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Properties

Label 9.1.6602349376.1
Degree $9$
Signature $[1, 4]$
Discriminant $2^{6}\cdot 7^{3}\cdot 67^{3}$
Root discriminant $12.33$
Ramified primes $2, 7, 67$
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, -2, -2, 3, -7, 5, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 2*x^8 + 5*x^7 - 7*x^6 + 3*x^5 - 2*x^4 - 2*x^3 + 5*x^2 + x + 1)
 
gp: K = bnfinit(x^9 - 2*x^8 + 5*x^7 - 7*x^6 + 3*x^5 - 2*x^4 - 2*x^3 + 5*x^2 + x + 1, 1)
 

Normalized defining polynomial

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

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

Invariants

Degree:  $9$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6602349376=2^{6}\cdot 7^{3}\cdot 67^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12.33$
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:  $2, 7, 67$
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 \(a\))

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

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:  $4$
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:  \( a \),  \( \frac{2}{7} a^{8} - \frac{2}{7} a^{7} + \frac{5}{7} a^{6} - \frac{2}{7} a^{5} - 2 a^{4} + \frac{9}{7} a^{3} - \frac{10}{7} a^{2} + \frac{10}{7} a + \frac{16}{7} \),  \( \frac{1}{7} a^{8} - \frac{1}{7} a^{6} + \frac{5}{7} a^{5} - \frac{16}{7} a^{4} + \frac{6}{7} a^{3} + \frac{1}{7} a^{2} - \frac{1}{7} a + 1 \),  \( \frac{1}{7} a^{7} - \frac{1}{7} a^{5} + \frac{5}{7} a^{4} - \frac{16}{7} a^{3} + \frac{6}{7} a^{2} + \frac{8}{7} a - \frac{1}{7} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 28.1795752405 \)
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

3.1.268.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$$[\ ]_{3}^{6}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$67$67.3.0.1$x^{3} - x + 16$$1$$3$$0$$C_3$$[\ ]^{3}$
67.6.3.2$x^{6} - 4489 x^{2} + 4812208$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.7.2t1.1c1$1$ $ 7 $ $x^{2} - x + 2$ $C_2$ (as 2T1) $1$ $-1$
1.67.2t1.1c1$1$ $ 67 $ $x^{2} - x + 17$ $C_2$ (as 2T1) $1$ $-1$
1.7_67.2t1.1c1$1$ $ 7 \cdot 67 $ $x^{2} - x - 117$ $C_2$ (as 2T1) $1$ $1$
2.2e2_7e2_67.6t3.1c1$2$ $ 2^{2} \cdot 7^{2} \cdot 67 $ $x^{6} - 2 x^{5} + 35 x^{4} - 108 x^{3} + 363 x^{2} - 1258 x - 507$ $D_{6}$ (as 6T3) $1$ $0$
2.7_67.6t3.2c1$2$ $ 7 \cdot 67 $ $x^{6} - 2 x^{5} + 3 x^{3} - 8 x + 8$ $D_{6}$ (as 6T3) $1$ $-2$
2.7_67.3t2.1c1$2$ $ 7 \cdot 67 $ $x^{3} - x^{2} - 5 x + 4$ $S_3$ (as 3T2) $1$ $2$
* 2.2e2_67.3t2.1c1$2$ $ 2^{2} \cdot 67 $ $x^{3} - x^{2} - 3 x + 5$ $S_3$ (as 3T2) $1$ $0$
4.2e4_7e2_67e2.6t9.1c1$4$ $ 2^{4} \cdot 7^{2} \cdot 67^{2}$ $x^{6} - 3 x^{5} + 5 x^{4} - 3 x^{3} + 6 x^{2} - 6 x + 22$ $S_3^2$ (as 6T9) $1$ $0$
6.2e4_7e3_67e4.18t51.2c1$6$ $ 2^{4} \cdot 7^{3} \cdot 67^{4}$ $x^{9} - 2 x^{8} + 5 x^{7} - 7 x^{6} + 3 x^{5} - 2 x^{4} - 2 x^{3} + 5 x^{2} + x + 1$ $C_3^2 : D_{6} $ (as 9T18) $1$ $0$
* 6.2e4_7e3_67e2.9t18.2c1$6$ $ 2^{4} \cdot 7^{3} \cdot 67^{2}$ $x^{9} - 2 x^{8} + 5 x^{7} - 7 x^{6} + 3 x^{5} - 2 x^{4} - 2 x^{3} + 5 x^{2} + x + 1$ $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 *.