Properties

Label 9.3.412698468096.1
Degree $9$
Signature $[3, 3]$
Discriminant $-\,2^{8}\cdot 3^{4}\cdot 271^{3}$
Root discriminant $19.53$
Ramified primes $2, 3, 271$
Class number $1$
Class group Trivial
Galois group $S_3\wr S_3$ (as 9T31)

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Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 3*x^8 + 6*x^7 - 6*x^6 + 4*x^5 - 10*x^3 + 10*x^2 - 9*x - 3)
 
gp: K = bnfinit(x^9 - 3*x^8 + 6*x^7 - 6*x^6 + 4*x^5 - 10*x^3 + 10*x^2 - 9*x - 3, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, -9, 10, -10, 0, 4, -6, 6, -3, 1]);
 

Normalized defining polynomial

\( x^{9} - 3 x^{8} + 6 x^{7} - 6 x^{6} + 4 x^{5} - 10 x^{3} + 10 x^{2} - 9 x - 3 \)

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

Invariants

Degree:  $9$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[3, 3]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-412698468096=-\,2^{8}\cdot 3^{4}\cdot 271^{3}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $19.53$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 3, 271$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{227} a^{8} + \frac{68}{227} a^{7} + \frac{67}{227} a^{6} - \frac{16}{227} a^{5} + \frac{3}{227} a^{4} - \frac{14}{227} a^{3} - \frac{96}{227} a^{2} + \frac{4}{227} a + \frac{48}{227}$

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $5$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 938.85217272 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Galois group

$S_3\wr S_3$ (as 9T31):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 1296
The 22 conjugacy class representatives for $S_3\wr S_3$
Character table for $S_3\wr S_3$ is not computed

