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Properties

Label 9.9.53038958912.1
Degree $9$
Signature $[9, 0]$
Discriminant $2^{6}\cdot 37^{3}\cdot 16361$
Root discriminant $15.55$
Ramified primes $2, 37, 16361$
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, 4, 5, -17, -9, 22, 6, -9, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - x^8 - 9*x^7 + 6*x^6 + 22*x^5 - 9*x^4 - 17*x^3 + 5*x^2 + 4*x - 1)
 
gp: K = bnfinit(x^9 - x^8 - 9*x^7 + 6*x^6 + 22*x^5 - 9*x^4 - 17*x^3 + 5*x^2 + 4*x - 1, 1)
 

Normalized defining polynomial

\( x^{9} - x^{8} - 9 x^{7} + 6 x^{6} + 22 x^{5} - 9 x^{4} - 17 x^{3} + 5 x^{2} + 4 x - 1 \)

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

Invariants

Degree:  $9$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[9, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(53038958912=2^{6}\cdot 37^{3}\cdot 16361\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.55$
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:  $2, 37, 16361$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$

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:  $8$
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:  \( 177.201889286 \)
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

3.3.148.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ ${\href{/LocalNumberField/7.9.0.1}{9} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ R ${\href{/LocalNumberField/41.9.0.1}{9} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
37Data not computed
16361Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.37_16361.2t1.1c1$1$ $ 37 \cdot 16361 $ $x^{2} - x - 151339$ $C_2$ (as 2T1) $1$ $1$
1.37.2t1.1c1$1$ $ 37 $ $x^{2} - x - 9$ $C_2$ (as 2T1) $1$ $1$
1.16361.2t1.1c1$1$ $ 16361 $ $x^{2} - x - 4090$ $C_2$ (as 2T1) $1$ $1$
2.2e2_37_16361e2.6t3.1c1$2$ $ 2^{2} \cdot 37 \cdot 16361^{2}$ $x^{6} - 3 x^{5} - 61352 x^{4} + 114529 x^{3} + 786298410 x^{2} - 1154386145 x - 2530911292529$ $D_{6}$ (as 6T3) $1$ $2$
* 2.2e2_37.3t2.1c1$2$ $ 2^{2} \cdot 37 $ $x^{3} - x^{2} - 3 x + 1$ $S_3$ (as 3T2) $1$ $2$
3.2e2_37e2_16361e2.6t8.1c1$3$ $ 2^{2} \cdot 37^{2} \cdot 16361^{2}$ $x^{4} - x^{3} - 346 x^{2} - 1872 x + 3130$ $S_4$ (as 4T5) $1$ $3$
3.2e2_37_16361e2.4t5.1c1$3$ $ 2^{2} \cdot 37 \cdot 16361^{2}$ $x^{4} - x^{3} - 346 x^{2} - 1872 x + 3130$ $S_4$ (as 4T5) $1$ $3$
3.2e2_37e2_16361.6t11.1c1$3$ $ 2^{2} \cdot 37^{2} \cdot 16361 $ $x^{6} - 3 x^{5} - 23 x^{4} + 51 x^{3} + 134 x^{2} - 160 x - 214$ $S_4\times C_2$ (as 6T11) $1$ $3$
