Properties

Label 9.1.1967079625.1
Degree $9$
Signature $[1, 4]$
Discriminant $5^{3}\cdot 7\cdot 131^{3}$
Root discriminant $10.78$
Ramified primes $5, 7, 131$
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, 1, -4, 0, 8, -6, -2, 6, -4, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 4*x^8 + 6*x^7 - 2*x^6 - 6*x^5 + 8*x^4 - 4*x^2 + x - 1)
gp: K = bnfinit(x^9 - 4*x^8 + 6*x^7 - 2*x^6 - 6*x^5 + 8*x^4 - 4*x^2 + x - 1, 1)

Normalized defining polynomial

\(x^{9} \) \(\mathstrut -\mathstrut 4 x^{8} \) \(\mathstrut +\mathstrut 6 x^{7} \) \(\mathstrut -\mathstrut 2 x^{6} \) \(\mathstrut -\mathstrut 6 x^{5} \) \(\mathstrut +\mathstrut 8 x^{4} \) \(\mathstrut -\mathstrut 4 x^{2} \) \(\mathstrut +\mathstrut x \) \(\mathstrut -\mathstrut 1 \)

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

Invariants

Degree:  $9$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[1, 4]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(1967079625=5^{3}\cdot 7\cdot 131^{3}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $10.78$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $5, 7, 131$
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 \(a\))

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

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:  \( a \),  \( \frac{3}{7} a^{8} - \frac{13}{7} a^{7} + \frac{20}{7} a^{6} - \frac{8}{7} a^{5} - \frac{20}{7} a^{4} + \frac{26}{7} a^{3} + \frac{3}{7} a^{2} - \frac{20}{7} a + \frac{5}{7} \),  \( a - 1 \),  \( \frac{3}{7} a^{8} - \frac{13}{7} a^{7} + \frac{20}{7} a^{6} - \frac{8}{7} a^{5} - \frac{20}{7} a^{4} + \frac{26}{7} a^{3} + \frac{3}{7} a^{2} - \frac{13}{7} a + \frac{5}{7} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 10.4388303534 \)
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

3.1.655.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }$ ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ R R ${\href{/LocalNumberField/11.9.0.1}{9} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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

