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Properties

Label 9.9.198371070650...7841.2
Degree $9$
Signature $[9, 0]$
Discriminant $3^{22}\cdot 43^{6}$
Root discriminant $180.00$
Ramified primes $3, 43$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
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![-5644997, 30769209, 0, -2385210, 0, 49923, 0, -387, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 387*x^7 + 49923*x^5 - 2385210*x^3 + 30769209*x - 5644997)
 
gp: K = bnfinit(x^9 - 387*x^7 + 49923*x^5 - 2385210*x^3 + 30769209*x - 5644997, 1)
 

Normalized defining polynomial

\( x^{9} - 387 x^{7} + 49923 x^{5} - 2385210 x^{3} + 30769209 x - 5644997 \)

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:  \(198371070650798987841=3^{22}\cdot 43^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $180.00$
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:  $3, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1161=3^{3}\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{1161}(1,·)$, $\chi_{1161}(388,·)$, $\chi_{1161}(775,·)$, $\chi_{1161}(79,·)$, $\chi_{1161}(49,·)$, $\chi_{1161}(466,·)$, $\chi_{1161}(436,·)$, $\chi_{1161}(853,·)$, $\chi_{1161}(823,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{43} a^{3}$, $\frac{1}{43} a^{4}$, $\frac{1}{13889} a^{5} + \frac{48}{13889} a^{4} + \frac{108}{13889} a^{3} + \frac{131}{323} a^{2} - \frac{108}{323} a - \frac{71}{323}$, $\frac{1}{597227} a^{6} - \frac{6}{13889} a^{4} + \frac{48}{13889} a^{3} + \frac{9}{323} a^{2} - \frac{144}{323} a - \frac{86}{323}$, $\frac{1}{597227} a^{7} + \frac{13}{13889} a^{4} + \frac{66}{13889} a^{3} - \frac{4}{323} a^{2} - \frac{88}{323} a - \frac{103}{323}$, $\frac{1}{597227} a^{8} + \frac{88}{13889} a^{4} + \frac{39}{13889} a^{3} + \frac{147}{323} a^{2} + \frac{9}{323} a - \frac{46}{323}$

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

Class group and class number

$C_{3}$, which has order $3$ (assuming GRH)

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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 11739683.2351 \) (assuming GRH)
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 ${\href{/LocalNumberField/5.9.0.1}{9} }$ ${\href{/LocalNumberField/7.9.0.1}{9} }$ ${\href{/LocalNumberField/11.9.0.1}{9} }$ ${\href{/LocalNumberField/13.9.0.1}{9} }$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/23.9.0.1}{9} }$ ${\href{/LocalNumberField/29.9.0.1}{9} }$ ${\href{/LocalNumberField/31.9.0.1}{9} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/41.9.0.1}{9} }$ R ${\href{/LocalNumberField/47.9.0.1}{9} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/59.9.0.1}{9} }$

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$3.9.22.2$x^{9} + 9 x^{7} + 3 x^{6} + 18 x^{5} + 51$$9$$1$$22$$C_9$$[2, 3]$
$43$43.9.6.3$x^{9} - 86 x^{6} + 1849 x^{3} - 7950700$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$