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Properties

Label 9.3.2156689088.1
Degree $9$
Signature $[3, 3]$
Discriminant $-\,2^{6}\cdot 17^{3}\cdot 19^{3}$
Root discriminant $10.89$
Ramified primes $2, 17, 19$
Class number $1$
Class group Trivial
Galois group $C_3^2 : D_{6} $ (as 9T18)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, -5, -6, -10, -5, -5, 1, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 + x^7 - 5*x^6 - 5*x^5 - 10*x^4 - 6*x^3 - 5*x^2 - x + 1)
 
gp: K = bnfinit(x^9 + x^7 - 5*x^6 - 5*x^5 - 10*x^4 - 6*x^3 - 5*x^2 - x + 1, 1)
 

Normalized defining polynomial

\( x^{9} + x^{7} - 5 x^{6} - 5 x^{5} - 10 x^{4} - 6 x^{3} - 5 x^{2} - x + 1 \)

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

Invariants

Degree:  $9$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-2156689088=-\,2^{6}\cdot 17^{3}\cdot 19^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $10.89$
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:  $2, 17, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{23} a^{8} + \frac{7}{23} a^{7} + \frac{4}{23} a^{6} - \frac{5}{23} a^{4} + \frac{1}{23} a^{3} + \frac{1}{23} a^{2} + \frac{2}{23} a - \frac{10}{23}$

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:  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:  \( 15.5635061347 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3.S_3^2$ (as 9T18):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 11 conjugacy class representatives for $C_3^2 : D_{6} $
Character table for $C_3^2 : D_{6} $

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 9 sibling: data not computed
Degree 18 siblings: data not computed
Degree 27 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 ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ R R ${\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
$17$17.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
17.6.3.1$x^{6} - 34 x^{4} + 289 x^{2} - 44217$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$19$19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
19.6.3.2$x^{6} - 361 x^{2} + 27436$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.17.2t1.1c1$1$ $ 17 $ $x^{2} - x - 4$ $C_2$ (as 2T1) $1$ $1$
1.19.2t1.1c1$1$ $ 19 $ $x^{2} - x + 5$ $C_2$ (as 2T1) $1$ $-1$
1.17_19.2t1.1c1$1$ $ 17 \cdot 19 $ $x^{2} - x + 81$ $C_2$ (as 2T1) $1$ $-1$
2.2e2_17e2_19.6t3.2c1$2$ $ 2^{2} \cdot 17^{2} \cdot 19 $ $x^{6} - 3 x^{5} + 24 x^{4} - 35 x^{3} + 248 x^{2} - 475 x + 1653$ $D_{6}$ (as 6T3) $1$ $0$
2.2e2_17_19.6t3.2c1$2$ $ 2^{2} \cdot 17 \cdot 19 $ $x^{6} - x^{5} + 8 x^{4} - 7 x^{3} + 24 x^{2} - 13 x + 23$ $D_{6}$ (as 6T3) $1$ $0$
2.2e2_17_19.3t2.1c1$2$ $ 2^{2} \cdot 17 \cdot 19 $ $x^{3} - 10 x - 14$ $S_3$ (as 3T2) $1$ $0$
* 2.2e2_19.3t2.1c1$2$ $ 2^{2} \cdot 19 $ $x^{3} - 2 x - 2$ $S_3$ (as 3T2) $1$ $0$
4.2e2_17e2_19e2.6t9.1c1$4$ $ 2^{2} \cdot 17^{2} \cdot 19^{2}$ $x^{6} - 2 x^{5} + 3 x^{4} + x^{3} - 2 x^{2} + 3 x - 2$ $S_3^2$ (as 6T9) $1$ $0$
6.2e4_17e3_19e4.18t51.2c1$6$ $ 2^{4} \cdot 17^{3} \cdot 19^{4}$ $x^{9} + x^{7} - 5 x^{6} - 5 x^{5} - 10 x^{4} - 6 x^{3} - 5 x^{2} - x + 1$ $C_3^2 : D_{6} $ (as 9T18) $1$ $-2$
* 6.2e4_17e3_19e2.9t18.2c1$6$ $ 2^{4} \cdot 17^{3} \cdot 19^{2}$ $x^{9} + x^{7} - 5 x^{6} - 5 x^{5} - 10 x^{4} - 6 x^{3} - 5 x^{2} - x + 1$ $C_3^2 : D_{6} $ (as 9T18) $1$ $2$

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 *.