# 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)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

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)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -6, -3, 16, 7, 7, -5, 6, -1, 1]);

## Normalizeddefining polynomial

$$x^{9} - x^{8} + 6 x^{7} - 5 x^{6} + 7 x^{5} + 7 x^{4} + 16 x^{3} - 3 x^{2} - 6 x + 2$$

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: $$490197028608=2^{8}\cdot 3^{4}\cdot 7^{3}\cdot 41^{3}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $19.90$ 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: $2, 3, 7, 41$ 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 $$a$$)

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

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: 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 sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$1001.06570617$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## 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

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

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 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.

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$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2] 2.3.0.1x^{3} - x + 1$$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$$[\ ]_{2} 3.6.3.1x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2} 7.3.0.1x^{3} - x + 2$$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$$[\ ]_{2} 41.2.0.1x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2} 41.2.1.1x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.3444.2t1.a.a$1$ $2^{2} \cdot 3 \cdot 7 \cdot 41$ $x^{2} + 861$ $C_2$ (as 2T1) $1$ $-1$
1.3.2t1.a.a$1$ $3$ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
1.1148.2t1.a.a$1$ $2^{2} \cdot 7 \cdot 41$ $x^{2} - 287$ $C_2$ (as 2T1) $1$ $1$
2.3444.6t3.l.a$2$ $2^{2} \cdot 3 \cdot 7 \cdot 41$ $x^{6} - 68 x^{3} + 8$ $D_{6}$ (as 6T3) $1$ $0$
* 2.3444.3t2.a.a$2$ $2^{2} \cdot 3 \cdot 7 \cdot 41$ $x^{3} - x^{2} - 12 x - 24$ $S_3$ (as 3T2) $1$ $0$
3.13776.4t5.a.a$3$ $2^{4} \cdot 3 \cdot 7 \cdot 41$ $x^{4} + x^{2} - 6 x + 3$ $S_4$ (as 4T5) $1$ $1$
3.15814848.6t11.a.a$3$ $2^{6} \cdot 3 \cdot 7^{2} \cdot 41^{2}$ $x^{6} + 7 x^{4} + 4 x^{2} + 12$ $S_4\times C_2$ (as 6T11) $1$ $1$
3.47444544.6t8.a.a$3$ $2^{6} \cdot 3^{2} \cdot 7^{2} \cdot 41^{2}$ $x^{4} + x^{2} - 6 x + 3$ $S_4$ (as 4T5) $1$ $-1$
3.41328.6t11.a.a$3$ $2^{4} \cdot 3^{2} \cdot 7 \cdot 41$ $x^{6} + 7 x^{4} + 4 x^{2} + 12$ $S_4\times C_2$ (as 6T11) $1$ $-1$
* 6.142333632.9t31.a.a$6$ $2^{6} \cdot 3^{3} \cdot 7^{2} \cdot 41^{2}$ $x^{9} - x^{8} + 6 x^{7} - 5 x^{6} + 7 x^{5} + 7 x^{4} + 16 x^{3} - 3 x^{2} - 6 x + 2$ $S_3\wr S_3$ (as 9T31) $1$ $0$
6.187582062947328.18t300.a.a$6$ $2^{10} \cdot 3^{3} \cdot 7^{4} \cdot 41^{4}$ $x^{9} - x^{8} + 6 x^{7} - 5 x^{6} + 7 x^{5} + 7 x^{4} + 16 x^{3} - 3 x^{2} - 6 x + 2$ $S_3\wr S_3$ (as 9T31) $1$ $0$
6.187582062947328.18t319.a.a$6$ $2^{10} \cdot 3^{3} \cdot 7^{4} \cdot 41^{4}$ $x^{9} - x^{8} + 6 x^{7} - 5 x^{6} + 7 x^{5} + 7 x^{4} + 16 x^{3} - 3 x^{2} - 6 x + 2$ $S_3\wr S_3$ (as 9T31) $1$ $0$
6.142333632.18t311.a.a$6$ $2^{6} \cdot 3^{3} \cdot 7^{2} \cdot 41^{2}$ $x^{9} - x^{8} + 6 x^{7} - 5 x^{6} + 7 x^{5} + 7 x^{4} + 16 x^{3} - 3 x^{2} - 6 x + 2$ $S_3\wr S_3$ (as 9T31) $1$ $0$
8.2966581813038424326144.24t2893.a.a$8$ $2^{16} \cdot 3^{4} \cdot 7^{6} \cdot 41^{6}$ $x^{9} - x^{8} + 6 x^{7} - 5 x^{6} + 7 x^{5} + 7 x^{4} + 16 x^{3} - 3 x^{2} - 6 x + 2$ $S_3\wr S_3$ (as 9T31) $1$ $0$
8.1708003584.12t213.a.a$8$ $2^{8} \cdot 3^{4} \cdot 7^{2} \cdot 41^{2}$ $x^{9} - x^{8} + 6 x^{7} - 5 x^{6} + 7 x^{5} + 7 x^{4} + 16 x^{3} - 3 x^{2} - 6 x + 2$ $S_3\wr S_3$ (as 9T31) $1$ $0$
12.40867631056417333516959744.36t2217.a.a$12$ $2^{20} \cdot 3^{5} \cdot 7^{7} \cdot 41^{7}$ $x^{9} - x^{8} + 6 x^{7} - 5 x^{6} + 7 x^{5} + 7 x^{4} + 16 x^{3} - 3 x^{2} - 6 x + 2$ $S_3\wr S_3$ (as 9T31) $1$ $2$
12.279086093909538177024.36t2214.a.a$12$ $2^{16} \cdot 3^{7} \cdot 7^{5} \cdot 41^{5}$ $x^{9} - x^{8} + 6 x^{7} - 5 x^{6} + 7 x^{5} + 7 x^{4} + 16 x^{3} - 3 x^{2} - 6 x + 2$ $S_3\wr S_3$ (as 9T31) $1$ $-2$
12.427187781077533102964736.36t2210.a.a$12$ $2^{20} \cdot 3^{6} \cdot 7^{6} \cdot 41^{6}$ $x^{9} - x^{8} + 6 x^{7} - 5 x^{6} + 7 x^{5} + 7 x^{4} + 16 x^{3} - 3 x^{2} - 6 x + 2$ $S_3\wr S_3$ (as 9T31) $1$ $0$
12.367808679507756001652637696.36t2216.a.a$12$ $2^{20} \cdot 3^{7} \cdot 7^{7} \cdot 41^{7}$ $x^{9} - x^{8} + 6 x^{7} - 5 x^{6} + 7 x^{5} + 7 x^{4} + 16 x^{3} - 3 x^{2} - 6 x + 2$ $S_3\wr S_3$ (as 9T31) $1$ $-2$
12.31009565989948686336.18t315.a.a$12$ $2^{16} \cdot 3^{5} \cdot 7^{5} \cdot 41^{5}$ $x^{9} - x^{8} + 6 x^{7} - 5 x^{6} + 7 x^{5} + 7 x^{4} + 16 x^{3} - 3 x^{2} - 6 x + 2$ $S_3\wr S_3$ (as 9T31) $1$ $2$
16.5066932368898846678766736900096.24t2912.a.a$16$ $2^{24} \cdot 3^{8} \cdot 7^{8} \cdot 41^{8}$ $x^{9} - x^{8} + 6 x^{7} - 5 x^{6} + 7 x^{5} + 7 x^{4} + 16 x^{3} - 3 x^{2} - 6 x + 2$ $S_3\wr S_3$ (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 *.