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Properties

Label 9.1.8279544064.1
Degree $9$
Signature $[1, 4]$
Discriminant $2^{8}\cdot 11^{4}\cdot 47^{2}$
Root discriminant $12.65$
Ramified primes $2, 11, 47$
Class number $1$
Class group Trivial
Galois group $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, 4, -12, 6, -12, 2, -6, 2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 + 2*x^7 - 6*x^6 + 2*x^5 - 12*x^4 + 6*x^3 - 12*x^2 + 4*x - 2)
 
gp: K = bnfinit(x^9 + 2*x^7 - 6*x^6 + 2*x^5 - 12*x^4 + 6*x^3 - 12*x^2 + 4*x - 2, 1)
 

Normalized defining polynomial

\( x^{9} + 2 x^{7} - 6 x^{6} + 2 x^{5} - 12 x^{4} + 6 x^{3} - 12 x^{2} + 4 x - 2 \)

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

Invariants

Degree:  $9$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(8279544064=2^{8}\cdot 11^{4}\cdot 47^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12.65$
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:  $2, 11, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{101} a^{8} - \frac{42}{101} a^{7} + \frac{49}{101} a^{6} - \frac{44}{101} a^{5} + \frac{32}{101} a^{4} - \frac{43}{101} a^{3} - \frac{6}{101} a^{2} + \frac{38}{101} a + \frac{24}{101}$

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:  $4$
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:  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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 30.1962278736 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 648
The 14 conjugacy class representatives for $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$
Character table for $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$

Intermediate fields

3.1.44.1

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 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 27 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 ${\href{/LocalNumberField/3.9.0.1}{9} }$ ${\href{/LocalNumberField/5.9.0.1}{9} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ R ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.9.0.1}{9} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.9.0.1}{9} }$ ${\href{/LocalNumberField/37.9.0.1}{9} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ R ${\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{3}$

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$2.9.8.1$x^{9} - 2$$9$$1$$8$$(C_9:C_3):C_2$$[\ ]_{9}^{6}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
47Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.11.2t1.1c1$1$ $ 11 $ $x^{2} - x + 3$ $C_2$ (as 2T1) $1$ $-1$
* 2.2e2_11.3t2.1c1$2$ $ 2^{2} \cdot 11 $ $x^{3} - x^{2} + x + 1$ $S_3$ (as 3T2) $1$ $0$
3.2e2_11e2_47e2.6t8.3c1$3$ $ 2^{2} \cdot 11^{2} \cdot 47^{2}$ $x^{4} - x^{3} - 8 x^{2} + 10 x + 6$ $S_4$ (as 4T5) $1$ $-1$
3.2e2_11_47e2.4t5.1c1$3$ $ 2^{2} \cdot 11 \cdot 47^{2}$ $x^{4} - x^{3} - 8 x^{2} + 10 x + 6$ $S_4$ (as 4T5) $1$ $1$
6.2e6_11e3_47e2.18t222.1c1$6$ $ 2^{6} \cdot 11^{3} \cdot 47^{2}$ $x^{9} + 2 x^{7} - 6 x^{6} + 2 x^{5} - 12 x^{4} + 6 x^{3} - 12 x^{2} + 4 x - 2$ $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $0$
* 6.2e6_11e3_47e2.9t30.1c1$6$ $ 2^{6} \cdot 11^{3} \cdot 47^{2}$ $x^{9} + 2 x^{7} - 6 x^{6} + 2 x^{5} - 12 x^{4} + 6 x^{3} - 12 x^{2} + 4 x - 2$ $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $0$
6.2e6_11e3_47e4.36t1130.1c1$6$ $ 2^{6} \cdot 11^{3} \cdot 47^{4}$ $x^{9} + 2 x^{7} - 6 x^{6} + 2 x^{5} - 12 x^{4} + 6 x^{3} - 12 x^{2} + 4 x - 2$ $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $0$
6.2e6_11e3_47e4.36t1130.1c2$6$ $ 2^{6} \cdot 11^{3} \cdot 47^{4}$ $x^{9} + 2 x^{7} - 6 x^{6} + 2 x^{5} - 12 x^{4} + 6 x^{3} - 12 x^{2} + 4 x - 2$ $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $0$
8.2e8_11e4_47e4.12t177.1c1$8$ $ 2^{8} \cdot 11^{4} \cdot 47^{4}$ $x^{9} + 2 x^{7} - 6 x^{6} + 2 x^{5} - 12 x^{4} + 6 x^{3} - 12 x^{2} + 4 x - 2$ $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $0$
8.2e6_11e4_47e4.12t177.1c1$8$ $ 2^{6} \cdot 11^{4} \cdot 47^{4}$ $x^{9} + 2 x^{7} - 6 x^{6} + 2 x^{5} - 12 x^{4} + 6 x^{3} - 12 x^{2} + 4 x - 2$ $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $0$
8.2e8_11e4_47e4.12t178.1c1$8$ $ 2^{8} \cdot 11^{4} \cdot 47^{4}$ $x^{9} + 2 x^{7} - 6 x^{6} + 2 x^{5} - 12 x^{4} + 6 x^{3} - 12 x^{2} + 4 x - 2$ $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $0$
12.2e10_11e7_47e6.36t1124.1c1$12$ $ 2^{10} \cdot 11^{7} \cdot 47^{6}$ $x^{9} + 2 x^{7} - 6 x^{6} + 2 x^{5} - 12 x^{4} + 6 x^{3} - 12 x^{2} + 4 x - 2$ $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $-2$
12.2e10_11e5_47e6.18t218.1c1$12$ $ 2^{10} \cdot 11^{5} \cdot 47^{6}$ $x^{9} + 2 x^{7} - 6 x^{6} + 2 x^{5} - 12 x^{4} + 6 x^{3} - 12 x^{2} + 4 x - 2$ $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $2$

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 *.