Properties

Label 9.1.58773123072.1
Degree $9$
Signature $[1, 4]$
Discriminant $58773123072$
Root discriminant \(15.72\)
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 - 3*x^8 + 6*x^7 - 12*x^6 + 12*x^5 - 6*x^4 + 12*x^3 + 6*x^2 - 3*x - 17)
 
gp: K = bnfinit(x^9 - 3*x^8 + 6*x^7 - 12*x^6 + 12*x^5 - 6*x^4 + 12*x^3 + 6*x^2 - 3*x - 17, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-17, -3, 6, 12, -6, 12, -12, 6, -3, 1]);
 

\( x^{9} - 3x^{8} + 6x^{7} - 12x^{6} + 12x^{5} - 6x^{4} + 12x^{3} + 6x^{2} - 3x - 17 \) 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:   \(58773123072\) \(\medspace = 2^{12}\cdot 3^{15}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  \(15.72\)
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:   \(2\), \(3\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{3442}a^{8}-\frac{183}{3442}a^{7}+\frac{247}{3442}a^{6}-\frac{1447}{3442}a^{5}+\frac{601}{3442}a^{4}+\frac{237}{3442}a^{3}+\frac{377}{3442}a^{2}-\frac{735}{3442}a-\frac{110}{1721}$ 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:   $\frac{167}{1721}a^{8}-\frac{887}{3442}a^{7}+\frac{1611}{3442}a^{6}-\frac{3139}{3442}a^{5}+\frac{2819}{3442}a^{4}-\frac{1729}{3442}a^{3}+\frac{3727}{3442}a^{2}+\frac{613}{3442}a-\frac{2919}{3442}$, $\frac{173}{3442}a^{8}-\frac{681}{3442}a^{7}+\frac{1427}{3442}a^{6}-\frac{2507}{3442}a^{5}+\frac{4155}{3442}a^{4}-\frac{3745}{3442}a^{3}+\frac{3265}{3442}a^{2}-\frac{3243}{3442}a-\frac{99}{1721}$, $\frac{549}{3442}a^{8}-\frac{649}{3442}a^{7}+\frac{1365}{3442}a^{6}-\frac{2743}{3442}a^{5}-\frac{483}{3442}a^{4}-\frac{683}{3442}a^{3}+\frac{7337}{3442}a^{2}+\frac{9525}{3442}a+\frac{8450}{1721}$, $\frac{57}{3442}a^{8}-\frac{105}{3442}a^{7}+\frac{311}{3442}a^{6}+\frac{129}{3442}a^{5}-\frac{163}{3442}a^{4}-\frac{259}{3442}a^{3}-\frac{2605}{3442}a^{2}-\frac{4033}{3442}a-\frac{1107}{1721}$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 224.186932991 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{4}\cdot 224.186932991 \cdot 1}{2\sqrt{58773123072}}\approx 1.44125353219$

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

3.1.108.1

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 ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }$ ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.9.0.1}{9} }$ ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ ${\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ ${\href{/padicField/37.9.0.1}{9} }$ ${\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }$

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
\(2\) Copy content Toggle raw display 2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.10.5$x^{6} + 2 x^{5} + 6$$6$$1$$10$$S_4\times C_2$$[2, 8/3, 8/3]_{3}^{2}$
\(3\) Copy content Toggle raw display 3.9.15.20$x^{9} + 3 x^{8} + 3 x^{7} + 6 x^{6} + 3 x^{3} + 6$$9$$1$$15$$C_3^2 : D_{6} $$[3/2, 3/2, 2]_{2}^{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.3.2t1.a.a$1$ $ 3 $ \(\Q(\sqrt{-3}) \) $C_2$ (as 2T1) $1$ $-1$
1.4.2t1.a.a$1$ $ 2^{2}$ \(\Q(\sqrt{-1}) \) $C_2$ (as 2T1) $1$ $-1$
1.12.2t1.a.a$1$ $ 2^{2} \cdot 3 $ \(\Q(\sqrt{3}) \) $C_2$ (as 2T1) $1$ $1$
2.432.6t3.a.a$2$ $ 2^{4} \cdot 3^{3}$ 6.2.559872.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.108.3t2.b.a$2$ $ 2^{2} \cdot 3^{3}$ 3.1.108.1 $S_3$ (as 3T2) $1$ $0$
3.6912.4t5.a.a$3$ $ 2^{8} \cdot 3^{3}$ 4.2.6912.2 $S_4$ (as 4T5) $1$ $1$
3.6912.6t11.b.a$3$ $ 2^{8} \cdot 3^{3}$ 6.0.746496.2 $S_4\times C_2$ (as 6T11) $1$ $-1$
3.20736.6t8.a.a$3$ $ 2^{8} \cdot 3^{4}$ 4.2.6912.2 $S_4$ (as 4T5) $1$ $-1$
3.20736.6t11.b.a$3$ $ 2^{8} \cdot 3^{4}$ 6.0.746496.2 $S_4\times C_2$ (as 6T11) $1$ $1$
* 6.544195584.9t31.a.a$6$ $ 2^{10} \cdot 3^{12}$ 9.1.58773123072.1 $S_3\wr S_3$ (as 9T31) $1$ $0$
6.8707129344.18t300.a.a$6$ $ 2^{14} \cdot 3^{12}$ 9.1.58773123072.1 $S_3\wr S_3$ (as 9T31) $1$ $0$
6.544195584.18t319.a.a$6$ $ 2^{10} \cdot 3^{12}$ 9.1.58773123072.1 $S_3\wr S_3$ (as 9T31) $1$ $0$
6.8707129344.18t311.a.a$6$ $ 2^{14} \cdot 3^{12}$ 9.1.58773123072.1 $S_3\wr S_3$ (as 9T31) $1$ $0$
8.125...536.24t2893.a.a$8$ $ 2^{18} \cdot 3^{14}$ 9.1.58773123072.1 $S_3\wr S_3$ (as 9T31) $1$ $0$
8.139314069504.12t213.a.a$8$ $ 2^{18} \cdot 3^{12}$ 9.1.58773123072.1 $S_3\wr S_3$ (as 9T31) $1$ $0$
12.112...272.36t2217.a.a$12$ $ 2^{30} \cdot 3^{21}$ 9.1.58773123072.1 $S_3\wr S_3$ (as 9T31) $1$ $2$
12.112...272.36t2214.a.a$12$ $ 2^{30} \cdot 3^{21}$ 9.1.58773123072.1 $S_3\wr S_3$ (as 9T31) $1$ $-2$
12.121...376.36t2210.a.a$12$ $ 2^{32} \cdot 3^{24}$ 9.1.58773123072.1 $S_3\wr S_3$ (as 9T31) $1$ $0$
12.701...392.36t2216.a.a$12$ $ 2^{26} \cdot 3^{21}$ 9.1.58773123072.1 $S_3\wr S_3$ (as 9T31) $1$ $-2$
12.701...392.18t315.a.a$12$ $ 2^{26} \cdot 3^{21}$ 9.1.58773123072.1 $S_3\wr S_3$ (as 9T31) $1$ $2$
16.565...656.24t2912.a.a$16$ $ 2^{38} \cdot 3^{30}$ 9.1.58773123072.1 $S_3\wr S_3$ (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 *.