# Properties

 Label 9.1.2366667072.1 Degree $9$ Signature $[1, 4]$ Discriminant $2366667072$ Root discriminant $11.00$ Ramified primes $2, 3, 7, 11$ Class number $1$ Class group trivial Galois group $S_3\wr S_3$ (as 9T31)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^9 - x^8 - 3*x^6 + x^5 + 3*x^4 + 3*x^3 - 4*x^2 + 1)

gp: K = bnfinit(x^9 - x^8 - 3*x^6 + x^5 + 3*x^4 + 3*x^3 - 4*x^2 + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -4, 3, 3, 1, -3, 0, -1, 1]);

$$x^{9} - x^{8} - 3 x^{6} + x^{5} + 3 x^{4} + 3 x^{3} - 4 x^{2} + 1$$

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: $$2366667072$$$$\medspace = 2^{6}\cdot 3^{4}\cdot 7^{3}\cdot 11^{3}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $11.00$ 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: $2, 3, 7, 11$ 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 $$a$$)

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

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: $$a$$,  $$a^{8} - \frac{2}{3} a^{7} - \frac{10}{3} a^{5} - \frac{1}{3} a^{4} + \frac{7}{3} a^{3} + \frac{14}{3} a^{2} - \frac{5}{3} a - \frac{2}{3}$$,  $$a^{8} - 3 a^{5} - 2 a^{4} + a^{3} + 4 a^{2}$$,  $$\frac{4}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - 4 a^{5} - \frac{7}{3} a^{4} + \frac{10}{3} a^{3} + \frac{17}{3} a^{2} - \frac{1}{3}$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$16.5992891922$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{4}\cdot 16.5992891922 \cdot 1}{2\sqrt{2366667072}}\approx 0.531790262147$

