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Properties

Label 9.1.86537154816.1
Degree $9$
Signature $[1, 4]$
Discriminant $2^{8}\cdot 3^{4}\cdot 7^{3}\cdot 23^{3}$
Root discriminant $16.42$
Ramified primes $2, 3, 7, 23$
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, -2, -2, 2, 0, 4, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 3*x^8 + 4*x^7 + 2*x^5 - 2*x^4 - 2*x^3 + 4*x^2 + x + 1)
 
gp: K = bnfinit(x^9 - 3*x^8 + 4*x^7 + 2*x^5 - 2*x^4 - 2*x^3 + 4*x^2 + x + 1, 1)
 

Normalized defining polynomial

\( x^{9} - 3 x^{8} + 4 x^{7} + 2 x^{5} - 2 x^{4} - 2 x^{3} + 4 x^{2} + x + 1 \)

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:  \(86537154816=2^{8}\cdot 3^{4}\cdot 7^{3}\cdot 23^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.42$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 7, 23$
magma: PrimeDivisors(Discriminant(Integers(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 \(a\))

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

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:  \( 205.305820981 \)
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

3.1.1932.1

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 ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.9.0.1}{9} }$ R ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$2$2.9.8.1$x^{9} - 2$$9$$1$$8$$(C_9:C_3):C_2$$[\ ]_{9}^{6}$
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.4.3.2$x^{4} - 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$7$7.2.1.1$x^{2} - 7$$2$$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$$[\ ]_{2}^{2}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.6.3.1$x^{6} - 46 x^{4} + 529 x^{2} - 194672$$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$1$ $1$ $x$ $C_1$ $1$ $1$
1.3_7_23.2t1.1c1$1$ $ 3 \cdot 7 \cdot 23 $ $x^{2} - x + 121$ $C_2$ (as 2T1) $1$ $-1$
1.3.2t1.1c1$1$ $ 3 $ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
1.7_23.2t1.1c1$1$ $ 7 \cdot 23 $ $x^{2} - x - 40$ $C_2$ (as 2T1) $1$ $1$
2.2e2_3_7_23.6t3.1c1$2$ $ 2^{2} \cdot 3 \cdot 7 \cdot 23 $ $x^{6} - x^{5} - 7 x^{4} + 37 x^{3} + 4 x^{2} - 204 x - 180$ $D_{6}$ (as 6T3) $1$ $0$
* 2.2e2_3_7_23.3t2.1c1$2$ $ 2^{2} \cdot 3 \cdot 7 \cdot 23 $ $x^{3} - x^{2} + 7 x + 3$ $S_3$ (as 3T2) $1$ $0$
3.2e2_3e3_7_23.4t5.1c1$3$ $ 2^{2} \cdot 3^{3} \cdot 7 \cdot 23 $ $x^{4} - x^{3} + 8 x + 4$ $S_4$ (as 4T5) $1$ $1$
3.2e2_3e2_7_23.6t11.1c1$3$ $ 2^{2} \cdot 3^{2} \cdot 7 \cdot 23 $ $x^{6} - 3 x^{5} + 2 x^{4} + x^{3} + 6 x^{2} - 7 x + 3$ $S_4\times C_2$ (as 6T11) $1$ $-1$
3.2e2_3e2_7e2_23e2.6t8.1c1$3$ $ 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 23^{2}$ $x^{4} - x^{3} + 8 x + 4$ $S_4$ (as 4T5) $1$ $-1$
