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Properties

Label 9.1.11090466000.1
Degree $9$
Signature $[1, 4]$
Discriminant $2^{4}\cdot 3^{3}\cdot 5^{3}\cdot 59^{3}$
Root discriminant $13.06$
Ramified primes $2, 3, 5, 59$
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, 4, -6, 7, -14, 13, -12, 7, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 3*x^8 + 7*x^7 - 12*x^6 + 13*x^5 - 14*x^4 + 7*x^3 - 6*x^2 + 4*x - 1)
 
gp: K = bnfinit(x^9 - 3*x^8 + 7*x^7 - 12*x^6 + 13*x^5 - 14*x^4 + 7*x^3 - 6*x^2 + 4*x - 1, 1)
 

Normalized defining polynomial

\( x^{9} - 3 x^{8} + 7 x^{7} - 12 x^{6} + 13 x^{5} - 14 x^{4} + 7 x^{3} - 6 x^{2} + 4 x - 1 \)

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

Invariants

Degree:  $9$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(11090466000=2^{4}\cdot 3^{3}\cdot 5^{3}\cdot 59^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.06$
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:  $2, 3, 5, 59$
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 \(a\))

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

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:  \( a^{8} - 3 a^{7} + 7 a^{6} - 12 a^{5} + 13 a^{4} - 14 a^{3} + 7 a^{2} - 6 a + 4 \),  \( \frac{3}{2} a^{8} - 4 a^{7} + 9 a^{6} - \frac{29}{2} a^{5} + 14 a^{4} - \frac{31}{2} a^{3} + \frac{9}{2} a^{2} - 7 a + 3 \),  \( \frac{3}{2} a^{8} - 4 a^{7} + 9 a^{6} - \frac{29}{2} a^{5} + 14 a^{4} - \frac{31}{2} a^{3} + \frac{11}{2} a^{2} - 7 a + 4 \),  \( \frac{3}{2} a^{8} - 4 a^{7} + 9 a^{6} - \frac{29}{2} a^{5} + 14 a^{4} - \frac{31}{2} a^{3} + \frac{9}{2} a^{2} - 8 a + 3 \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 24.5507315161 \)
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

3.1.59.1

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 R R ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{3}$ R

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
$3$3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$59$$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$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$1$ $1$ $x$ $C_1$ $1$ $1$
1.3_5.2t1.1c1$1$ $ 3 \cdot 5 $ $x^{2} - x + 4$ $C_2$ (as 2T1) $1$ $-1$
1.59.2t1.1c1$1$ $ 59 $ $x^{2} - x + 15$ $C_2$ (as 2T1) $1$ $-1$
1.3_5_59.2t1.1c1$1$ $ 3 \cdot 5 \cdot 59 $ $x^{2} - x - 221$ $C_2$ (as 2T1) $1$ $1$
2.3e2_5e2_59.6t3.1c1$2$ $ 3^{2} \cdot 5^{2} \cdot 59 $ $x^{6} - 2 x^{5} + 31 x^{4} - 94 x^{3} + 289 x^{2} - 960 x - 2516$ $D_{6}$ (as 6T3) $1$ $0$
2.2e2_3_5_59.6t3.11c1$2$ $ 2^{2} \cdot 3 \cdot 5 \cdot 59 $ $x^{6} - 3 x^{5} + 13 x^{4} - 21 x^{3} + 28 x^{2} - 18 x + 6$ $D_{6}$ (as 6T3) $1$ $-2$
2.2e2_3_5_59.3t2.2c1$2$ $ 2^{2} \cdot 3 \cdot 5 \cdot 59 $ $x^{3} - x^{2} - 15 x - 15$ $S_3$ (as 3T2) $1$ $2$
* 2.59.3t2.1c1$2$ $ 59 $ $x^{3} + 2 x - 1$ $S_3$ (as 3T2) $1$ $0$
4.2e4_3e2_5e2_59e2.6t9.1c1$4$ $ 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 59^{2}$ $x^{6} - 3 x^{5} + 7 x^{4} - 7 x^{3} + 16 x^{2} - 14 x + 46$ $S_3^2$ (as 6T9) $1$ $0$
6.2e4_3e3_5e3_59e4.18t51.1c1$6$ $ 2^{4} \cdot 3^{3} \cdot 5^{3} \cdot 59^{4}$ $x^{9} - 3 x^{8} + 7 x^{7} - 12 x^{6} + 13 x^{5} - 14 x^{4} + 7 x^{3} - 6 x^{2} + 4 x - 1$ $C_3^2 : D_{6} $ (as 9T18) $1$ $0$
* 6.2e4_3e3_5e3_59e2.9t18.1c1$6$ $ 2^{4} \cdot 3^{3} \cdot 5^{3} \cdot 59^{2}$ $x^{9} - 3 x^{8} + 7 x^{7} - 12 x^{6} + 13 x^{5} - 14 x^{4} + 7 x^{3} - 6 x^{2} + 4 x - 1$ $C_3^2 : D_{6} $ (as 9T18) $1$ $0$

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 *.