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Properties

Label 9.9.532962204162...0969.6
Degree $9$
Signature $[9, 0]$
Discriminant $3^{22}\cdot 19^{8}$
Root discriminant $200.89$
Ramified primes $3, 19$
Class number $9$ (GRH)
Class group $[9]$ (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![-2375, 68913, -18468, -26904, 5130, 3591, -342, -171, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 171*x^7 - 342*x^6 + 3591*x^5 + 5130*x^4 - 26904*x^3 - 18468*x^2 + 68913*x - 2375)
 
gp: K = bnfinit(x^9 - 171*x^7 - 342*x^6 + 3591*x^5 + 5130*x^4 - 26904*x^3 - 18468*x^2 + 68913*x - 2375, 1)
 

Normalized defining polynomial

\( x^{9} - 171 x^{7} - 342 x^{6} + 3591 x^{5} + 5130 x^{4} - 26904 x^{3} - 18468 x^{2} + 68913 x - 2375 \)

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:  \(532962204162830310969=3^{22}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $200.89$
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:  $3, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(513=3^{3}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{513}(64,·)$, $\chi_{513}(1,·)$, $\chi_{513}(139,·)$, $\chi_{513}(358,·)$, $\chi_{513}(427,·)$, $\chi_{513}(175,·)$, $\chi_{513}(340,·)$, $\chi_{513}(214,·)$, $\chi_{513}(505,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} - \frac{1}{5} a$, $\frac{1}{5} a^{6} - \frac{1}{5} a^{2}$, $\frac{1}{25} a^{7} + \frac{2}{25} a^{6} + \frac{2}{25} a^{5} - \frac{1}{5} a^{4} - \frac{1}{25} a^{3} + \frac{8}{25} a^{2} - \frac{7}{25} a$, $\frac{1}{743828021875} a^{8} + \frac{6870012161}{743828021875} a^{7} - \frac{94889092}{5950624175} a^{6} - \frac{27424038717}{743828021875} a^{5} - \frac{151862467596}{743828021875} a^{4} + \frac{135638788924}{743828021875} a^{3} - \frac{28414832928}{148765604375} a^{2} + \frac{360081196992}{743828021875} a + \frac{39870279}{5950624175}$

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

Class group and class number

$C_{9}$, which has order $9$ (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:  \( 22548077.9628 \) (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

3.3.29241.2

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 ${\href{/LocalNumberField/5.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/7.9.0.1}{9} }$ ${\href{/LocalNumberField/11.9.0.1}{9} }$ ${\href{/LocalNumberField/13.9.0.1}{9} }$ ${\href{/LocalNumberField/17.9.0.1}{9} }$ R ${\href{/LocalNumberField/23.9.0.1}{9} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/31.9.0.1}{9} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/43.9.0.1}{9} }$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/53.9.0.1}{9} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{3}$

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$3.9.22.9$x^{9} + 21 x^{6} + 18 x^{5} + 9 x^{3} + 6$$9$$1$$22$$C_9$$[2, 3]$
$19$19.9.8.3$x^{9} - 77824$$9$$1$$8$$C_9$$[\ ]_{9}$