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Properties

Label 9.1.24321026304.1
Degree $9$
Signature $[1, 4]$
Discriminant $2^{8}\cdot 3^{6}\cdot 19^{4}$
Root discriminant $14.26$
Ramified primes $2, 3, 19$
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![-8, -4, -6, 21, 4, 3, -6, -1, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - x^7 - 6*x^6 + 3*x^5 + 4*x^4 + 21*x^3 - 6*x^2 - 4*x - 8)
 
gp: K = bnfinit(x^9 - x^7 - 6*x^6 + 3*x^5 + 4*x^4 + 21*x^3 - 6*x^2 - 4*x - 8, 1)
 

Normalized defining polynomial

\( x^{9} - x^{7} - 6 x^{6} + 3 x^{5} + 4 x^{4} + 21 x^{3} - 6 x^{2} - 4 x - 8 \)

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:  \(24321026304=2^{8}\cdot 3^{6}\cdot 19^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.26$
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:  $2, 3, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{12} a^{6} + \frac{1}{6} a^{4} + \frac{1}{6} a^{3} + \frac{1}{4} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{12} a^{7} + \frac{1}{6} a^{5} + \frac{1}{6} a^{4} + \frac{1}{4} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{24} a^{8} - \frac{1}{24} a^{6} - \frac{1}{6} a^{5} + \frac{1}{8} a^{4} + \frac{1}{3} a^{3} + \frac{1}{24} a^{2} - \frac{1}{2}$

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:  \( 94.454255671 \)
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

3.1.76.1

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 R ${\href{/LocalNumberField/5.9.0.1}{9} }$ ${\href{/LocalNumberField/7.9.0.1}{9} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{3}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.9.0.1}{9} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.6.8$x^{6} + 2 x + 2$$6$$1$$6$$S_4$$[4/3, 4/3]_{3}^{2}$
$3$3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.6.6.2$x^{6} + 6 x^{4} + 6 x^{3} + 18$$3$$2$$6$$C_3^2:C_4$$[3/2, 3/2]_{2}^{2}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.19.2t1.1c1$1$ $ 19 $ $x^{2} - x + 5$ $C_2$ (as 2T1) $1$ $-1$
* 2.2e2_19.3t2.1c1$2$ $ 2^{2} \cdot 19 $ $x^{3} - 2 x - 2$ $S_3$ (as 3T2) $1$ $0$
3.2e4_3e2_19e2.6t8.1c1$3$ $ 2^{4} \cdot 3^{2} \cdot 19^{2}$ $x^{4} - 2 x^{3} + 2 x^{2} + 2 x - 2$ $S_4$ (as 4T5) $1$ $-1$
3.2e4_3e2_19.4t5.1c1$3$ $ 2^{4} \cdot 3^{2} \cdot 19 $ $x^{4} - 2 x^{3} + 2 x^{2} + 2 x - 2$ $S_4$ (as 4T5) $1$ $1$
6.2e6_3e6_19e3.18t222.1c1$6$ $ 2^{6} \cdot 3^{6} \cdot 19^{3}$ $x^{9} - x^{7} - 6 x^{6} + 3 x^{5} + 4 x^{4} + 21 x^{3} - 6 x^{2} - 4 x - 8$ $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $0$
* 6.2e6_3e6_19e3.9t30.1c1$6$ $ 2^{6} \cdot 3^{6} \cdot 19^{3}$ $x^{9} - x^{7} - 6 x^{6} + 3 x^{5} + 4 x^{4} + 21 x^{3} - 6 x^{2} - 4 x - 8$ $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $0$
6.2e8_3e8_19e3.36t1130.1c1$6$ $ 2^{8} \cdot 3^{8} \cdot 19^{3}$ $x^{9} - x^{7} - 6 x^{6} + 3 x^{5} + 4 x^{4} + 21 x^{3} - 6 x^{2} - 4 x - 8$ $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $0$
6.2e8_3e8_19e3.36t1130.1c2$6$ $ 2^{8} \cdot 3^{8} \cdot 19^{3}$ $x^{9} - x^{7} - 6 x^{6} + 3 x^{5} + 4 x^{4} + 21 x^{3} - 6 x^{2} - 4 x - 8$ $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $0$
8.2e10_3e12_19e4.12t177.1c1$8$ $ 2^{10} \cdot 3^{12} \cdot 19^{4}$ $x^{9} - x^{7} - 6 x^{6} + 3 x^{5} + 4 x^{4} + 21 x^{3} - 6 x^{2} - 4 x - 8$ $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $0$
8.2e10_3e12_19e4.12t177.2c1$8$ $ 2^{10} \cdot 3^{12} \cdot 19^{4}$ $x^{9} - x^{7} - 6 x^{6} + 3 x^{5} + 4 x^{4} + 21 x^{3} - 6 x^{2} - 4 x - 8$ $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $0$
8.2e8_3e12_19e4.12t178.1c1$8$ $ 2^{8} \cdot 3^{12} \cdot 19^{4}$ $x^{9} - x^{7} - 6 x^{6} + 3 x^{5} + 4 x^{4} + 21 x^{3} - 6 x^{2} - 4 x - 8$ $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $0$
12.2e14_3e18_19e7.36t1124.1c1$12$ $ 2^{14} \cdot 3^{18} \cdot 19^{7}$ $x^{9} - x^{7} - 6 x^{6} + 3 x^{5} + 4 x^{4} + 21 x^{3} - 6 x^{2} - 4 x - 8$ $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $-2$
12.2e14_3e18_19e5.18t218.1c1$12$ $ 2^{14} \cdot 3^{18} \cdot 19^{5}$ $x^{9} - x^{7} - 6 x^{6} + 3 x^{5} + 4 x^{4} + 21 x^{3} - 6 x^{2} - 4 x - 8$ $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (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 *.