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Properties

Label 9.3.2210782784.1
Degree $9$
Signature $[3, 3]$
Discriminant $-\,2^{6}\cdot 7^{2}\cdot 89^{3}$
Root discriminant $10.92$
Ramified primes $2, 7, 89$
Class number $1$
Class group Trivial
Galois group $C_3 \wr S_3 $ (as 9T20)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 1, -5, 0, 12, -9, -3, 7, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 4*x^8 + 7*x^7 - 3*x^6 - 9*x^5 + 12*x^4 - 5*x^2 + x - 1)
 
gp: K = bnfinit(x^9 - 4*x^8 + 7*x^7 - 3*x^6 - 9*x^5 + 12*x^4 - 5*x^2 + x - 1, 1)
 

Normalized defining polynomial

\( x^{9} - 4 x^{8} + 7 x^{7} - 3 x^{6} - 9 x^{5} + 12 x^{4} - 5 x^{2} + x - 1 \)

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

Invariants

Degree:  $9$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-2210782784=-\,2^{6}\cdot 7^{2}\cdot 89^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $10.92$
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:  $2, 7, 89$
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 \(a\))

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

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:  \( \frac{1}{2} a^{6} - 2 a^{5} + 4 a^{4} - 3 a^{3} - \frac{3}{2} a^{2} + 4 a - \frac{3}{2} \),  \( a \),  \( a^{7} - \frac{7}{2} a^{6} + 6 a^{5} - 3 a^{4} - 5 a^{3} + \frac{11}{2} a^{2} + \frac{1}{2} \),  \( \frac{1}{2} a^{8} - \frac{3}{2} a^{7} + 2 a^{6} + a^{5} - \frac{11}{2} a^{4} + \frac{9}{2} a^{3} + \frac{3}{2} a^{2} - \frac{5}{2} a + 2 \),  \( \frac{1}{2} a^{8} - 2 a^{7} + \frac{7}{2} a^{6} - 2 a^{5} - \frac{7}{2} a^{4} + 5 a^{3} - a^{2} - a + \frac{3}{2} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 13.875517777 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\wr S_3$ (as 9T20):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 162
The 22 conjugacy class representatives for $C_3 \wr S_3 $
Character table for $C_3 \wr S_3 $ is not computed

