Properties

Label 9.7.236372930487.1
Degree $9$
Signature $[7, 1]$
Discriminant $-\,3^{9}\cdot 229^{3}$
Root discriminant $18.35$
Ramified primes $3, 229$
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![-1, 12, 0, -31, 0, 27, 0, -9, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 9*x^7 + 27*x^5 - 31*x^3 + 12*x - 1)
gp: K = bnfinit(x^9 - 9*x^7 + 27*x^5 - 31*x^3 + 12*x - 1, 1)

Normalized defining polynomial

\(x^{9} \) \(\mathstrut -\mathstrut 9 x^{7} \) \(\mathstrut +\mathstrut 27 x^{5} \) \(\mathstrut -\mathstrut 31 x^{3} \) \(\mathstrut +\mathstrut 12 x \) \(\mathstrut -\mathstrut 1 \)

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

Invariants

Degree:  $9$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[7, 1]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(-236372930487=-\,3^{9}\cdot 229^{3}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $18.35$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 229$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$

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:  $7$
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:  \( a^{2} - 1 \),  \( a + 1 \),  \( a^{5} - 5 a^{3} + 4 a \),  \( a \),  \( a^{7} - 7 a^{5} + 14 a^{3} - 7 a + 1 \),  \( a^{7} - 8 a^{5} + 19 a^{3} - a^{2} - 12 a + 3 \),  \( a^{7} - 7 a^{5} - a^{4} + 14 a^{3} + 4 a^{2} - 7 a - 2 \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 397.137760612 \)
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

3.3.229.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ R ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.9.0.1}{9} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.9.0.1}{9} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$3$3.9.9.9$x^{9} + 18 x^{5} + 27 x^{2} + 54$$3$$3$$9$$(C_3^2:C_3):C_2$$[3/2, 3/2, 3/2]_{2}^{3}$
229Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.229.2t1.1c1$1$ $ 229 $ $x^{2} - x - 57$ $C_2$ (as 2T1) $1$ $1$
1.3.2t1.1c1$1$ $ 3 $ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
1.3_229.2t1.1c1$1$ $ 3 \cdot 229 $ $x^{2} - x + 172$ $C_2$ (as 2T1) $1$ $-1$
2.3e2_229.6t3.1c1$2$ $ 3^{2} \cdot 229 $ $x^{6} + 16 x^{4} - 7 x^{3} + 64 x^{2} - 56 x + 184$ $D_{6}$ (as 6T3) $1$ $-2$
* 2.229.3t2.1c1$2$ $ 229 $ $x^{3} - 4 x - 1$ $S_3$ (as 3T2) $1$ $2$
3.229.4t5.1c1$3$ $ 229 $ $x^{4} - x + 1$ $S_4$ (as 4T5) $1$ $-1$
3.3e3_229.6t11.1c1$3$ $ 3^{3} \cdot 229 $ $x^{6} - 5 x^{4} - 2 x^{3} + 7 x^{2} + 5 x - 2$ $S_4\times C_2$ (as 6T11) $1$ $1$
3.229e2.6t8.1c1$3$ $ 229^{2}$ $x^{4} - x + 1$ $S_4$ (as 4T5) $1$ $-1$
3.3e3_229e2.6t11.1c1$3$ $ 3^{3} \cdot 229^{2}$ $x^{6} - 5 x^{4} - 2 x^{3} + 7 x^{2} + 5 x - 2$ $S_4\times C_2$ (as 6T11) $1$ $1$
* 6.3e9_229e2.9t31.1c1$6$ $ 3^{9} \cdot 229^{2}$ $x^{9} - 9 x^{7} + 27 x^{5} - 31 x^{3} + 12 x - 1$ $S_3\wr S_3$ (as 9T31) $1$ $4$
6.3e9_229e4.18t303.1c1$6$ $ 3^{9} \cdot 229^{4}$ $x^{9} - 9 x^{7} + 27 x^{5} - 31 x^{3} + 12 x - 1$ $S_3\wr S_3$ (as 9T31) $1$ $-4$
6.3e9_229e4.18t320.1c1$6$ $ 3^{9} \cdot 229^{4}$ $x^{9} - 9 x^{7} + 27 x^{5} - 31 x^{3} + 12 x - 1$ $S_3\wr S_3$ (as 9T31) $1$ $4$
6.3e9_229e2.18t312.1c1$6$ $ 3^{9} \cdot 229^{2}$ $x^{9} - 9 x^{7} + 27 x^{5} - 31 x^{3} + 12 x - 1$ $S_3\wr S_3$ (as 9T31) $1$ $-4$
8.3e12_229e6.24t2895.1c1$8$ $ 3^{12} \cdot 229^{6}$ $x^{9} - 9 x^{7} + 27 x^{5} - 31 x^{3} + 12 x - 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
8.3e12_229e2.12t213.1c1$8$ $ 3^{12} \cdot 229^{2}$ $x^{9} - 9 x^{7} + 27 x^{5} - 31 x^{3} + 12 x - 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
12.3e18_229e7.36t2217.1c1$12$ $ 3^{18} \cdot 229^{7}$ $x^{9} - 9 x^{7} + 27 x^{5} - 31 x^{3} + 12 x - 1$ $S_3\wr S_3$ (as 9T31) $1$ $-4$
12.3e18_229e5.36t2215.1c1$12$ $ 3^{18} \cdot 229^{5}$ $x^{9} - 9 x^{7} + 27 x^{5} - 31 x^{3} + 12 x - 1$ $S_3\wr S_3$ (as 9T31) $1$ $-4$
12.3e18_229e6.36t2211.1c1$12$ $ 3^{18} \cdot 229^{6}$ $x^{9} - 9 x^{7} + 27 x^{5} - 31 x^{3} + 12 x - 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
12.3e18_229e7.36t2218.1c1$12$ $ 3^{18} \cdot 229^{7}$ $x^{9} - 9 x^{7} + 27 x^{5} - 31 x^{3} + 12 x - 1$ $S_3\wr S_3$ (as 9T31) $1$ $4$
12.3e18_229e5.18t315.1c1$12$ $ 3^{18} \cdot 229^{5}$ $x^{9} - 9 x^{7} + 27 x^{5} - 31 x^{3} + 12 x - 1$ $S_3\wr S_3$ (as 9T31) $1$ $4$
16.3e24_229e8.24t2912.1c1$16$ $ 3^{24} \cdot 229^{8}$ $x^{9} - 9 x^{7} + 27 x^{5} - 31 x^{3} + 12 x - 1$ $S_3\wr S_3$ (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 *.