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Properties

Label 9.9.110446108890554889.2
Degree $9$
Signature $[9, 0]$
Discriminant $3^{16}\cdot 37^{6}$
Root discriminant $78.29$
Ramified primes $3, 37$
Class number $3$
Class group $[3]$
Galois group $C_3^2:C_3$ (as 9T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9989, 15756, 465, -6717, -1029, 978, 126, -57, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 3*x^8 - 57*x^7 + 126*x^6 + 978*x^5 - 1029*x^4 - 6717*x^3 + 465*x^2 + 15756*x + 9989)
 
gp: K = bnfinit(x^9 - 3*x^8 - 57*x^7 + 126*x^6 + 978*x^5 - 1029*x^4 - 6717*x^3 + 465*x^2 + 15756*x + 9989, 1)
 

Normalized defining polynomial

\( x^{9} - 3 x^{8} - 57 x^{7} + 126 x^{6} + 978 x^{5} - 1029 x^{4} - 6717 x^{3} + 465 x^{2} + 15756 x + 9989 \)

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

Invariants

Degree:  $9$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[9, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(110446108890554889=3^{16}\cdot 37^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $78.29$
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:  $3, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{697911596} a^{8} - \frac{53839967}{697911596} a^{7} - \frac{80232463}{697911596} a^{6} + \frac{88217095}{697911596} a^{5} + \frac{21830491}{348955798} a^{4} - \frac{8979094}{174477899} a^{3} - \frac{70038465}{697911596} a^{2} - \frac{262341975}{697911596} a + \frac{777051}{348955798}$

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:  $8$
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:  \( 225037.842924 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$He_3$ (as 9T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 27
The 11 conjugacy class representatives for $C_3^2:C_3$
Character table for $C_3^2:C_3$

Intermediate fields

3.3.110889.1

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 9 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{3}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{3}$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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
$3$3.9.16.3$x^{9} + 3 x^{8} + 3 x^{6} + 12$$9$$1$$16$$C_3^2:C_3$$[2, 2]^{3}$
$37$37.9.6.1$x^{9} + 222 x^{6} + 15059 x^{3} + 405224$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.3e2_37.3t1.2c1$1$ $ 3^{2} \cdot 37 $ $x^{3} - 111 x - 37$ $C_3$ (as 3T1) $0$ $1$
1.37.3t1.1c1$1$ $ 37 $ $x^{3} - x^{2} - 12 x - 11$ $C_3$ (as 3T1) $0$ $1$
1.3e2.3t1.1c1$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
1.3e2.3t1.1c2$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
* 1.3e2_37.3t1.2c2$1$ $ 3^{2} \cdot 37 $ $x^{3} - 111 x - 37$ $C_3$ (as 3T1) $0$ $1$
1.3e2_37.3t1.1c1$1$ $ 3^{2} \cdot 37 $ $x^{3} - 111 x - 370$ $C_3$ (as 3T1) $0$ $1$
1.37.3t1.1c2$1$ $ 37 $ $x^{3} - x^{2} - 12 x - 11$ $C_3$ (as 3T1) $0$ $1$
1.3e2_37.3t1.1c2$1$ $ 3^{2} \cdot 37 $ $x^{3} - 111 x - 370$ $C_3$ (as 3T1) $0$ $1$
* 3.3e6_37e2.9t7.4c1$3$ $ 3^{6} \cdot 37^{2}$ $x^{9} - 3 x^{8} - 57 x^{7} + 126 x^{6} + 978 x^{5} - 1029 x^{4} - 6717 x^{3} + 465 x^{2} + 15756 x + 9989$ $C_3^2:C_3$ (as 9T7) $0$ $3$
* 3.3e6_37e2.9t7.4c2$3$ $ 3^{6} \cdot 37^{2}$ $x^{9} - 3 x^{8} - 57 x^{7} + 126 x^{6} + 978 x^{5} - 1029 x^{4} - 6717 x^{3} + 465 x^{2} + 15756 x + 9989$ $C_3^2:C_3$ (as 9T7) $0$ $3$

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 *.