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Properties

Label 9.9.3691950281939241.2
Degree $9$
Signature $[9, 0]$
Discriminant $3^{22}\cdot 7^{6}$
Root discriminant $53.67$
Ramified primes $3, 7$
Class number $3$
Class group $[3]$
Galois group $C_9$ (as 9T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5831, 21609, 0, -10290, 0, 1323, 0, -63, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 63*x^7 + 1323*x^5 - 10290*x^3 + 21609*x - 5831)
 
gp: K = bnfinit(x^9 - 63*x^7 + 1323*x^5 - 10290*x^3 + 21609*x - 5831, 1)
 

Normalized defining polynomial

\( x^{9} - 63 x^{7} + 1323 x^{5} - 10290 x^{3} + 21609 x - 5831 \)

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:  \(3691950281939241=3^{22}\cdot 7^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.67$
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:  $3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(189=3^{3}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{189}(64,·)$, $\chi_{189}(1,·)$, $\chi_{189}(184,·)$, $\chi_{189}(25,·)$, $\chi_{189}(151,·)$, $\chi_{189}(88,·)$, $\chi_{189}(121,·)$, $\chi_{189}(58,·)$, $\chi_{189}(127,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{7} a^{3}$, $\frac{1}{7} a^{4}$, $\frac{1}{133} a^{5} - \frac{8}{133} a^{4} + \frac{3}{133} a^{3} - \frac{6}{19} a^{2} - \frac{3}{19} a + \frac{2}{19}$, $\frac{1}{931} a^{6} - \frac{6}{133} a^{4} - \frac{8}{133} a^{3} + \frac{9}{19} a^{2} + \frac{5}{19} a + \frac{5}{19}$, $\frac{1}{931} a^{7} + \frac{1}{133} a^{4} + \frac{5}{133} a^{3} + \frac{7}{19} a^{2} + \frac{6}{19} a - \frac{7}{19}$, $\frac{1}{931} a^{8} - \frac{6}{133} a^{4} + \frac{8}{133} a^{3} - \frac{7}{19} a^{2} - \frac{4}{19} a - \frac{2}{19}$

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:  \( 40597.2195867 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_9$ (as 9T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 9
The 9 conjugacy class representatives for $C_9$
Character table for $C_9$

Intermediate fields

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

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

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 ${\href{/LocalNumberField/5.9.0.1}{9} }$ R ${\href{/LocalNumberField/11.9.0.1}{9} }$ ${\href{/LocalNumberField/13.9.0.1}{9} }$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/23.9.0.1}{9} }$ ${\href{/LocalNumberField/29.9.0.1}{9} }$ ${\href{/LocalNumberField/31.9.0.1}{9} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/41.9.0.1}{9} }$ ${\href{/LocalNumberField/43.9.0.1}{9} }$ ${\href{/LocalNumberField/47.9.0.1}{9} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/59.9.0.1}{9} }$

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$3.9.22.6$x^{9} + 9 x^{8} + 21 x^{6} + 18 x^{5} + 9 x^{3} + 24$$9$$1$$22$$C_9$$[2, 3]$
$7$7.9.6.3$x^{9} - 14 x^{6} + 49 x^{3} - 1372$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.3e3_7.9t1.2c1$1$ $ 3^{3} \cdot 7 $ $x^{9} - 63 x^{7} + 1323 x^{5} - 10290 x^{3} + 21609 x - 5831$ $C_9$ (as 9T1) $0$ $1$
* 1.3e3_7.9t1.2c2$1$ $ 3^{3} \cdot 7 $ $x^{9} - 63 x^{7} + 1323 x^{5} - 10290 x^{3} + 21609 x - 5831$ $C_9$ (as 9T1) $0$ $1$
* 1.3e2.3t1.1c1$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
* 1.3e3_7.9t1.2c3$1$ $ 3^{3} \cdot 7 $ $x^{9} - 63 x^{7} + 1323 x^{5} - 10290 x^{3} + 21609 x - 5831$ $C_9$ (as 9T1) $0$ $1$
* 1.3e3_7.9t1.2c4$1$ $ 3^{3} \cdot 7 $ $x^{9} - 63 x^{7} + 1323 x^{5} - 10290 x^{3} + 21609 x - 5831$ $C_9$ (as 9T1) $0$ $1$
* 1.3e2.3t1.1c2$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
* 1.3e3_7.9t1.2c5$1$ $ 3^{3} \cdot 7 $ $x^{9} - 63 x^{7} + 1323 x^{5} - 10290 x^{3} + 21609 x - 5831$ $C_9$ (as 9T1) $0$ $1$
* 1.3e3_7.9t1.2c6$1$ $ 3^{3} \cdot 7 $ $x^{9} - 63 x^{7} + 1323 x^{5} - 10290 x^{3} + 21609 x - 5831$ $C_9$ (as 9T1) $0$ $1$

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 *.