3.2e2_37_16361.6t11.1c1$3$ $ 2^{2} \cdot 37 \cdot 16361 $ $x^{6} - 3 x^{5} - 23 x^{4} + 51 x^{3} + 134 x^{2} - 160 x - 214$ $S_4\times C_2$ (as 6T11) $1$ $3$
6.2e4_37e4_16361.18t319.1c1$6$ $ 2^{4} \cdot 37^{4} \cdot 16361 $ $x^{9} - x^{8} - 9 x^{7} + 6 x^{6} + 22 x^{5} - 9 x^{4} - 17 x^{3} + 5 x^{2} + 4 x - 1$ $S_3\wr S_3$ (as 9T31) $1$ $6$
6.2e4_37e2_16361e5.18t312.1c1$6$ $ 2^{4} \cdot 37^{2} \cdot 16361^{5}$ $x^{9} - x^{8} - 9 x^{7} + 6 x^{6} + 22 x^{5} - 9 x^{4} - 17 x^{3} + 5 x^{2} + 4 x - 1$ $S_3\wr S_3$ (as 9T31) $1$ $6$
* 6.2e4_37e2_16361.9t31.1c1$6$ $ 2^{4} \cdot 37^{2} \cdot 16361 $ $x^{9} - x^{8} - 9 x^{7} + 6 x^{6} + 22 x^{5} - 9 x^{4} - 17 x^{3} + 5 x^{2} + 4 x - 1$ $S_3\wr S_3$ (as 9T31) $1$ $6$
6.2e4_37e4_16361e5.18t303.1c1$6$ $ 2^{4} \cdot 37^{4} \cdot 16361^{5}$ $x^{9} - x^{8} - 9 x^{7} + 6 x^{6} + 22 x^{5} - 9 x^{4} - 17 x^{3} + 5 x^{2} + 4 x - 1$ $S_3\wr S_3$ (as 9T31) $1$ $6$
8.2e4_37e6_16361e4.24t2895.1c1$8$ $ 2^{4} \cdot 37^{6} \cdot 16361^{4}$ $x^{9} - x^{8} - 9 x^{7} + 6 x^{6} + 22 x^{5} - 9 x^{4} - 17 x^{3} + 5 x^{2} + 4 x - 1$ $S_3\wr S_3$ (as 9T31) $1$ $8$
8.2e4_37e2_16361e4.12t213.1c1$8$ $ 2^{4} \cdot 37^{2} \cdot 16361^{4}$ $x^{9} - x^{8} - 9 x^{7} + 6 x^{6} + 22 x^{5} - 9 x^{4} - 17 x^{3} + 5 x^{2} + 4 x - 1$ $S_3\wr S_3$ (as 9T31) $1$ $8$
12.2e8_37e5_16361e8.36t2214.1c1$12$ $ 2^{8} \cdot 37^{5} \cdot 16361^{8}$ $x^{9} - x^{8} - 9 x^{7} + 6 x^{6} + 22 x^{5} - 9 x^{4} - 17 x^{3} + 5 x^{2} + 4 x - 1$ $S_3\wr S_3$ (as 9T31) $1$ $12$
12.2e8_37e7_16361e8.36t2219.1c1$12$ $ 2^{8} \cdot 37^{7} \cdot 16361^{8}$ $x^{9} - x^{8} - 9 x^{7} + 6 x^{6} + 22 x^{5} - 9 x^{4} - 17 x^{3} + 5 x^{2} + 4 x - 1$ $S_3\wr S_3$ (as 9T31) $1$ $12$
12.2e8_37e6_16361e6.36t2211.1c1$12$ $ 2^{8} \cdot 37^{6} \cdot 16361^{6}$ $x^{9} - x^{8} - 9 x^{7} + 6 x^{6} + 22 x^{5} - 9 x^{4} - 17 x^{3} + 5 x^{2} + 4 x - 1$ $S_3\wr S_3$ (as 9T31) $1$ $12$
12.2e8_37e5_16361e4.18t315.1c1$12$ $ 2^{8} \cdot 37^{5} \cdot 16361^{4}$ $x^{9} - x^{8} - 9 x^{7} + 6 x^{6} + 22 x^{5} - 9 x^{4} - 17 x^{3} + 5 x^{2} + 4 x - 1$ $S_3\wr S_3$ (as 9T31) $1$ $12$
12.2e8_37e7_16361e4.36t2216.1c1$12$ $ 2^{8} \cdot 37^{7} \cdot 16361^{4}$ $x^{9} - x^{8} - 9 x^{7} + 6 x^{6} + 22 x^{5} - 9 x^{4} - 17 x^{3} + 5 x^{2} + 4 x - 1$ $S_3\wr S_3$ (as 9T31) $1$ $12$
16.2e12_37e8_16361e8.24t2912.1c1$16$ $ 2^{12} \cdot 37^{8} \cdot 16361^{8}$ $x^{9} - x^{8} - 9 x^{7} + 6 x^{6} + 22 x^{5} - 9 x^{4} - 17 x^{3} + 5 x^{2} + 4 x - 1$ $S_3\wr S_3$ (as 9T31) $1$ $16$

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 *.