Properties

Label 9.1.490197028608.1
Degree $9$
Signature $[1, 4]$
Discriminant $2^{8}\cdot 3^{4}\cdot 7^{3}\cdot 41^{3}$
Root discriminant $19.90$
Ramified primes $2, 3, 7, 41$
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![2, -6, -3, 16, 7, 7, -5, 6, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - x^8 + 6*x^7 - 5*x^6 + 7*x^5 + 7*x^4 + 16*x^3 - 3*x^2 - 6*x + 2)
gp: K = bnfinit(x^9 - x^8 + 6*x^7 - 5*x^6 + 7*x^5 + 7*x^4 + 16*x^3 - 3*x^2 - 6*x + 2, 1)

Normalized defining polynomial

\(x^{9} \) \(\mathstrut -\mathstrut x^{8} \) \(\mathstrut +\mathstrut 6 x^{7} \) \(\mathstrut -\mathstrut 5 x^{6} \) \(\mathstrut +\mathstrut 7 x^{5} \) \(\mathstrut +\mathstrut 7 x^{4} \) \(\mathstrut +\mathstrut 16 x^{3} \) \(\mathstrut -\mathstrut 3 x^{2} \) \(\mathstrut -\mathstrut 6 x \) \(\mathstrut +\mathstrut 2 \)

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:  \(490197028608=2^{8}\cdot 3^{4}\cdot 7^{3}\cdot 41^{3}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $19.90$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 3, 7, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{38} a^{7} - \frac{7}{38} a^{6} - \frac{15}{38} a^{5} - \frac{3}{19} a^{4} + \frac{5}{38} a^{2} - \frac{7}{19} a - \frac{3}{19}$, $\frac{1}{76} a^{8} - \frac{13}{38} a^{6} - \frac{35}{76} a^{5} - \frac{1}{19} a^{4} - \frac{33}{76} a^{3} - \frac{17}{76} a^{2} - \frac{7}{19} a + \frac{17}{38}$

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:  \( 1001.06570617 \)
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