Intermediate fields

3.3.3252.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 12 sibling: 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 ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/7.9.0.1}{9} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.9.0.1}{9} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.9.8.1$x^{9} - 2$$9$$1$$8$$(C_9:C_3):C_2$$[\ ]_{9}^{6}$
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.4.3.2$x^{4} - 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
271Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.813.2t1.a.a$1$ $ 3 \cdot 271 $ $x^{2} - x - 203$ $C_2$ (as 2T1) $1$ $1$
1.3.2t1.a.a$1$ $ 3 $ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
1.271.2t1.a.a$1$ $ 271 $ $x^{2} - x + 68$ $C_2$ (as 2T1) $1$ $-1$
2.3252.6t3.f.a$2$ $ 2^{2} \cdot 3 \cdot 271 $ $x^{6} - 3 x^{5} + 33 x^{4} - 61 x^{3} + 246 x^{2} - 216 x + 158$ $D_{6}$ (as 6T3) $1$ $-2$
* 2.3252.3t2.a.a$2$ $ 2^{2} \cdot 3 \cdot 271 $ $x^{3} - x^{2} - 9 x + 3$ $S_3$ (as 3T2) $1$ $2$
3.29268.4t5.b.a$3$ $ 2^{2} \cdot 3^{3} \cdot 271 $ $x^{4} - x^{3} - 9 x^{2} + 5 x + 16$ $S_4$ (as 4T5) $1$ $3$
3.7931628.6t11.b.a$3$ $ 2^{2} \cdot 3^{3} \cdot 271^{2}$ $x^{6} - x^{5} - 3 x^{3} - 9 x + 27$ $S_4\times C_2$ (as 6T11) $1$ $-3$
3.2643876.6t8.a.a$3$ $ 2^{2} \cdot 3^{2} \cdot 271^{2}$ $x^{4} - x^{3} - 9 x^{2} + 5 x + 16$ $S_4$ (as 4T5) $1$ $3$
3.9756.6t11.c.a$3$ $ 2^{2} \cdot 3^{2} \cdot 271 $ $x^{6} - x^{5} - 3 x^{3} - 9 x + 27$ $S_4\times C_2$ (as 6T11) $1$ $-3$
* 6.126906048.9t31.a.a$6$ $ 2^{6} \cdot 3^{3} \cdot 271^{2}$ $x^{9} - 3 x^{8} + 6 x^{7} - 6 x^{6} + 4 x^{5} - 10 x^{3} + 10 x^{2} - 9 x - 3$ $S_3\wr S_3$ (as 9T31) $1$ $0$
6.9320107071168.18t300.a.a$6$ $ 2^{6} \cdot 3^{3} \cdot 271^{4}$ $x^{9} - 3 x^{8} + 6 x^{7} - 6 x^{6} + 4 x^{5} - 10 x^{3} + 10 x^{2} - 9 x - 3$ $S_3\wr S_3$ (as 9T31) $1$ $0$
6.83880963640512.18t319.a.a$6$ $ 2^{6} \cdot 3^{5} \cdot 271^{4}$ $x^{9} - 3 x^{8} + 6 x^{7} - 6 x^{6} + 4 x^{5} - 10 x^{3} + 10 x^{2} - 9 x - 3$ $S_3\wr S_3$ (as 9T31) $1$ $0$
6.1142154432.18t311.a.a$6$ $ 2^{6} \cdot 3^{5} \cdot 271^{2}$ $x^{9} - 3 x^{8} + 6 x^{7} - 6 x^{6} + 4 x^{5} - 10 x^{3} + 10 x^{2} - 9 x - 3$ $S_3\wr S_3$ (as 9T31) $1$ $0$
8.73923622208674101504.24t2893.a.a$8$ $ 2^{8} \cdot 3^{6} \cdot 271^{6}$ $x^{9} - 3 x^{8} + 6 x^{7} - 6 x^{6} + 4 x^{5} - 10 x^{3} + 10 x^{2} - 9 x - 3$ $S_3\wr S_3$ (as 9T31) $1$ $0$
8.13705853184.12t213.a.a$8$ $ 2^{8} \cdot 3^{6} \cdot 271^{2}$ $x^{9} - 3 x^{8} + 6 x^{7} - 6 x^{6} + 4 x^{5} - 10 x^{3} + 10 x^{2} - 9 x - 3$ $S_3\wr S_3$ (as 9T31) $1$ $0$
12.2163596574803473602819072.36t2217.a.a$12$ $ 2^{10} \cdot 3^{9} \cdot 271^{7}$ $x^{9} - 3 x^{8} + 6 x^{7} - 6 x^{6} + 4 x^{5} - 10 x^{3} + 10 x^{2} - 9 x - 3$ $S_3\wr S_3$ (as 9T31) $1$ $0$
12.29460336525966062592.36t2214.a.a$12$ $ 2^{10} \cdot 3^{9} \cdot 271^{5}$ $x^{9} - 3 x^{8} + 6 x^{7} - 6 x^{6} + 4 x^{5} - 10 x^{3} + 10 x^{2} - 9 x - 3$ $S_3\wr S_3$ (as 9T31) $1$ $0$
12.95805014382441635549184.36t2210.a.a$12$ $ 2^{12} \cdot 3^{10} \cdot 271^{6}$ $x^{9} - 3 x^{8} + 6 x^{7} - 6 x^{6} + 4 x^{5} - 10 x^{3} + 10 x^{2} - 9 x - 3$ $S_3\wr S_3$ (as 9T31) $1$ $0$
12.2163596574803473602819072.36t2216.a.a$12$ $ 2^{10} \cdot 3^{9} \cdot 271^{7}$ $x^{9} - 3 x^{8} + 6 x^{7} - 6 x^{6} + 4 x^{5} - 10 x^{3} + 10 x^{2} - 9 x - 3$ $S_3\wr S_3$ (as 9T31) $1$ $0$
12.29460336525966062592.18t315.a.a$12$ $ 2^{10} \cdot 3^{9} \cdot 271^{5}$ $x^{9} - 3 x^{8} + 6 x^{7} - 6 x^{6} + 4 x^{5} - 10 x^{3} + 10 x^{2} - 9 x - 3$ $S_3\wr S_3$ (as 9T31) $1$ $0$
16.253296578205392261629234397184.24t2912.a.a$16$ $ 2^{14} \cdot 3^{12} \cdot 271^{8}$ $x^{9} - 3 x^{8} + 6 x^{7} - 6 x^{6} + 4 x^{5} - 10 x^{3} + 10 x^{2} - 9 x - 3$ $S_3\wr S_3$ (as 9T31) $1$ $0$

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 *.