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Properties

Label 8.0.82301184.1
Degree $8$
Signature $[0, 4]$
Discriminant $2^{8}\cdot 3^{8}\cdot 7^{2}$
Root discriminant $9.76$
Ramified primes $2, 3, 7$
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![9, 0, 0, 12, -3, 0, 4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 + 4*x^6 - 3*x^4 + 12*x^3 + 9)
 
gp: K = bnfinit(x^8 - 2*x^7 + 4*x^6 - 3*x^4 + 12*x^3 + 9, 1)
 

Normalized defining polynomial

\( x^{8} - 2 x^{7} + 4 x^{6} - 3 x^{4} + 12 x^{3} + 9 \)

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:  \(82301184=2^{8}\cdot 3^{8}\cdot 7^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $9.76$
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:  $2, 3, 7$
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 \(a\))

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

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:  \( -\frac{2}{9} a^{7} + \frac{4}{9} a^{6} - \frac{8}{9} a^{5} + \frac{1}{3} a^{4} - \frac{4}{3} a^{2} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  \( \frac{1}{9} a^{7} - \frac{1}{3} a^{6} + a^{5} - \frac{10}{9} a^{4} + \frac{2}{3} a^{3} + \frac{4}{3} a^{2} - \frac{2}{3} a + 1 \),  \( \frac{1}{9} a^{7} + \frac{1}{3} a^{5} + \frac{2}{9} a^{4} + \frac{2}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + 1 \),  \( \frac{1}{3} a^{7} - \frac{4}{9} a^{6} + \frac{8}{9} a^{5} + \frac{5}{9} a^{4} - \frac{2}{3} a^{3} + \frac{8}{3} a^{2} + \frac{7}{3} a \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 23.0216827639 \)
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

\(\Q(\sqrt{-3}) \), 4.0.3024.1

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

Sibling fields

Degree 6 siblings: 6.4.6858432.1, 6.4.16003008.3
Degree 8 sibling: 8.0.448084224.9
Degree 12 siblings: 12.4.439022168653824.1, 12.0.47038089498624.1, 12.0.2304866385432576.1, 12.8.2304866385432576.1, 12.0.2304866385432576.5, 12.0.256096265048064.2
Degree 16 sibling: 16.0.16263137215612256256.1
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 ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.8.8.11$x^{8} + 20 x^{2} + 4$$4$$2$$8$$S_4$$[4/3, 4/3]_{3}^{2}$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.6.7.5$x^{6} + 6 x^{2} + 3$$6$$1$$7$$D_{6}$$[3/2]_{2}^{2}$
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$

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_7.2t1.1c1$1$ $ 3 \cdot 7 $ $x^{2} - x - 5$ $C_2$ (as 2T1) $1$ $1$
1.7.2t1.1c1$1$ $ 7 $ $x^{2} - x + 2$ $C_2$ (as 2T1) $1$ $-1$
2.2e2_3e3_7.6t3.6c1$2$ $ 2^{2} \cdot 3^{3} \cdot 7 $ $x^{6} - 3 x^{5} + 12 x^{4} - 19 x^{3} + 30 x^{2} - 21 x + 7$ $D_{6}$ (as 6T3) $1$ $-2$
2.2e2_3e3_7.3t2.2c1$2$ $ 2^{2} \cdot 3^{3} \cdot 7 $ $x^{3} - 6 x - 2$ $S_3$ (as 3T2) $1$ $2$
* 3.2e4_3e3_7.4t5.1c1$3$ $ 2^{4} \cdot 3^{3} \cdot 7 $ $x^{4} - 2 x^{3} + 3$ $S_4$ (as 4T5) $1$ $-1$
3.2e4_3e3_7e2.6t11.9c1$3$ $ 2^{4} \cdot 3^{3} \cdot 7^{2}$ $x^{8} - 2 x^{7} + 4 x^{6} - 3 x^{4} + 12 x^{3} + 9$ $S_4\times C_2$ (as 8T24) $1$ $1$
* 3.2e4_3e4_7.6t11.5c1$3$ $ 2^{4} \cdot 3^{4} \cdot 7 $ $x^{8} - 2 x^{7} + 4 x^{6} - 3 x^{4} + 12 x^{3} + 9$ $S_4\times C_2$ (as 8T24) $1$ $1$
3.2e4_3e4_7e2.6t8.1c1$3$ $ 2^{4} \cdot 3^{4} \cdot 7^{2}$ $x^{4} - 2 x^{3} + 3$ $S_4$ (as 4T5) $1$ $-1$

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 *.