3.1.3444.1

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 ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.9.0.1}{9} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.9.0.1}{9} }$ R ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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}$
2.4.6.9$x^{4} + 2 x^{3} + 6$$4$$1$$6$$D_{4}$$[2, 2]^{2}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.2.1.2$x^{2} + 14$$2$$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$$[\ ]_{2}^{2}$
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{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_41.2t1.1c1$1$ $ 2^{2} \cdot 3 \cdot 7 \cdot 41 $ $x^{2} + 861$ $C_2$ (as 2T1) $1$ $-1$
1.3.2t1.1c1$1$ $ 3 $ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
1.2e2_7_41.2t1.1c1$1$ $ 2^{2} \cdot 7 \cdot 41 $ $x^{2} - 287$ $C_2$ (as 2T1) $1$ $1$
2.2e2_3_7_41.6t3.5c1$2$ $ 2^{2} \cdot 3 \cdot 7 \cdot 41 $ $x^{6} - 68 x^{3} + 8$ $D_{6}$ (as 6T3) $1$ $0$
* 2.2e2_3_7_41.3t2.1c1$2$ $ 2^{2} \cdot 3 \cdot 7 \cdot 41 $ $x^{3} - x^{2} - 12 x - 24$ $S_3$ (as 3T2) $1$ $0$
3.2e4_3_7_41.4t5.1c1$3$ $ 2^{4} \cdot 3 \cdot 7 \cdot 41 $ $x^{4} + x^{2} - 6 x + 3$ $S_4$ (as 4T5) $1$ $1$
3.2e6_3_7e2_41e2.6t11.1c1$3$ $ 2^{6} \cdot 3 \cdot 7^{2} \cdot 41^{2}$ $x^{6} + 7 x^{4} + 4 x^{2} + 12$ $S_4\times C_2$ (as 6T11) $1$ $1$
3.2e6_3e2_7e2_41e2.6t8.2c1$3$ $ 2^{6} \cdot 3^{2} \cdot 7^{2} \cdot 41^{2}$ $x^{4} + x^{2} - 6 x + 3$ $S_4$ (as 4T5) $1$ $-1$
3.2e4_3e2_7_41.6t11.1c1$3$ $ 2^{4} \cdot 3^{2} \cdot 7 \cdot 41 $ $x^{6} + 7 x^{4} + 4 x^{2} + 12$ $S_4\times C_2$ (as 6T11) $1$ $-1$
* 6.2e6_3e3_7e2_41e2.9t31.1c1$6$ $ 2^{6} \cdot 3^{3} \cdot 7^{2} \cdot 41^{2}$ $x^{9} - x^{8} + 6 x^{7} - 5 x^{6} + 7 x^{5} + 7 x^{4} + 16 x^{3} - 3 x^{2} - 6 x + 2$ $S_3\wr S_3$ (as 9T31) $1$ $0$
6.2e10_3e3_7e4_41e4.18t303.1c1$6$ $ 2^{10} \cdot 3^{3} \cdot 7^{4} \cdot 41^{4}$ $x^{9} - x^{8} + 6 x^{7} - 5 x^{6} + 7 x^{5} + 7 x^{4} + 16 x^{3} - 3 x^{2} - 6 x + 2$ $S_3\wr S_3$ (as 9T31) $1$ $0$
6.2e10_3e3_7e4_41e4.18t320.1c1$6$ $ 2^{10} \cdot 3^{3} \cdot 7^{4} \cdot 41^{4}$ $x^{9} - x^{8} + 6 x^{7} - 5 x^{6} + 7 x^{5} + 7 x^{4} + 16 x^{3} - 3 x^{2} - 6 x + 2$ $S_3\wr S_3$ (as 9T31) $1$ $0$
6.2e6_3e3_7e2_41e2.18t312.1c1$6$ $ 2^{6} \cdot 3^{3} \cdot 7^{2} \cdot 41^{2}$ $x^{9} - x^{8} + 6 x^{7} - 5 x^{6} + 7 x^{5} + 7 x^{4} + 16 x^{3} - 3 x^{2} - 6 x + 2$ $S_3\wr S_3$ (as 9T31) $1$ $0$
8.2e16_3e4_7e6_41e6.24t2895.1c1$8$ $ 2^{16} \cdot 3^{4} \cdot 7^{6} \cdot 41^{6}$ $x^{9} - x^{8} + 6 x^{7} - 5 x^{6} + 7 x^{5} + 7 x^{4} + 16 x^{3} - 3 x^{2} - 6 x + 2$ $S_3\wr S_3$ (as 9T31) $1$ $0$
8.2e8_3e4_7e2_41e2.12t213.1c1$8$ $ 2^{8} \cdot 3^{4} \cdot 7^{2} \cdot 41^{2}$ $x^{9} - x^{8} + 6 x^{7} - 5 x^{6} + 7 x^{5} + 7 x^{4} + 16 x^{3} - 3 x^{2} - 6 x + 2$ $S_3\wr S_3$ (as 9T31) $1$ $0$
12.2e20_3e5_7e7_41e7.36t2219.1c1$12$ $ 2^{20} \cdot 3^{5} \cdot 7^{7} \cdot 41^{7}$ $x^{9} - x^{8} + 6 x^{7} - 5 x^{6} + 7 x^{5} + 7 x^{4} + 16 x^{3} - 3 x^{2} - 6 x + 2$ $S_3\wr S_3$ (as 9T31) $1$ $2$
12.2e16_3e7_7e5_41e5.36t2214.1c1$12$ $ 2^{16} \cdot 3^{7} \cdot 7^{5} \cdot 41^{5}$ $x^{9} - x^{8} + 6 x^{7} - 5 x^{6} + 7 x^{5} + 7 x^{4} + 16 x^{3} - 3 x^{2} - 6 x + 2$ $S_3\wr S_3$ (as 9T31) $1$ $-2$
12.2e20_3e6_7e6_41e6.36t2210.1c1$12$ $ 2^{20} \cdot 3^{6} \cdot 7^{6} \cdot 41^{6}$ $x^{9} - x^{8} + 6 x^{7} - 5 x^{6} + 7 x^{5} + 7 x^{4} + 16 x^{3} - 3 x^{2} - 6 x + 2$ $S_3\wr S_3$ (as 9T31) $1$ $0$
12.2e20_3e7_7e7_41e7.36t2216.1c1$12$ $ 2^{20} \cdot 3^{7} \cdot 7^{7} \cdot 41^{7}$ $x^{9} - x^{8} + 6 x^{7} - 5 x^{6} + 7 x^{5} + 7 x^{4} + 16 x^{3} - 3 x^{2} - 6 x + 2$ $S_3\wr S_3$ (as 9T31) $1$ $-2$
12.2e16_3e5_7e5_41e5.18t315.1c1$12$ $ 2^{16} \cdot 3^{5} \cdot 7^{5} \cdot 41^{5}$ $x^{9} - x^{8} + 6 x^{7} - 5 x^{6} + 7 x^{5} + 7 x^{4} + 16 x^{3} - 3 x^{2} - 6 x + 2$ $S_3\wr S_3$ (as 9T31) $1$ $2$
16.2e24_3e8_7e8_41e8.24t2912.1c1$16$ $ 2^{24} \cdot 3^{8} \cdot 7^{8} \cdot 41^{8}$ $x^{9} - x^{8} + 6 x^{7} - 5 x^{6} + 7 x^{5} + 7 x^{4} + 16 x^{3} - 3 x^{2} - 6 x + 2$ $S_3\wr S_3$ (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 *.