Properties

Label 9.1.2366667072.1
Degree $9$
Signature $[1, 4]$
Discriminant $2366667072$
Root discriminant $11.00$
Ramified primes $2, 3, 7, 11$
Class number $1$
Class group trivial
Galois group $S_3\wr S_3$ (as 9T31)

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Normalized defining polynomial

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

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

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

Invariants

Degree:  $9$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[1, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(2366667072\)\(\medspace = 2^{6}\cdot 3^{4}\cdot 7^{3}\cdot 11^{3}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $11.00$
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:  $2, 3, 7, 11$
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 \(a\))

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

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:  $4$
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:  \( a \),  \( a^{8} - \frac{2}{3} a^{7} - \frac{10}{3} a^{5} - \frac{1}{3} a^{4} + \frac{7}{3} a^{3} + \frac{14}{3} a^{2} - \frac{5}{3} a - \frac{2}{3} \),  \( a^{8} - 3 a^{5} - 2 a^{4} + a^{3} + 4 a^{2} \),  \( \frac{4}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - 4 a^{5} - \frac{7}{3} a^{4} + \frac{10}{3} a^{3} + \frac{17}{3} a^{2} - \frac{1}{3} \)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 16.5992891922 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{4}\cdot 16.5992891922 \cdot 1}{2\sqrt{2366667072}}\approx 0.531790262147$

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

3.1.231.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R R ${\href{/LocalNumberField/13.9.0.1}{9} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.9.0.1}{9} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.9.0.1}{9} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.6.2$x^{6} - x^{4} - 5$$2$$3$$6$$A_4\times C_2$$[2, 2]^{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}$$[\ ]_{4}^{2}$
$7$7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.231.2t1.a.a$1$ $ 3 \cdot 7 \cdot 11 $ $x^{2} - x + 58$ $C_2$ (as 2T1) $1$ $-1$
1.3.2t1.a.a$1$ $ 3 $ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
1.77.2t1.a.a$1$ $ 7 \cdot 11 $ $x^{2} - x - 19$ $C_2$ (as 2T1) $1$ $1$
2.231.6t3.f.a$2$ $ 3 \cdot 7 \cdot 11 $ $x^{6} - 9 x^{3} + 1$ $D_{6}$ (as 6T3) $1$ $0$
* 2.231.3t2.a.a$2$ $ 3 \cdot 7 \cdot 11 $ $x^{3} - x^{2} + 3$ $S_3$ (as 3T2) $1$ $0$
3.133056.4t5.a.a$3$ $ 2^{6} \cdot 3^{3} \cdot 7 \cdot 11 $ $x^{4} - 2 x^{3} + 6 x^{2} + 4 x - 2$ $S_4$ (as 4T5) $1$ $1$
3.10245312.6t11.a.a$3$ $ 2^{6} \cdot 3^{3} \cdot 7^{2} \cdot 11^{2}$ $x^{6} + 2 x^{4} + 15 x^{2} + 3$ $S_4\times C_2$ (as 6T11) $1$ $1$
3.3415104.6t8.d.a$3$ $ 2^{6} \cdot 3^{2} \cdot 7^{2} \cdot 11^{2}$ $x^{4} - 2 x^{3} + 6 x^{2} + 4 x - 2$ $S_4$ (as 4T5) $1$ $-1$
3.44352.6t11.a.a$3$ $ 2^{6} \cdot 3^{2} \cdot 7 \cdot 11 $ $x^{6} + 2 x^{4} + 15 x^{2} + 3$ $S_4\times C_2$ (as 6T11) $1$ $-1$
* 6.10245312.9t31.a.a$6$ $ 2^{6} \cdot 3^{3} \cdot 7^{2} \cdot 11^{2}$ $x^{9} - x^{8} - 3 x^{6} + x^{5} + 3 x^{4} + 3 x^{3} - 4 x^{2} + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
6.60744454848.18t300.a.a$6$ $ 2^{6} \cdot 3^{3} \cdot 7^{4} \cdot 11^{4}$ $x^{9} - x^{8} - 3 x^{6} + x^{5} + 3 x^{4} + 3 x^{3} - 4 x^{2} + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
6.546700093632.18t319.a.a$6$ $ 2^{6} \cdot 3^{5} \cdot 7^{4} \cdot 11^{4}$ $x^{9} - x^{8} - 3 x^{6} + x^{5} + 3 x^{4} + 3 x^{3} - 4 x^{2} + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
6.92207808.18t311.a.a$6$ $ 2^{6} \cdot 3^{5} \cdot 7^{2} \cdot 11^{2}$ $x^{9} - x^{8} - 3 x^{6} + x^{5} + 3 x^{4} + 3 x^{3} - 4 x^{2} + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
8.622...576.24t2893.b.a$8$ $ 2^{12} \cdot 3^{6} \cdot 7^{6} \cdot 11^{6}$ $x^{9} - x^{8} - 3 x^{6} + x^{5} + 3 x^{4} + 3 x^{3} - 4 x^{2} + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
8.17703899136.12t213.a.a$8$ $ 2^{12} \cdot 3^{6} \cdot 7^{2} \cdot 11^{2}$ $x^{9} - x^{8} - 3 x^{6} + x^{5} + 3 x^{4} + 3 x^{3} - 4 x^{2} + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
12.828...256.36t2217.a.a$12$ $ 2^{18} \cdot 3^{9} \cdot 7^{7} \cdot 11^{7}$ $x^{9} - x^{8} - 3 x^{6} + x^{5} + 3 x^{4} + 3 x^{3} - 4 x^{2} + 1$ $S_3\wr S_3$ (as 9T31) $1$ $2$
12.139...264.36t2214.a.a$12$ $ 2^{18} \cdot 3^{9} \cdot 7^{5} \cdot 11^{5}$ $x^{9} - x^{8} - 3 x^{6} + x^{5} + 3 x^{4} + 3 x^{3} - 4 x^{2} + 1$ $S_3\wr S_3$ (as 9T31) $1$ $-2$
12.206...976.36t2210.a.a$12$ $ 2^{24} \cdot 3^{10} \cdot 7^{6} \cdot 11^{6}$ $x^{9} - x^{8} - 3 x^{6} + x^{5} + 3 x^{4} + 3 x^{3} - 4 x^{2} + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
12.828...256.36t2216.a.a$12$ $ 2^{18} \cdot 3^{9} \cdot 7^{7} \cdot 11^{7}$ $x^{9} - x^{8} - 3 x^{6} + x^{5} + 3 x^{4} + 3 x^{3} - 4 x^{2} + 1$ $S_3\wr S_3$ (as 9T31) $1$ $-2$
12.139...264.18t315.a.a$12$ $ 2^{18} \cdot 3^{9} \cdot 7^{5} \cdot 11^{5}$ $x^{9} - x^{8} - 3 x^{6} + x^{5} + 3 x^{4} + 3 x^{3} - 4 x^{2} + 1$ $S_3\wr S_3$ (as 9T31) $1$ $2$
16.110...336.24t2912.a.a$16$ $ 2^{24} \cdot 3^{12} \cdot 7^{8} \cdot 11^{8}$ $x^{9} - x^{8} - 3 x^{6} + x^{5} + 3 x^{4} + 3 x^{3} - 4 x^{2} + 1$ $S_3\wr S_3$ (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 *.