3.2e2_3e3_7e2_23e2.6t11.1c1$3$ $ 2^{2} \cdot 3^{3} \cdot 7^{2} \cdot 23^{2}$ $x^{6} - 3 x^{5} + 2 x^{4} + x^{3} + 6 x^{2} - 7 x + 3$ $S_4\times C_2$ (as 6T11) $1$ $1$
* 6.2e6_3e3_7e2_23e2.9t31.1c1$6$ $ 2^{6} \cdot 3^{3} \cdot 7^{2} \cdot 23^{2}$ $x^{9} - 3 x^{8} + 4 x^{7} + 2 x^{5} - 2 x^{4} - 2 x^{3} + 4 x^{2} + x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
6.2e6_3e3_7e4_23e4.18t303.1c1$6$ $ 2^{6} \cdot 3^{3} \cdot 7^{4} \cdot 23^{4}$ $x^{9} - 3 x^{8} + 4 x^{7} + 2 x^{5} - 2 x^{4} - 2 x^{3} + 4 x^{2} + x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
6.2e6_3e5_7e4_23e4.18t320.1c1$6$ $ 2^{6} \cdot 3^{5} \cdot 7^{4} \cdot 23^{4}$ $x^{9} - 3 x^{8} + 4 x^{7} + 2 x^{5} - 2 x^{4} - 2 x^{3} + 4 x^{2} + x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
6.2e6_3e5_7e2_23e2.18t312.1c1$6$ $ 2^{6} \cdot 3^{5} \cdot 7^{2} \cdot 23^{2}$ $x^{9} - 3 x^{8} + 4 x^{7} + 2 x^{5} - 2 x^{4} - 2 x^{3} + 4 x^{2} + x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
8.2e8_3e6_7e6_23e6.24t2895.1c1$8$ $ 2^{8} \cdot 3^{6} \cdot 7^{6} \cdot 23^{6}$ $x^{9} - 3 x^{8} + 4 x^{7} + 2 x^{5} - 2 x^{4} - 2 x^{3} + 4 x^{2} + x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
8.2e8_3e6_7e2_23e2.12t213.1c1$8$ $ 2^{8} \cdot 3^{6} \cdot 7^{2} \cdot 23^{2}$ $x^{9} - 3 x^{8} + 4 x^{7} + 2 x^{5} - 2 x^{4} - 2 x^{3} + 4 x^{2} + x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
12.2e10_3e9_7e7_23e7.36t2219.1c1$12$ $ 2^{10} \cdot 3^{9} \cdot 7^{7} \cdot 23^{7}$ $x^{9} - 3 x^{8} + 4 x^{7} + 2 x^{5} - 2 x^{4} - 2 x^{3} + 4 x^{2} + x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $2$
12.2e10_3e9_7e5_23e5.36t2214.1c1$12$ $ 2^{10} \cdot 3^{9} \cdot 7^{5} \cdot 23^{5}$ $x^{9} - 3 x^{8} + 4 x^{7} + 2 x^{5} - 2 x^{4} - 2 x^{3} + 4 x^{2} + x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $-2$
12.2e12_3e10_7e6_23e6.36t2210.1c1$12$ $ 2^{12} \cdot 3^{10} \cdot 7^{6} \cdot 23^{6}$ $x^{9} - 3 x^{8} + 4 x^{7} + 2 x^{5} - 2 x^{4} - 2 x^{3} + 4 x^{2} + x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
12.2e10_3e9_7e7_23e7.36t2216.1c1$12$ $ 2^{10} \cdot 3^{9} \cdot 7^{7} \cdot 23^{7}$ $x^{9} - 3 x^{8} + 4 x^{7} + 2 x^{5} - 2 x^{4} - 2 x^{3} + 4 x^{2} + x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $-2$
12.2e10_3e9_7e5_23e5.18t315.1c1$12$ $ 2^{10} \cdot 3^{9} \cdot 7^{5} \cdot 23^{5}$ $x^{9} - 3 x^{8} + 4 x^{7} + 2 x^{5} - 2 x^{4} - 2 x^{3} + 4 x^{2} + x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $2$
16.2e14_3e12_7e8_23e8.24t2912.1c1$16$ $ 2^{14} \cdot 3^{12} \cdot 7^{8} \cdot 23^{8}$ $x^{9} - 3 x^{8} + 4 x^{7} + 2 x^{5} - 2 x^{4} - 2 x^{3} + 4 x^{2} + x + 1$ $S_3\wr S_3$ (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 *.