Properties

Label 9.1.83319616.1
Degree $9$
Signature $[1, 4]$
Discriminant $2^{6}\cdot 23^{3}\cdot 107$
Root discriminant $7.59$
Ramified primes $2, 23, 107$
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, -1, 8, -14, 17, -15, 9, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 4*x^8 + 9*x^7 - 15*x^6 + 17*x^5 - 14*x^4 + 8*x^3 - x^2 - x + 1)
 
gp: K = bnfinit(x^9 - 4*x^8 + 9*x^7 - 15*x^6 + 17*x^5 - 14*x^4 + 8*x^3 - x^2 - x + 1, 1)
 

Normalized defining polynomial

\( x^{9} - 4 x^{8} + 9 x^{7} - 15 x^{6} + 17 x^{5} - 14 x^{4} + 8 x^{3} - x^{2} - x + 1 \)

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:  \(83319616=2^{6}\cdot 23^{3}\cdot 107\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $7.59$
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:  $2, 23, 107$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

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

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:  $4$
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:  \( a^{8} - 4 a^{7} + 8 a^{6} - 12 a^{5} + 12 a^{4} - 8 a^{3} + 4 a^{2} - 1 \),  \( a^{8} - 3 a^{7} + 5 a^{6} - 7 a^{5} + 6 a^{4} - 4 a^{3} + 3 a^{2} \),  \( a^{8} - 3 a^{7} + 5 a^{6} - 7 a^{5} + 5 a^{4} - 2 a^{3} + 3 a \),  \( a^{8} - 3 a^{7} + 5 a^{6} - 7 a^{5} + 5 a^{4} - 2 a^{3} + 3 a - 1 \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1.34933902572 \)
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

3.1.23.1

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 ${\href{/LocalNumberField/3.9.0.1}{9} }$ ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ R ${\href{/LocalNumberField/29.9.0.1}{9} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.9.0.1}{9} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$$[\ ]_{3}^{6}$
$23$23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
107Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.23.2t1.a.a$1$ $ 23 $ $x^{2} - x + 6$ $C_2$ (as 2T1) $1$ $-1$
1.107.2t1.a.a$1$ $ 107 $ $x^{2} - x + 27$ $C_2$ (as 2T1) $1$ $-1$
1.2461.2t1.a.a$1$ $ 23 \cdot 107 $ $x^{2} - x - 615$ $C_2$ (as 2T1) $1$ $1$
2.263327.6t3.b.a$2$ $ 23 \cdot 107^{2}$ $x^{6} - 2 x^{5} + 55 x^{4} - 295 x^{3} + 970 x^{2} - 6507 x - 433997$ $D_{6}$ (as 6T3) $1$ $0$
* 2.23.3t2.b.a$2$ $ 23 $ $x^{3} - x^{2} + 1$ $S_3$ (as 3T2) $1$ $0$
3.263327.4t5.a.a$3$ $ 23 \cdot 107^{2}$ $x^{4} - x^{3} + 9 x^{2} + 9 x - 26$ $S_4$ (as 4T5) $1$ $1$
3.56603.6t11.a.a$3$ $ 23^{2} \cdot 107 $ $x^{6} - 2 x^{5} + 3 x^{4} + x^{2} + 3 x + 1$ $S_4\times C_2$ (as 6T11) $1$ $1$
3.6056521.6t8.a.a$3$ $ 23^{2} \cdot 107^{2}$ $x^{4} - x^{3} + 9 x^{2} + 9 x - 26$ $S_4$ (as 4T5) $1$ $-1$
3.2461.6t11.a.a$3$ $ 23 \cdot 107 $ $x^{6} - 2 x^{5} + 3 x^{4} + x^{2} + 3 x + 1$ $S_4\times C_2$ (as 6T11) $1$ $-1$
* 6.3622592.9t31.a.a$6$ $ 2^{6} \cdot 23^{2} \cdot 107 $ $x^{9} - 4 x^{8} + 9 x^{7} - 15 x^{6} + 17 x^{5} - 14 x^{4} + 8 x^{3} - x^{2} - x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
6.251194546477323968.18t300.a.a$6$ $ 2^{6} \cdot 23^{4} \cdot 107^{5}$ $x^{9} - 4 x^{8} + 9 x^{7} - 15 x^{6} + 17 x^{5} - 14 x^{4} + 8 x^{3} - x^{2} - x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
6.1916351168.18t319.a.a$6$ $ 2^{6} \cdot 23^{4} \cdot 107 $ $x^{9} - 4 x^{8} + 9 x^{7} - 15 x^{6} + 17 x^{5} - 14 x^{4} + 8 x^{3} - x^{2} - x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
6.474847913945792.18t311.a.a$6$ $ 2^{6} \cdot 23^{2} \cdot 107^{5}$ $x^{9} - 4 x^{8} + 9 x^{7} - 15 x^{6} + 17 x^{5} - 14 x^{4} + 8 x^{3} - x^{2} - x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
8.1241887056883218496.24t2893.a.a$8$ $ 2^{6} \cdot 23^{6} \cdot 107^{4}$ $x^{9} - 4 x^{8} + 9 x^{7} - 15 x^{6} + 17 x^{5} - 14 x^{4} + 8 x^{3} - x^{2} - x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
8.4437830971456.12t213.a.a$8$ $ 2^{6} \cdot 23^{2} \cdot 107^{4}$ $x^{9} - 4 x^{8} + 9 x^{7} - 15 x^{6} + 17 x^{5} - 14 x^{4} + 8 x^{3} - x^{2} - x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
12.3744079377776281433184502208.36t2217.a.a$12$ $ 2^{6} \cdot 23^{7} \cdot 107^{8}$ $x^{9} - 4 x^{8} + 9 x^{7} - 15 x^{6} + 17 x^{5} - 14 x^{4} + 8 x^{3} - x^{2} - x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $2$
12.7077654778405068871804352.36t2214.a.a$12$ $ 2^{6} \cdot 23^{5} \cdot 107^{8}$ $x^{9} - 4 x^{8} + 9 x^{7} - 15 x^{6} + 17 x^{5} - 14 x^{4} + 8 x^{3} - x^{2} - x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $-2$
12.909975354512381987885056.36t2210.a.a$12$ $ 2^{12} \cdot 23^{6} \cdot 107^{6}$ $x^{9} - 4 x^{8} + 9 x^{7} - 15 x^{6} + 17 x^{5} - 14 x^{4} + 8 x^{3} - x^{2} - x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
12.28563402308314025408.36t2216.a.a$12$ $ 2^{6} \cdot 23^{7} \cdot 107^{4}$ $x^{9} - 4 x^{8} + 9 x^{7} - 15 x^{6} + 17 x^{5} - 14 x^{4} + 8 x^{3} - x^{2} - x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $-2$
12.53995089429705152.18t315.a.a$12$ $ 2^{6} \cdot 23^{5} \cdot 107^{4}$ $x^{9} - 4 x^{8} + 9 x^{7} - 15 x^{6} + 17 x^{5} - 14 x^{4} + 8 x^{3} - x^{2} - x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $2$
16.5511284844086686269647587250176.24t2912.a.a$16$ $ 2^{12} \cdot 23^{8} \cdot 107^{8}$ $x^{9} - 4 x^{8} + 9 x^{7} - 15 x^{6} + 17 x^{5} - 14 x^{4} + 8 x^{3} - x^{2} - x + 1$ $S_3\wr S_3$ (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 *.