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Properties

Label 9.9.428585957696...0721.1
Degree $9$
Signature $[9, 0]$
Discriminant $3^{12}\cdot 73^{8}$
Root discriminant $196.09$
Ramified primes $3, 73$
Class number $9$ (GRH)
Class group $[3, 3]$ (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![810081, 609039, -484866, -148482, 36792, 9855, -730, -219, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 219*x^7 - 730*x^6 + 9855*x^5 + 36792*x^4 - 148482*x^3 - 484866*x^2 + 609039*x + 810081)
 
gp: K = bnfinit(x^9 - 219*x^7 - 730*x^6 + 9855*x^5 + 36792*x^4 - 148482*x^3 - 484866*x^2 + 609039*x + 810081, 1)
 

Normalized defining polynomial

\( x^{9} - 219 x^{7} - 730 x^{6} + 9855 x^{5} + 36792 x^{4} - 148482 x^{3} - 484866 x^{2} + 609039 x + 810081 \)

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:  \(428585957696282300721=3^{12}\cdot 73^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $196.09$
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:  $3, 73$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(657=3^{2}\cdot 73\)
Dirichlet character group:    $\lbrace$$\chi_{657}(256,·)$, $\chi_{657}(1,·)$, $\chi_{657}(4,·)$, $\chi_{657}(64,·)$, $\chi_{657}(493,·)$, $\chi_{657}(367,·)$, $\chi_{657}(16,·)$, $\chi_{657}(616,·)$, $\chi_{657}(154,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} - \frac{1}{3} a^{2}$, $\frac{1}{27} a^{6} + \frac{2}{9} a^{4} + \frac{8}{27} a^{3} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{81} a^{7} + \frac{2}{27} a^{5} + \frac{35}{81} a^{4} + \frac{1}{3} a^{3} + \frac{1}{9} a^{2} + \frac{4}{9} a$, $\frac{1}{413916568623801} a^{8} + \frac{192458090140}{137972189541267} a^{7} + \frac{1995748914698}{137972189541267} a^{6} - \frac{32833978338466}{413916568623801} a^{5} + \frac{32795724327563}{137972189541267} a^{4} + \frac{14660325714680}{45990729847089} a^{3} - \frac{14652602610068}{45990729847089} a^{2} - \frac{4762405218476}{15330243282363} a + \frac{1381028858582}{5110081094121}$

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

Class group and class number

$C_{3}\times C_{3}$, 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:  \( 4758026.42602 \) (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

3.3.5329.1

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 ${\href{/LocalNumberField/5.9.0.1}{9} }$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/11.9.0.1}{9} }$ ${\href{/LocalNumberField/13.9.0.1}{9} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/19.9.0.1}{9} }$ ${\href{/LocalNumberField/23.9.0.1}{9} }$ ${\href{/LocalNumberField/29.9.0.1}{9} }$ ${\href{/LocalNumberField/31.9.0.1}{9} }$ ${\href{/LocalNumberField/37.9.0.1}{9} }$ ${\href{/LocalNumberField/41.9.0.1}{9} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }$ ${\href{/LocalNumberField/53.9.0.1}{9} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }$

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$3.3.4.1$x^{3} - 3 x^{2} + 21$$3$$1$$4$$C_3$$[2]$
3.3.4.1$x^{3} - 3 x^{2} + 21$$3$$1$$4$$C_3$$[2]$
3.3.4.1$x^{3} - 3 x^{2} + 21$$3$$1$$4$$C_3$$[2]$
$73$73.9.8.1$x^{9} - 73$$9$$1$$8$$C_9$$[\ ]_{9}$