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
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.6.0.1$x^{6} + 3 x^{2} - x + 5$$1$$6$$0$$C_6$$[\ ]^{6}$
$131$$\Q_{131}$$x + 3$$1$$1$$0$Trivial$[\ ]$
131.2.0.1$x^{2} - x + 14$$1$$2$$0$$C_2$$[\ ]^{2}$
131.6.3.2$x^{6} - 17161 x^{2} + 20232819$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.5_131.2t1.1c1$1$ $ 5 \cdot 131 $ $x^{2} - x + 164$ $C_2$ (as 2T1) $1$ $-1$
1.7.2t1.1c1$1$ $ 7 $ $x^{2} - x + 2$ $C_2$ (as 2T1) $1$ $-1$
1.5_7_131.2t1.1c1$1$ $ 5 \cdot 7 \cdot 131 $ $x^{2} - x - 1146$ $C_2$ (as 2T1) $1$ $1$
2.5_7e2_131.6t3.1c1$2$ $ 5 \cdot 7^{2} \cdot 131 $ $x^{6} - 2 x^{5} + 5 x^{4} - 84 x^{3} + 84 x^{2} - 160 x - 2985$ $D_{6}$ (as 6T3) $1$ $0$
* 2.5_131.3t2.1c1$2$ $ 5 \cdot 131 $ $x^{3} - x^{2} + 5$ $S_3$ (as 3T2) $1$ $0$
3.5_7e2_131.4t5.1c1$3$ $ 5 \cdot 7^{2} \cdot 131 $ $x^{4} - x^{3} - 5 x^{2} + 7$ $S_4$ (as 4T5) $1$ $1$
3.5e2_7_131e2.6t11.2c1$3$ $ 5^{2} \cdot 7 \cdot 131^{2}$ $x^{6} - x^{5} + 3 x^{4} + 3 x^{3} + 3 x^{2} - x + 1$ $S_4\times C_2$ (as 6T11) $1$ $1$
3.5e2_7e2_131e2.6t8.1c1$3$ $ 5^{2} \cdot 7^{2} \cdot 131^{2}$ $x^{4} - x^{3} - 5 x^{2} + 7$ $S_4$ (as 4T5) $1$ $-1$
3.5_7_131.6t11.2c1$3$ $ 5 \cdot 7 \cdot 131 $ $x^{6} - x^{5} + 3 x^{4} + 3 x^{3} + 3 x^{2} - x + 1$ $S_4\times C_2$ (as 6T11) $1$ $-1$
* 6.5e2_7_131e2.9t31.1c1$6$ $ 5^{2} \cdot 7 \cdot 131^{2}$ $x^{9} - 4 x^{8} + 6 x^{7} - 2 x^{6} - 6 x^{5} + 8 x^{4} - 4 x^{2} + x - 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
6.5e4_7e5_131e4.18t303.1c1$6$ $ 5^{4} \cdot 7^{5} \cdot 131^{4}$ $x^{9} - 4 x^{8} + 6 x^{7} - 2 x^{6} - 6 x^{5} + 8 x^{4} - 4 x^{2} + x - 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
6.5e4_7_131e4.18t320.1c1$6$ $ 5^{4} \cdot 7 \cdot 131^{4}$ $x^{9} - 4 x^{8} + 6 x^{7} - 2 x^{6} - 6 x^{5} + 8 x^{4} - 4 x^{2} + x - 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
6.5e2_7e5_131e2.18t312.1c1$6$ $ 5^{2} \cdot 7^{5} \cdot 131^{2}$ $x^{9} - 4 x^{8} + 6 x^{7} - 2 x^{6} - 6 x^{5} + 8 x^{4} - 4 x^{2} + x - 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
8.5e6_7e4_131e6.24t2895.1c1$8$ $ 5^{6} \cdot 7^{4} \cdot 131^{6}$ $x^{9} - 4 x^{8} + 6 x^{7} - 2 x^{6} - 6 x^{5} + 8 x^{4} - 4 x^{2} + x - 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
8.5e2_7e4_131e2.12t213.1c1$8$ $ 5^{2} \cdot 7^{4} \cdot 131^{2}$ $x^{9} - 4 x^{8} + 6 x^{7} - 2 x^{6} - 6 x^{5} + 8 x^{4} - 4 x^{2} + x - 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
12.5e7_7e8_131e7.36t2219.1c1$12$ $ 5^{7} \cdot 7^{8} \cdot 131^{7}$ $x^{9} - 4 x^{8} + 6 x^{7} - 2 x^{6} - 6 x^{5} + 8 x^{4} - 4 x^{2} + x - 1$ $S_3\wr S_3$ (as 9T31) $1$ $2$
12.5e5_7e8_131e5.36t2214.1c1$12$ $ 5^{5} \cdot 7^{8} \cdot 131^{5}$ $x^{9} - 4 x^{8} + 6 x^{7} - 2 x^{6} - 6 x^{5} + 8 x^{4} - 4 x^{2} + x - 1$ $S_3\wr S_3$ (as 9T31) $1$ $-2$
12.5e6_7e6_131e6.36t2210.1c1$12$ $ 5^{6} \cdot 7^{6} \cdot 131^{6}$ $x^{9} - 4 x^{8} + 6 x^{7} - 2 x^{6} - 6 x^{5} + 8 x^{4} - 4 x^{2} + x - 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
12.5e7_7e4_131e7.36t2216.1c1$12$ $ 5^{7} \cdot 7^{4} \cdot 131^{7}$ $x^{9} - 4 x^{8} + 6 x^{7} - 2 x^{6} - 6 x^{5} + 8 x^{4} - 4 x^{2} + x - 1$ $S_3\wr S_3$ (as 9T31) $1$ $-2$
12.5e5_7e4_131e5.18t315.1c1$12$ $ 5^{5} \cdot 7^{4} \cdot 131^{5}$ $x^{9} - 4 x^{8} + 6 x^{7} - 2 x^{6} - 6 x^{5} + 8 x^{4} - 4 x^{2} + x - 1$ $S_3\wr S_3$ (as 9T31) $1$ $2$
16.5e8_7e8_131e8.24t2912.1c1$16$ $ 5^{8} \cdot 7^{8} \cdot 131^{8}$ $x^{9} - 4 x^{8} + 6 x^{7} - 2 x^{6} - 6 x^{5} + 8 x^{4} - 4 x^{2} + x - 1$ $S_3\wr S_3$ (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 *.