## 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

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type R R ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R R ${\href{/LocalNumberField/13.9.0.1}{9} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.9.0.1}{9} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.9.0.1}{9} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3} 2.6.6.2x^{6} - x^{4} - 5$$2$$3$$6$$A_4\times C_2$$[2, 2]^{6}$
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2} 3.3.0.1x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.4.3.2$x^{4} - 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2} 77.3.0.1x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3} 11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2} 11.6.3.1x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.231.2t1.a.a$1$ $3 \cdot 7 \cdot 11$ $x^{2} - x + 58$ $C_2$ (as 2T1) $1$ $-1$
1.3.2t1.a.a$1$ $3$ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
1.77.2t1.a.a$1$ $7 \cdot 11$ $x^{2} - x - 19$ $C_2$ (as 2T1) $1$ $1$
2.231.6t3.f.a$2$ $3 \cdot 7 \cdot 11$ $x^{6} - 9 x^{3} + 1$ $D_{6}$ (as 6T3) $1$ $0$
* 2.231.3t2.a.a$2$ $3 \cdot 7 \cdot 11$ $x^{3} - x^{2} + 3$ $S_3$ (as 3T2) $1$ $0$
3.133056.4t5.a.a$3$ $2^{6} \cdot 3^{3} \cdot 7 \cdot 11$ $x^{4} - 2 x^{3} + 6 x^{2} + 4 x - 2$ $S_4$ (as 4T5) $1$ $1$
3.10245312.6t11.a.a$3$ $2^{6} \cdot 3^{3} \cdot 7^{2} \cdot 11^{2}$ $x^{6} + 2 x^{4} + 15 x^{2} + 3$ $S_4\times C_2$ (as 6T11) $1$ $1$
3.3415104.6t8.d.a$3$ $2^{6} \cdot 3^{2} \cdot 7^{2} \cdot 11^{2}$ $x^{4} - 2 x^{3} + 6 x^{2} + 4 x - 2$ $S_4$ (as 4T5) $1$ $-1$
3.44352.6t11.a.a$3$ $2^{6} \cdot 3^{2} \cdot 7 \cdot 11$ $x^{6} + 2 x^{4} + 15 x^{2} + 3$ $S_4\times C_2$ (as 6T11) $1$ $-1$
* 6.10245312.9t31.a.a$6$ $2^{6} \cdot 3^{3} \cdot 7^{2} \cdot 11^{2}$ $x^{9} - x^{8} - 3 x^{6} + x^{5} + 3 x^{4} + 3 x^{3} - 4 x^{2} + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
6.60744454848.18t300.a.a$6$ $2^{6} \cdot 3^{3} \cdot 7^{4} \cdot 11^{4}$ $x^{9} - x^{8} - 3 x^{6} + x^{5} + 3 x^{4} + 3 x^{3} - 4 x^{2} + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
6.546700093632.18t319.a.a$6$ $2^{6} \cdot 3^{5} \cdot 7^{4} \cdot 11^{4}$ $x^{9} - x^{8} - 3 x^{6} + x^{5} + 3 x^{4} + 3 x^{3} - 4 x^{2} + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
6.92207808.18t311.a.a$6$ $2^{6} \cdot 3^{5} \cdot 7^{2} \cdot 11^{2}$ $x^{9} - x^{8} - 3 x^{6} + x^{5} + 3 x^{4} + 3 x^{3} - 4 x^{2} + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
8.622...576.24t2893.b.a$8$ $2^{12} \cdot 3^{6} \cdot 7^{6} \cdot 11^{6}$ $x^{9} - x^{8} - 3 x^{6} + x^{5} + 3 x^{4} + 3 x^{3} - 4 x^{2} + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
8.17703899136.12t213.a.a$8$ $2^{12} \cdot 3^{6} \cdot 7^{2} \cdot 11^{2}$ $x^{9} - x^{8} - 3 x^{6} + x^{5} + 3 x^{4} + 3 x^{3} - 4 x^{2} + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
12.828...256.36t2217.a.a$12$ $2^{18} \cdot 3^{9} \cdot 7^{7} \cdot 11^{7}$ $x^{9} - x^{8} - 3 x^{6} + x^{5} + 3 x^{4} + 3 x^{3} - 4 x^{2} + 1$ $S_3\wr S_3$ (as 9T31) $1$ $2$
12.139...264.36t2214.a.a$12$ $2^{18} \cdot 3^{9} \cdot 7^{5} \cdot 11^{5}$ $x^{9} - x^{8} - 3 x^{6} + x^{5} + 3 x^{4} + 3 x^{3} - 4 x^{2} + 1$ $S_3\wr S_3$ (as 9T31) $1$ $-2$
12.206...976.36t2210.a.a$12$ $2^{24} \cdot 3^{10} \cdot 7^{6} \cdot 11^{6}$ $x^{9} - x^{8} - 3 x^{6} + x^{5} + 3 x^{4} + 3 x^{3} - 4 x^{2} + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
12.828...256.36t2216.a.a$12$ $2^{18} \cdot 3^{9} \cdot 7^{7} \cdot 11^{7}$ $x^{9} - x^{8} - 3 x^{6} + x^{5} + 3 x^{4} + 3 x^{3} - 4 x^{2} + 1$ $S_3\wr S_3$ (as 9T31) $1$ $-2$
12.139...264.18t315.a.a$12$ $2^{18} \cdot 3^{9} \cdot 7^{5} \cdot 11^{5}$ $x^{9} - x^{8} - 3 x^{6} + x^{5} + 3 x^{4} + 3 x^{3} - 4 x^{2} + 1$ $S_3\wr S_3$ (as 9T31) $1$ $2$
16.110...336.24t2912.a.a$16$ $2^{24} \cdot 3^{12} \cdot 7^{8} \cdot 11^{8}$ $x^{9} - x^{8} - 3 x^{6} + x^{5} + 3 x^{4} + 3 x^{3} - 4 x^{2} + 1$ $S_3\wr S_3$ (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 *.