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Properties

Label 9.9.433512750098961.1
Degree $9$
Signature $[9, 0]$
Discriminant $3^{12}\cdot 13^{8}$
Root discriminant $42.30$
Ramified primes $3, 13$
Class number $1$
Class group Trivial
Galois group $C_9:C_3$ (as 9T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-65, 117, 234, -429, -39, 234, -13, -39, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 39*x^7 - 13*x^6 + 234*x^5 - 39*x^4 - 429*x^3 + 234*x^2 + 117*x - 65)
 
gp: K = bnfinit(x^9 - 39*x^7 - 13*x^6 + 234*x^5 - 39*x^4 - 429*x^3 + 234*x^2 + 117*x - 65, 1)
 

Normalized defining polynomial

\( x^{9} - 39 x^{7} - 13 x^{6} + 234 x^{5} - 39 x^{4} - 429 x^{3} + 234 x^{2} + 117 x - 65 \)

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:  \(433512750098961=3^{12}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.30$
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:  $3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4898321} a^{8} - \frac{937284}{4898321} a^{7} + \frac{2120230}{4898321} a^{6} + \frac{971009}{4898321} a^{5} + \frac{1740799}{4898321} a^{4} - \frac{2121497}{4898321} a^{3} + \frac{1173695}{4898321} a^{2} - \frac{1020682}{4898321} a + \frac{2324900}{4898321}$

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:  $8$
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:  \( 28649.1218433 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_9:C_3$ (as 9T6):

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 $C_9:C_3$
Character table for $C_9:C_3$

Intermediate fields

3.3.169.1

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

Sibling fields

Galois closure: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.9.0.1}{9} }$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/7.9.0.1}{9} }$ ${\href{/LocalNumberField/11.9.0.1}{9} }$ R ${\href{/LocalNumberField/17.9.0.1}{9} }$ ${\href{/LocalNumberField/19.9.0.1}{9} }$ ${\href{/LocalNumberField/23.9.0.1}{9} }$ ${\href{/LocalNumberField/29.9.0.1}{9} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.9.0.1}{9} }$ ${\href{/LocalNumberField/41.9.0.1}{9} }$ ${\href{/LocalNumberField/43.9.0.1}{9} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/59.9.0.1}{9} }$

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$3.9.12.6$x^{9} + 3 x^{8} + 6 x^{6} + 27$$3$$3$$12$$C_9:C_3$$[2, 2]^{3}$
$13$13.9.8.2$x^{9} - 13$$9$$1$$8$$C_9:C_3$$[\ ]_{9}^{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_13.3t1.2c1$1$ $ 3^{2} \cdot 13 $ $x^{3} - 39 x - 26$ $C_3$ (as 3T1) $0$ $1$
1.3e2.3t1.1c1$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
1.3e2_13.3t1.1c1$1$ $ 3^{2} \cdot 13 $ $x^{3} - 39 x - 91$ $C_3$ (as 3T1) $0$ $1$
1.3e2_13.3t1.2c2$1$ $ 3^{2} \cdot 13 $ $x^{3} - 39 x - 26$ $C_3$ (as 3T1) $0$ $1$
* 1.13.3t1.1c1$1$ $ 13 $ $x^{3} - x^{2} - 4 x - 1$ $C_3$ (as 3T1) $0$ $1$
* 1.13.3t1.1c2$1$ $ 13 $ $x^{3} - x^{2} - 4 x - 1$ $C_3$ (as 3T1) $0$ $1$
1.3e2_13.3t1.1c2$1$ $ 3^{2} \cdot 13 $ $x^{3} - 39 x - 91$ $C_3$ (as 3T1) $0$ $1$
1.3e2.3t1.1c2$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
* 3.3e6_13e3.9t6.1c1$3$ $ 3^{6} \cdot 13^{3}$ $x^{9} - 39 x^{7} - 13 x^{6} + 234 x^{5} - 39 x^{4} - 429 x^{3} + 234 x^{2} + 117 x - 65$ $C_9:C_3$ (as 9T6) $0$ $3$
* 3.3e6_13e3.9t6.1c2$3$ $ 3^{6} \cdot 13^{3}$ $x^{9} - 39 x^{7} - 13 x^{6} + 234 x^{5} - 39 x^{4} - 429 x^{3} + 234 x^{2} + 117 x - 65$ $C_9:C_3$ (as 9T6) $0$ $3$

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 *.