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Properties

Label 8.0.283855104.1
Degree $8$
Signature $[0, 4]$
Discriminant $2^{8}\cdot 3^{8}\cdot 13^{2}$
Root discriminant $11.39$
Ramified primes $2, 3, 13$
Class number $1$
Class group Trivial
Galois group $A_4\wr C_2$ (as 8T42)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13, 4, -20, -4, 11, 2, -2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 - 2*x^6 + 2*x^5 + 11*x^4 - 4*x^3 - 20*x^2 + 4*x + 13)
 
gp: K = bnfinit(x^8 - 2*x^7 - 2*x^6 + 2*x^5 + 11*x^4 - 4*x^3 - 20*x^2 + 4*x + 13, 1)
 

Normalized defining polynomial

\( x^{8} - 2 x^{7} - 2 x^{6} + 2 x^{5} + 11 x^{4} - 4 x^{3} - 20 x^{2} + 4 x + 13 \)

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:  \(283855104=2^{8}\cdot 3^{8}\cdot 13^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $11.39$
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:  $2, 3, 13$
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 \(a\))

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

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:  \( 18 a^{7} - 10 a^{6} - 50 a^{5} - 37 a^{4} + 144 a^{3} + 136 a^{2} - 160 a - 159 \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  \( 10 a^{7} - 6 a^{6} - 27 a^{5} - 20 a^{4} + 80 a^{3} + 72 a^{2} - 88 a - 85 \),  \( 34 a^{7} - 18 a^{6} - 96 a^{5} - 71 a^{4} + 272 a^{3} + 265 a^{2} - 303 a - 310 \),  \( 4 a^{7} - 3 a^{6} - 10 a^{5} - 7 a^{4} + 33 a^{3} + 24 a^{2} - 36 a - 30 \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 46.620513672 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_4\wr C_2$ (as 8T42):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 288
The 14 conjugacy class representatives for $A_4\wr C_2$
Character table for $A_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \)

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 sibling: data not computed
Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 sibling: 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ R ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.8.8.12$x^{8} + 2 x^{5} + 2 x^{4} + 4$$4$$2$$8$$A_4\wr C_2$$[4/3, 4/3, 4/3, 4/3]_{3}^{6}$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.6.7.6$x^{6} + 3 x^{3} + 3 x^{2} + 3$$6$$1$$7$$S_3\times C_3$$[3/2]_{2}^{3}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.3.2t1.1c1$1$ $ 3 $ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
1.3_13.6t1.1c1$1$ $ 3 \cdot 13 $ $x^{6} - x^{5} + 5 x^{4} + 6 x^{3} + 15 x^{2} + 4 x + 1$ $C_6$ (as 6T1) $0$ $-1$
1.13.3t1.1c1$1$ $ 13 $ $x^{3} - x^{2} - 4 x - 1$ $C_3$ (as 3T1) $0$ $1$
1.13.3t1.1c2$1$ $ 13 $ $x^{3} - x^{2} - 4 x - 1$ $C_3$ (as 3T1) $0$ $1$
1.3_13.6t1.1c2$1$ $ 3 \cdot 13 $ $x^{6} - x^{5} + 5 x^{4} + 6 x^{3} + 15 x^{2} + 4 x + 1$ $C_6$ (as 6T1) $0$ $-1$
2.2e2_3e3_13e2.3t2.2c1$2$ $ 2^{2} \cdot 3^{3} \cdot 13^{2}$ $x^{3} - 52$ $S_3$ (as 3T2) $1$ $0$
2.2e2_3e3_13.6t5.2c1$2$ $ 2^{2} \cdot 3^{3} \cdot 13 $ $x^{6} - 10 x^{3} + 52$ $S_3\times C_3$ (as 6T5) $0$ $0$
2.2e2_3e3_13.6t5.2c2$2$ $ 2^{2} \cdot 3^{3} \cdot 13 $ $x^{6} - 10 x^{3} + 52$ $S_3\times C_3$ (as 6T5) $0$ $0$
* 6.2e8_3e7_13e2.8t42.1c1$6$ $ 2^{8} \cdot 3^{7} \cdot 13^{2}$ $x^{8} - 2 x^{7} - 2 x^{6} + 2 x^{5} + 11 x^{4} - 4 x^{3} - 20 x^{2} + 4 x + 13$ $A_4\wr C_2$ (as 8T42) $1$ $0$
6.2e8_3e7_13e5.24t703.1c1$6$ $ 2^{8} \cdot 3^{7} \cdot 13^{5}$ $x^{8} - 2 x^{7} - 2 x^{6} + 2 x^{5} + 11 x^{4} - 4 x^{3} - 20 x^{2} + 4 x + 13$ $A_4\wr C_2$ (as 8T42) $0$ $0$
6.2e8_3e7_13e5.24t703.1c2$6$ $ 2^{8} \cdot 3^{7} \cdot 13^{5}$ $x^{8} - 2 x^{7} - 2 x^{6} + 2 x^{5} + 11 x^{4} - 4 x^{3} - 20 x^{2} + 4 x + 13$ $A_4\wr C_2$ (as 8T42) $0$ $0$
9.2e12_3e9_13e6.12t128.1c1$9$ $ 2^{12} \cdot 3^{9} \cdot 13^{6}$ $x^{8} - 2 x^{7} - 2 x^{6} + 2 x^{5} + 11 x^{4} - 4 x^{3} - 20 x^{2} + 4 x + 13$ $A_4\wr C_2$ (as 8T42) $1$ $3$
9.2e12_3e12_13e6.18t112.2c1$9$ $ 2^{12} \cdot 3^{12} \cdot 13^{6}$ $x^{8} - 2 x^{7} - 2 x^{6} + 2 x^{5} + 11 x^{4} - 4 x^{3} - 20 x^{2} + 4 x + 13$ $A_4\wr C_2$ (as 8T42) $1$ $-3$

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 *.