Intermediate fields

3.1.356.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 9 siblings: data not computed
Degree 18 siblings: data not computed
Degree 27 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 ${\href{/LocalNumberField/3.9.0.1}{9} }$ ${\href{/LocalNumberField/5.9.0.1}{9} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.9.0.1}{9} }$ ${\href{/LocalNumberField/19.9.0.1}{9} }$ ${\href{/LocalNumberField/23.9.0.1}{9} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.9.0.1}{9} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.9.0.1}{9} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
$7$7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
89Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.2e2_89.2t1.1c1$1$ $ 2^{2} \cdot 89 $ $x^{2} + 89$ $C_2$ (as 2T1) $1$ $-1$
1.7.3t1.1c1$1$ $ 7 $ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
1.2e2_7_89.6t1.1c1$1$ $ 2^{2} \cdot 7 \cdot 89 $ $x^{6} - 2 x^{5} + 264 x^{4} - 350 x^{3} + 23943 x^{2} - 16024 x + 745109$ $C_6$ (as 6T1) $0$ $-1$
1.2e2_7_89.6t1.1c2$1$ $ 2^{2} \cdot 7 \cdot 89 $ $x^{6} - 2 x^{5} + 264 x^{4} - 350 x^{3} + 23943 x^{2} - 16024 x + 745109$ $C_6$ (as 6T1) $0$ $-1$
1.7.3t1.1c2$1$ $ 7 $ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
* 2.2e2_89.3t2.1c1$2$ $ 2^{2} \cdot 89 $ $x^{3} - x^{2} + x + 7$ $S_3$ (as 3T2) $1$ $0$
2.2e2_7e2_89.6t5.1c1$2$ $ 2^{2} \cdot 7^{2} \cdot 89 $ $x^{6} - 3 x^{4} - 56 x^{3} + 514 x^{2} + 2576 x + 7993$ $S_3\times C_3$ (as 6T5) $0$ $0$
2.2e2_7e2_89.6t5.1c2$2$ $ 2^{2} \cdot 7^{2} \cdot 89 $ $x^{6} - 3 x^{4} - 56 x^{3} + 514 x^{2} + 2576 x + 7993$ $S_3\times C_3$ (as 6T5) $0$ $0$
3.2e2_7e2_89.9t20.2c1$3$ $ 2^{2} \cdot 7^{2} \cdot 89 $ $x^{9} - 4 x^{8} + 7 x^{7} - 3 x^{6} - 9 x^{5} + 12 x^{4} - 5 x^{2} + x - 1$ $C_3 \wr S_3 $ (as 9T20) $0$ $1$
3.2e4_7e2_89e2.18t86.2c1$3$ $ 2^{4} \cdot 7^{2} \cdot 89^{2}$ $x^{9} - 4 x^{8} + 7 x^{7} - 3 x^{6} - 9 x^{5} + 12 x^{4} - 5 x^{2} + x - 1$ $C_3 \wr S_3 $ (as 9T20) $0$ $-1$
3.2e4_7_89e2.18t86.2c1$3$ $ 2^{4} \cdot 7 \cdot 89^{2}$ $x^{9} - 4 x^{8} + 7 x^{7} - 3 x^{6} - 9 x^{5} + 12 x^{4} - 5 x^{2} + x - 1$ $C_3 \wr S_3 $ (as 9T20) $0$ $-1$
3.2e4_7e3_89e2.18t86.2c1$3$ $ 2^{4} \cdot 7^{3} \cdot 89^{2}$ $x^{9} - 4 x^{8} + 7 x^{7} - 3 x^{6} - 9 x^{5} + 12 x^{4} - 5 x^{2} + x - 1$ $C_3 \wr S_3 $ (as 9T20) $0$ $-1$
* 3.2e2_7_89.9t20.2c1$3$ $ 2^{2} \cdot 7 \cdot 89 $ $x^{9} - 4 x^{8} + 7 x^{7} - 3 x^{6} - 9 x^{5} + 12 x^{4} - 5 x^{2} + x - 1$ $C_3 \wr S_3 $ (as 9T20) $0$ $1$
3.2e4_7e2_89e2.18t86.2c2$3$ $ 2^{4} \cdot 7^{2} \cdot 89^{2}$ $x^{9} - 4 x^{8} + 7 x^{7} - 3 x^{6} - 9 x^{5} + 12 x^{4} - 5 x^{2} + x - 1$ $C_3 \wr S_3 $ (as 9T20) $0$ $-1$
* 3.2e2_7_89.9t20.2c2$3$ $ 2^{2} \cdot 7 \cdot 89 $ $x^{9} - 4 x^{8} + 7 x^{7} - 3 x^{6} - 9 x^{5} + 12 x^{4} - 5 x^{2} + x - 1$ $C_3 \wr S_3 $ (as 9T20) $0$ $1$
3.2e2_7e2_89.9t20.2c2$3$ $ 2^{2} \cdot 7^{2} \cdot 89 $ $x^{9} - 4 x^{8} + 7 x^{7} - 3 x^{6} - 9 x^{5} + 12 x^{4} - 5 x^{2} + x - 1$ $C_3 \wr S_3 $ (as 9T20) $0$ $1$
3.2e4_7_89e2.18t86.2c2$3$ $ 2^{4} \cdot 7 \cdot 89^{2}$ $x^{9} - 4 x^{8} + 7 x^{7} - 3 x^{6} - 9 x^{5} + 12 x^{4} - 5 x^{2} + x - 1$ $C_3 \wr S_3 $ (as 9T20) $0$ $-1$
3.2e2_7e3_89.9t20.2c1$3$ $ 2^{2} \cdot 7^{3} \cdot 89 $ $x^{9} - 4 x^{8} + 7 x^{7} - 3 x^{6} - 9 x^{5} + 12 x^{4} - 5 x^{2} + x - 1$ $C_3 \wr S_3 $ (as 9T20) $0$ $1$
3.2e2_7e3_89.9t20.2c2$3$ $ 2^{2} \cdot 7^{3} \cdot 89 $ $x^{9} - 4 x^{8} + 7 x^{7} - 3 x^{6} - 9 x^{5} + 12 x^{4} - 5 x^{2} + x - 1$ $C_3 \wr S_3 $ (as 9T20) $0$ $1$
3.2e4_7e3_89e2.18t86.2c2$3$ $ 2^{4} \cdot 7^{3} \cdot 89^{2}$ $x^{9} - 4 x^{8} + 7 x^{7} - 3 x^{6} - 9 x^{5} + 12 x^{4} - 5 x^{2} + x - 1$ $C_3 \wr S_3 $ (as 9T20) $0$ $-1$
6.2e6_7e4_89e3.9t13.1c1$6$ $ 2^{6} \cdot 7^{4} \cdot 89^{3}$ $x^{9} - 2 x^{8} + 2 x^{7} - 32 x^{6} + 79 x^{5} - 74 x^{4} + 147 x^{3} + 280 x^{2} - 75 x + 500$ $C_3^2 : C_6$ (as 9T11) $1$ $0$

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 *.