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Properties

Label 9.3.67121414144.1
Degree $9$
Signature $[3, 3]$
Discriminant $-\,2^{15}\cdot 127^{3}$
Root discriminant $15.96$
Ramified primes $2, 127$
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![-4, -8, -2, 5, 12, -7, -6, 9, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 4*x^8 + 9*x^7 - 6*x^6 - 7*x^5 + 12*x^4 + 5*x^3 - 2*x^2 - 8*x - 4)
 
gp: K = bnfinit(x^9 - 4*x^8 + 9*x^7 - 6*x^6 - 7*x^5 + 12*x^4 + 5*x^3 - 2*x^2 - 8*x - 4, 1)
 

Normalized defining polynomial

\( x^{9} - 4 x^{8} + 9 x^{7} - 6 x^{6} - 7 x^{5} + 12 x^{4} + 5 x^{3} - 2 x^{2} - 8 x - 4 \)

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

Invariants

Degree:  $9$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-67121414144=-\,2^{15}\cdot 127^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.96$
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:  $2, 127$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{346} a^{8} + \frac{49}{346} a^{7} + \frac{11}{346} a^{6} + \frac{29}{173} a^{5} - \frac{47}{346} a^{4} - \frac{57}{346} a^{3} - \frac{75}{346} a^{2} + \frac{1}{173} a + \frac{49}{173}$

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:  $5$
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:  \( 404.737164807 \)
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

