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Properties

Label 9.3.2344156490304.2
Degree $9$
Signature $[3, 3]$
Discriminant $-\,2^{6}\cdot 3^{12}\cdot 41^{3}$
Root discriminant $23.68$
Ramified primes $2, 3, 41$
Class number $3$
Class group $[3]$
Galois group $S_3\times C_3$ (as 9T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![111, 72, -255, 341, -249, 132, -48, 9, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 3*x^8 + 9*x^7 - 48*x^6 + 132*x^5 - 249*x^4 + 341*x^3 - 255*x^2 + 72*x + 111)
 
gp: K = bnfinit(x^9 - 3*x^8 + 9*x^7 - 48*x^6 + 132*x^5 - 249*x^4 + 341*x^3 - 255*x^2 + 72*x + 111, 1)
 

Normalized defining polynomial

\( x^{9} - 3 x^{8} + 9 x^{7} - 48 x^{6} + 132 x^{5} - 249 x^{4} + 341 x^{3} - 255 x^{2} + 72 x + 111 \)

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:  \(-2344156490304=-\,2^{6}\cdot 3^{12}\cdot 41^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.68$
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:  $2, 3, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{42} a^{7} + \frac{1}{6} a^{6} - \frac{3}{14} a^{5} - \frac{19}{42} a^{4} + \frac{10}{21} a^{3} + \frac{1}{7} a^{2} - \frac{5}{14} a - \frac{3}{14}$, $\frac{1}{175518} a^{8} + \frac{737}{175518} a^{7} + \frac{18835}{175518} a^{6} - \frac{15829}{175518} a^{5} + \frac{23189}{87759} a^{4} + \frac{2851}{87759} a^{3} + \frac{821}{19502} a^{2} + \frac{8849}{58506} a - \frac{2194}{29253}$

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

Class group and class number

$C_{3}$, which has order $3$

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:  \( 455.796503383 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 9T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3\times C_3$
Character table for $S_3\times C_3$

Intermediate fields

\(\Q(\zeta_{9})^+\), 3.1.13284.1

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

Sibling fields

Galois closure: data not computed
Degree 6 sibling: 6.0.357286464.1

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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{9}$ R ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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}$
$3$3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
$41$41.3.0.1$x^{3} - x + 13$$1$$3$$0$$C_3$$[\ ]^{3}$
41.6.3.1$x^{6} - 82 x^{4} + 1681 x^{2} - 11647649$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$

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_41.2t1.1c1$1$ $ 2^{2} \cdot 41 $ $x^{2} + 41$ $C_2$ (as 2T1) $1$ $-1$
* 1.3e2.3t1.1c1$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
1.2e2_3e2_41.6t1.1c1$1$ $ 2^{2} \cdot 3^{2} \cdot 41 $ $x^{6} + 117 x^{4} - 2 x^{3} + 5052 x^{2} + 252 x + 79377$ $C_6$ (as 6T1) $0$ $-1$
1.2e2_3e2_41.6t1.1c2$1$ $ 2^{2} \cdot 3^{2} \cdot 41 $ $x^{6} + 117 x^{4} - 2 x^{3} + 5052 x^{2} + 252 x + 79377$ $C_6$ (as 6T1) $0$ $-1$
* 1.3e2.3t1.1c2$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
* 2.2e2_3e4_41.3t2.1c1$2$ $ 2^{2} \cdot 3^{4} \cdot 41 $ $x^{3} + 6 x - 44$ $S_3$ (as 3T2) $1$ $0$
* 2.2e2_3e2_41.6t5.2c1$2$ $ 2^{2} \cdot 3^{2} \cdot 41 $ $x^{9} - 3 x^{8} + 9 x^{7} - 48 x^{6} + 132 x^{5} - 249 x^{4} + 341 x^{3} - 255 x^{2} + 72 x + 111$ $S_3\times C_3$ (as 9T4) $0$ $0$
* 2.2e2_3e2_41.6t5.2c2$2$ $ 2^{2} \cdot 3^{2} \cdot 41 $ $x^{9} - 3 x^{8} + 9 x^{7} - 48 x^{6} + 132 x^{5} - 249 x^{4} + 341 x^{3} - 255 x^{2} + 72 x + 111$ $S_3\times C_3$ (as 9T4) $0$ $0$

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 *.