + 8 x^{6} - 4 x^{5} - 8 x^{4} + 12 x^{3} + 4 x^{2} - 4 x - 1$ $(A_4\wr C_2):C_2$ (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 *.