3.3.1016.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ ${\href{/LocalNumberField/5.9.0.1}{9} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.9.0.1}{9} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.9.0.1}{9} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.9.0.1}{9} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.4.10.7$x^{4} - 2 x^{2} + 3$$4$$1$$10$$D_{4}$$[2, 3, 7/2]$
$127$$\Q_{127}$$x + 9$$1$$1$$0$Trivial$[\ ]$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.2e3_127.2t1.2c1$1$ $ 2^{3} \cdot 127 $ $x^{2} + 254$ $C_2$ (as 2T1) $1$ $-1$
1.2e3_127.2t1.1c1$1$ $ 2^{3} \cdot 127 $ $x^{2} - 254$ $C_2$ (as 2T1) $1$ $1$
1.2e2.2t1.1c1$1$ $ 2^{2}$ $x^{2} + 1$ $C_2$ (as 2T1) $1$ $-1$
2.2e5_127.6t3.2c1$2$ $ 2^{5} \cdot 127 $ $x^{6} + 14 x^{4} - 40 x^{3} + 49 x^{2} - 280 x + 654$ $D_{6}$ (as 6T3) $1$ $-2$
* 2.2e3_127.3t2.1c1$2$ $ 2^{3} \cdot 127 $ $x^{3} - x^{2} - 6 x + 2$ $S_3$ (as 3T2) $1$ $2$
3.2e10_127e2.6t8.1c1$3$ $ 2^{10} \cdot 127^{2}$ $x^{4} - 8 x^{2} - 4 x + 1$ $S_4$ (as 4T5) $1$ $3$
3.2e9_127.4t5.1c1$3$ $ 2^{9} \cdot 127 $ $x^{4} - 8 x^{2} - 4 x + 1$ $S_4$ (as 4T5) $1$ $3$
3.2e10_127e2.6t11.1c1$3$ $ 2^{10} \cdot 127^{2}$ $x^{6} + 9 x^{4} + 13 x^{2} + 1$ $S_4\times C_2$ (as 6T11) $1$ $-3$
3.2e7_127.6t11.1c1$3$ $ 2^{7} \cdot 127 $ $x^{6} + 9 x^{4} + 13 x^{2} + 1$ $S_4\times C_2$ (as 6T11) $1$ $-3$
6.2e16_127e4.18t320.1c1$6$ $ 2^{16} \cdot 127^{4}$ $x^{9} - 4 x^{8} + 9 x^{7} - 6 x^{6} - 7 x^{5} + 12 x^{4} + 5 x^{3} - 2 x^{2} - 8 x - 4$ $S_3\wr S_3$ (as 9T31) $1$ $0$
6.2e16_127e2.18t312.1c1$6$ $ 2^{16} \cdot 127^{2}$ $x^{9} - 4 x^{8} + 9 x^{7} - 6 x^{6} - 7 x^{5} + 12 x^{4} + 5 x^{3} - 2 x^{2} - 8 x - 4$ $S_3\wr S_3$ (as 9T31) $1$ $0$
* 6.2e12_127e2.9t31.1c1$6$ $ 2^{12} \cdot 127^{2}$ $x^{9} - 4 x^{8} + 9 x^{7} - 6 x^{6} - 7 x^{5} + 12 x^{4} + 5 x^{3} - 2 x^{2} - 8 x - 4$ $S_3\wr S_3$ (as 9T31) $1$ $0$
6.2e18_127e4.18t303.1c1$6$ $ 2^{18} \cdot 127^{4}$ $x^{9} - 4 x^{8} + 9 x^{7} - 6 x^{6} - 7 x^{5} + 12 x^{4} + 5 x^{3} - 2 x^{2} - 8 x - 4$ $S_3\wr S_3$ (as 9T31) $1$ $0$
8.2e26_127e6.24t2895.1c1$8$ $ 2^{26} \cdot 127^{6}$ $x^{9} - 4 x^{8} + 9 x^{7} - 6 x^{6} - 7 x^{5} + 12 x^{4} + 5 x^{3} - 2 x^{2} - 8 x - 4$ $S_3\wr S_3$ (as 9T31) $1$ $0$
8.2e20_127e2.12t213.1c1$8$ $ 2^{20} \cdot 127^{2}$ $x^{9} - 4 x^{8} + 9 x^{7} - 6 x^{6} - 7 x^{5} + 12 x^{4} + 5 x^{3} - 2 x^{2} - 8 x - 4$ $S_3\wr S_3$ (as 9T31) $1$ $0$
12.2e37_127e7.36t2219.1c1$12$ $ 2^{37} \cdot 127^{7}$ $x^{9} - 4 x^{8} + 9 x^{7} - 6 x^{6} - 7 x^{5} + 12 x^{4} + 5 x^{3} - 2 x^{2} - 8 x - 4$ $S_3\wr S_3$ (as 9T31) $1$ $0$
12.2e36_127e6.36t2210.2c1$12$ $ 2^{36} \cdot 127^{6}$ $x^{9} - 4 x^{8} + 9 x^{7} - 6 x^{6} - 7 x^{5} + 12 x^{4} + 5 x^{3} - 2 x^{2} - 8 x - 4$ $S_3\wr S_3$ (as 9T31) $1$ $0$
12.2e33_127e5.36t2214.1c1$12$ $ 2^{33} \cdot 127^{5}$ $x^{9} - 4 x^{8} + 9 x^{7} - 6 x^{6} - 7 x^{5} + 12 x^{4} + 5 x^{3} - 2 x^{2} - 8 x - 4$ $S_3\wr S_3$ (as 9T31) $1$ $0$
12.2e35_127e7.36t2216.1c1$12$ $ 2^{35} \cdot 127^{7}$ $x^{9} - 4 x^{8} + 9 x^{7} - 6 x^{6} - 7 x^{5} + 12 x^{4} + 5 x^{3} - 2 x^{2} - 8 x - 4$ $S_3\wr S_3$ (as 9T31) $1$ $0$
12.2e31_127e5.18t315.1c1$12$ $ 2^{31} \cdot 127^{5}$ $x^{9} - 4 x^{8} + 9 x^{7} - 6 x^{6} - 7 x^{5} + 12 x^{4} + 5 x^{3} - 2 x^{2} - 8 x - 4$ $S_3\wr S_3$ (as 9T31) $1$ $0$
16.2e46_127e8.24t2912.1c1$16$ $ 2^{46} \cdot 127^{8}$ $x^{9} - 4 x^{8} + 9 x^{7} - 6 x^{6} - 7 x^{5} + 12 x^{4} + 5 x^{3} - 2 x^{2} - 8 x - 4$ $S_3\wr S_3$ (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 *.