Properties

Label 9.1.134573648589.1
Degree $9$
Signature $[1, 4]$
Discriminant $134573648589$
Root discriminant \(17.24\)
Ramified primes see page
Class number $1$
Class group trivial
Galois group $S_3\wr S_3$ (as 9T31)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^9 + 9*x^7 - 5*x^6 + 27*x^5 - 30*x^4 + 31*x^3 - 45*x^2 + 12*x + 9)
 
gp: K = bnfinit(x^9 + 9*x^7 - 5*x^6 + 27*x^5 - 30*x^4 + 31*x^3 - 45*x^2 + 12*x + 9, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 12, -45, 31, -30, 27, -5, 9, 0, 1]);
 

\( x^{9} + 9x^{7} - 5x^{6} + 27x^{5} - 30x^{4} + 31x^{3} - 45x^{2} + 12x + 9 \) Copy content Toggle raw display

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:   \(134573648589\) \(\medspace = 3^{8}\cdot 29^{5}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  \(17.24\)
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:   \(3\), \(29\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$\card{ \Aut(K/\Q) }$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{4}$, $\frac{1}{9}a^{8}+\frac{1}{9}a^{7}+\frac{1}{9}a^{6}-\frac{1}{9}a^{5}-\frac{1}{9}a^{4}-\frac{4}{9}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

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 $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:   $\frac{1}{3}a^{6}+2a^{4}-\frac{1}{3}a^{3}+3a^{2}-a-1$, $\frac{2}{9}a^{8}+\frac{2}{9}a^{7}+\frac{17}{9}a^{6}+\frac{7}{9}a^{5}+\frac{43}{9}a^{4}-\frac{5}{9}a^{3}+\frac{10}{3}a^{2}-\frac{8}{3}a-1$, $\frac{2}{9}a^{8}+\frac{2}{9}a^{7}+\frac{20}{9}a^{6}+\frac{13}{9}a^{5}+\frac{61}{9}a^{4}+\frac{19}{9}a^{3}+\frac{14}{3}a^{2}-\frac{2}{3}a-6$, $\frac{1}{3}a^{8}+\frac{10}{3}a^{6}-a^{5}+11a^{4}-\frac{19}{3}a^{3}+\frac{35}{3}a^{2}-10a-2$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 177.351346168 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{4}\cdot 177.351346168 \cdot 1}{2\sqrt{134573648589}}\approx 0.753483851587$

Galois group

$C_3^3.S_4.C_2$ (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

3.1.87.1

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 ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.3.0.1}{3} }$ R ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ ${\href{/padicField/7.9.0.1}{9} }$ ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ ${\href{/padicField/13.9.0.1}{9} }$ ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ ${\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ R ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$

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
\(3\) Copy content Toggle raw display $\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.6.7.3$x^{6} + 6 x^{3} + 6 x^{2} + 6$$6$$1$$7$$S_3\times C_3$$[3/2]_{2}^{3}$
\(29\) Copy content Toggle raw display $\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.87.2t1.a.a$1$ $ 3 \cdot 29 $ \(\Q(\sqrt{-87}) \) $C_2$ (as 2T1) $1$ $-1$
1.3.2t1.a.a$1$ $ 3 $ \(\Q(\sqrt{-3}) \) $C_2$ (as 2T1) $1$ $-1$
1.29.2t1.a.a$1$ $ 29 $ \(\Q(\sqrt{29}) \) $C_2$ (as 2T1) $1$ $1$
2.87.6t3.a.a$2$ $ 3 \cdot 29 $ 6.2.219501.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.87.3t2.a.a$2$ $ 3 \cdot 29 $ 3.1.87.1 $S_3$ (as 3T2) $1$ $0$
3.73167.4t5.a.a$3$ $ 3 \cdot 29^{3}$ 4.2.73167.1 $S_4$ (as 4T5) $1$ $1$
3.2523.6t11.b.a$3$ $ 3 \cdot 29^{2}$ 6.0.19096587.1 $S_4\times C_2$ (as 6T11) $1$ $1$
3.7569.6t8.a.a$3$ $ 3^{2} \cdot 29^{2}$ 4.2.73167.1 $S_4$ (as 4T5) $1$ $-1$
3.219501.6t11.b.a$3$ $ 3^{2} \cdot 29^{3}$ 6.0.19096587.1 $S_4\times C_2$ (as 6T11) $1$ $-1$
* 6.1546823547.9t31.a.a$6$ $ 3^{7} \cdot 29^{4}$ 9.1.134573648589.1 $S_3\wr S_3$ (as 9T31) $1$ $0$
6.1546823547.18t300.a.a$6$ $ 3^{7} \cdot 29^{4}$ 9.1.134573648589.1 $S_3\wr S_3$ (as 9T31) $1$ $0$
6.1546823547.18t319.a.a$6$ $ 3^{7} \cdot 29^{4}$ 9.1.134573648589.1 $S_3\wr S_3$ (as 9T31) $1$ $0$
6.1546823547.18t311.a.a$6$ $ 3^{7} \cdot 29^{4}$ 9.1.134573648589.1 $S_3\wr S_3$ (as 9T31) $1$ $0$
8.390...081.24t2893.a.a$8$ $ 3^{8} \cdot 29^{6}$ 9.1.134573648589.1 $S_3\wr S_3$ (as 9T31) $1$ $0$
8.390...081.12t213.a.a$8$ $ 3^{8} \cdot 29^{6}$ 9.1.134573648589.1 $S_3\wr S_3$ (as 9T31) $1$ $0$
12.208...183.36t2217.a.a$12$ $ 3^{15} \cdot 29^{9}$ 9.1.134573648589.1 $S_3\wr S_3$ (as 9T31) $1$ $2$
12.187...647.36t2214.a.a$12$ $ 3^{17} \cdot 29^{9}$ 9.1.134573648589.1 $S_3\wr S_3$ (as 9T31) $1$ $-2$
12.201...769.36t2210.a.a$12$ $ 3^{14} \cdot 29^{10}$ 9.1.134573648589.1 $S_3\wr S_3$ (as 9T31) $1$ $0$
12.187...647.36t2216.a.a$12$ $ 3^{17} \cdot 29^{9}$ 9.1.134573648589.1 $S_3\wr S_3$ (as 9T31) $1$ $-2$
12.208...183.18t315.a.a$12$ $ 3^{15} \cdot 29^{9}$ 9.1.134573648589.1 $S_3\wr S_3$ (as 9T31) $1$ $2$
16.152...561.24t2912.a.a$16$ $ 3^{16} \cdot 29^{12}$ 9.1.134573648589.1 $S_3\wr S_3$ (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 *.