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Properties

Label 9.1.186659085312.1
Degree $9$
Signature $[1, 4]$
Discriminant $2^{10}\cdot 3^{12}\cdot 7^{3}$
Root discriminant $17.88$
Ramified primes $2, 3, 7$
Class number $1$
Class group Trivial
Galois group $S_3\wr S_3$ (as 9T31)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12, 0, -9, 15, -3, -3, 3, 3, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 3*x^8 + 3*x^7 + 3*x^6 - 3*x^5 - 3*x^4 + 15*x^3 - 9*x^2 + 12)
 
gp: K = bnfinit(x^9 - 3*x^8 + 3*x^7 + 3*x^6 - 3*x^5 - 3*x^4 + 15*x^3 - 9*x^2 + 12, 1)
 

Normalized defining polynomial

\( x^{9} - 3 x^{8} + 3 x^{7} + 3 x^{6} - 3 x^{5} - 3 x^{4} + 15 x^{3} - 9 x^{2} + 12 \)

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:  \(186659085312=2^{10}\cdot 3^{12}\cdot 7^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.88$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{56} a^{8} + \frac{1}{28} a^{7} - \frac{1}{56} a^{6} - \frac{1}{28} a^{5} - \frac{27}{56} a^{4} + \frac{1}{28} a^{3} + \frac{11}{56} a^{2} - \frac{5}{28} a - \frac{1}{7}$

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:  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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 440.372217029 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
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

