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Properties

Label 9.1.17353862469.1
Degree $9$
Signature $[1, 4]$
Discriminant $3^{3}\cdot 863^{3}$
Root discriminant $13.73$
Ramified primes $3, 863$
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, 0, 0, 7, 0, 0, -4, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 4*x^6 + 7*x^3 + 1)
 
gp: K = bnfinit(x^9 - 4*x^6 + 7*x^3 + 1, 1)
 

Normalized defining polynomial

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

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:  \(17353862469=3^{3}\cdot 863^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.73$
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:  $3, 863$
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 \(a\))

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

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:  \( a \),  \( \frac{1}{3} a^{8} - \frac{5}{3} a^{5} - \frac{1}{3} a^{4} + \frac{10}{3} a^{2} + \frac{1}{3} a + \frac{1}{3} \),  \( \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - a^{4} - \frac{4}{3} a^{3} - \frac{1}{3} a^{2} + \frac{4}{3} a + 1 \),  \( \frac{2}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{8}{3} a^{5} + \frac{4}{3} a^{4} - \frac{2}{3} a^{3} + \frac{13}{3} a^{2} - \frac{7}{3} a + \frac{4}{3} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 28.7944000624 \)
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

3.1.863.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }$ R ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$3$3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
863Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.3.2t1.1c1$1$ $ 3 $ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
1.863.2t1.1c1$1$ $ 863 $ $x^{2} - x + 216$ $C_2$ (as 2T1) $1$ $-1$
1.3_863.2t1.1c1$1$ $ 3 \cdot 863 $ $x^{2} - x - 647$ $C_2$ (as 2T1) $1$ $1$
2.3e2_863.6t3.1c1$2$ $ 3^{2} \cdot 863 $ $x^{6} - 2 x^{5} - 5 x^{4} - 31 x^{3} + 46 x^{2} + 111 x - 305$ $D_{6}$ (as 6T3) $1$ $0$
2.3_863.6t3.1c1$2$ $ 3 \cdot 863 $ $x^{6} - 3 x^{5} + 16 x^{4} - 27 x^{3} + 55 x^{2} - 42 x + 12$ $D_{6}$ (as 6T3) $1$ $-2$
2.3_863.3t2.1c1$2$ $ 3 \cdot 863 $ $x^{3} - x^{2} - 14 x + 12$ $S_3$ (as 3T2) $1$ $2$
* 2.863.3t2.1c1$2$ $ 863 $ $x^{3} - x^{2} + 2 x + 5$ $S_3$ (as 3T2) $1$ $0$
4.3e2_863e2.6t9.1c1$4$ $ 3^{2} \cdot 863^{2}$ $x^{6} - 2 x^{5} - 12 x^{4} + 6 x^{3} + 86 x^{2} + 119 x + 49$ $S_3^2$ (as 6T9) $1$ $0$
6.3e3_863e4.18t51.1c1$6$ $ 3^{3} \cdot 863^{4}$ $x^{9} - 4 x^{6} + 7 x^{3} + 1$ $C_3^2 : D_{6} $ (as 9T18) $1$ $0$
* 6.3e3_863e2.9t18.1c1$6$ $ 3^{3} \cdot 863^{2}$ $x^{9} - 4 x^{6} + 7 x^{3} + 1$ $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 *.