3.1.756.1

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 ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.9.0.1}{9} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.9.0.1}{9} }$ ${\href{/LocalNumberField/37.9.0.1}{9} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.4.8.5$x^{4} + 2 x^{2} + 4 x + 6$$4$$1$$8$$D_{4}$$[2, 3]^{2}$
$3$3.9.12.22$x^{9} + 6 x^{4} + 6 x^{3} + 3$$9$$1$$12$$C_3^2 : C_6$$[3/2, 3/2]_{2}^{3}$
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$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$1$ $1$ $x$ $C_1$ $1$ $1$
1.2e2_3_7.2t1.1c1$1$ $ 2^{2} \cdot 3 \cdot 7 $ $x^{2} + 21$ $C_2$ (as 2T1) $1$ $-1$
1.3.2t1.1c1$1$ $ 3 $ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
1.2e2_7.2t1.1c1$1$ $ 2^{2} \cdot 7 $ $x^{2} - 7$ $C_2$ (as 2T1) $1$ $1$
2.2e2_3e3_7.6t3.7c1$2$ $ 2^{2} \cdot 3^{3} \cdot 7 $ $x^{6} - 12 x^{3} + 8$ $D_{6}$ (as 6T3) $1$ $0$
* 2.2e2_3e3_7.3t2.1c1$2$ $ 2^{2} \cdot 3^{3} \cdot 7 $ $x^{3} - 6 x - 12$ $S_3$ (as 3T2) $1$ $0$
3.2e6_3e3_7.4t5.1c1$3$ $ 2^{6} \cdot 3^{3} \cdot 7 $ $x^{4} - 2 x^{3} - 3 x^{2} + 2 x + 5$ $S_4$ (as 4T5) $1$ $1$
3.2e8_3e3_7e2.6t11.3c1$3$ $ 2^{8} \cdot 3^{3} \cdot 7^{2}$ $x^{6} - 3 x^{5} + 12 x^{4} - 18 x^{3} + 30 x^{2} - 18 x + 12$ $S_4\times C_2$ (as 6T11) $1$ $1$
3.2e8_3e4_7e2.6t8.3c1$3$ $ 2^{8} \cdot 3^{4} \cdot 7^{2}$ $x^{4} - 2 x^{3} - 3 x^{2} + 2 x + 5$ $S_4$ (as 4T5) $1$ $-1$
3.2e6_3e4_7.6t11.5c1$3$ $ 2^{6} \cdot 3^{4} \cdot 7 $ $x^{6} - 3 x^{5} + 12 x^{4} - 18 x^{3} + 30 x^{2} - 18 x + 12$ $S_4\times C_2$ (as 6T11) $1$ $-1$
* 6.2e8_3e9_7e2.9t31.1c1$6$ $ 2^{8} \cdot 3^{9} \cdot 7^{2}$ $x^{9} - 3 x^{8} + 3 x^{7} + 3 x^{6} - 3 x^{5} - 3 x^{4} + 15 x^{3} - 9 x^{2} + 12$ $S_3\wr S_3$ (as 9T31) $1$ $0$
6.2e12_3e9_7e4.18t303.1c1$6$ $ 2^{12} \cdot 3^{9} \cdot 7^{4}$ $x^{9} - 3 x^{8} + 3 x^{7} + 3 x^{6} - 3 x^{5} - 3 x^{4} + 15 x^{3} - 9 x^{2} + 12$ $S_3\wr S_3$ (as 9T31) $1$ $0$
6.2e12_3e9_7e4.18t320.1c1$6$ $ 2^{12} \cdot 3^{9} \cdot 7^{4}$ $x^{9} - 3 x^{8} + 3 x^{7} + 3 x^{6} - 3 x^{5} - 3 x^{4} + 15 x^{3} - 9 x^{2} + 12$ $S_3\wr S_3$ (as 9T31) $1$ $0$
6.2e8_3e9_7e2.18t312.1c1$6$ $ 2^{8} \cdot 3^{9} \cdot 7^{2}$ $x^{9} - 3 x^{8} + 3 x^{7} + 3 x^{6} - 3 x^{5} - 3 x^{4} + 15 x^{3} - 9 x^{2} + 12$ $S_3\wr S_3$ (as 9T31) $1$ $0$
8.2e20_3e10_7e6.24t2895.1c1$8$ $ 2^{20} \cdot 3^{10} \cdot 7^{6}$ $x^{9} - 3 x^{8} + 3 x^{7} + 3 x^{6} - 3 x^{5} - 3 x^{4} + 15 x^{3} - 9 x^{2} + 12$ $S_3\wr S_3$ (as 9T31) $1$ $0$
8.2e12_3e10_7e2.12t213.1c1$8$ $ 2^{12} \cdot 3^{10} \cdot 7^{2}$ $x^{9} - 3 x^{8} + 3 x^{7} + 3 x^{6} - 3 x^{5} - 3 x^{4} + 15 x^{3} - 9 x^{2} + 12$ $S_3\wr S_3$ (as 9T31) $1$ $0$
12.2e26_3e15_7e7.36t2305.1c1$12$ $ 2^{26} \cdot 3^{15} \cdot 7^{7}$ $x^{9} - 3 x^{8} + 3 x^{7} + 3 x^{6} - 3 x^{5} - 3 x^{4} + 15 x^{3} - 9 x^{2} + 12$ $S_3\wr S_3$ (as 9T31) $1$ $2$
12.2e22_3e17_7e5.36t2215.1c1$12$ $ 2^{22} \cdot 3^{17} \cdot 7^{5}$ $x^{9} - 3 x^{8} + 3 x^{7} + 3 x^{6} - 3 x^{5} - 3 x^{4} + 15 x^{3} - 9 x^{2} + 12$ $S_3\wr S_3$ (as 9T31) $1$ $-2$
12.2e28_3e18_7e6.36t2210.1c1$12$ $ 2^{28} \cdot 3^{18} \cdot 7^{6}$ $x^{9} - 3 x^{8} + 3 x^{7} + 3 x^{6} - 3 x^{5} - 3 x^{4} + 15 x^{3} - 9 x^{2} + 12$ $S_3\wr S_3$ (as 9T31) $1$ $0$
12.2e26_3e17_7e7.36t2218.1c1$12$ $ 2^{26} \cdot 3^{17} \cdot 7^{7}$ $x^{9} - 3 x^{8} + 3 x^{7} + 3 x^{6} - 3 x^{5} - 3 x^{4} + 15 x^{3} - 9 x^{2} + 12$ $S_3\wr S_3$ (as 9T31) $1$ $-2$
12.2e22_3e15_7e5.18t315.1c1$12$ $ 2^{22} \cdot 3^{15} \cdot 7^{5}$ $x^{9} - 3 x^{8} + 3 x^{7} + 3 x^{6} - 3 x^{5} - 3 x^{4} + 15 x^{3} - 9 x^{2} + 12$ $S_3\wr S_3$ (as 9T31) $1$ $2$
16.2e32_3e24_7e8.24t2912.1c1$16$ $ 2^{32} \cdot 3^{24} \cdot 7^{8}$ $x^{9} - 3 x^{8} + 3 x^{7} + 3 x^{6} - 3 x^{5} - 3 x^{4} + 15 x^{3} - 9 x^{2} + 12$ $S_3\wr S_3$ (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 *.