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Properties

Label 9.9.4429692664637819049.4
Degree $9$
Signature $[9, 0]$
Discriminant $3^{22}\cdot 109^{4}$
Root discriminant $117.98$
Ramified primes $3, 109$
Class number $1$ (GRH)
Class group Trivial (GRH)
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![6213763, 213858, -1603935, -300513, 57879, 14715, -495, -216, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 216*x^7 - 495*x^6 + 14715*x^5 + 57879*x^4 - 300513*x^3 - 1603935*x^2 + 213858*x + 6213763)
 
gp: K = bnfinit(x^9 - 216*x^7 - 495*x^6 + 14715*x^5 + 57879*x^4 - 300513*x^3 - 1603935*x^2 + 213858*x + 6213763, 1)
 

Normalized defining polynomial

\( x^{9} - 216 x^{7} - 495 x^{6} + 14715 x^{5} + 57879 x^{4} - 300513 x^{3} - 1603935 x^{2} + 213858 x + 6213763 \)

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:  \(4429692664637819049=3^{22}\cdot 109^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $117.98$
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:  $3, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{109} a^{6} + \frac{2}{109} a^{4} + \frac{50}{109} a^{3}$, $\frac{1}{2071} a^{7} + \frac{1}{2071} a^{6} - \frac{325}{2071} a^{5} - \frac{493}{2071} a^{4} + \frac{268}{2071} a^{3} - \frac{6}{19} a^{2} - \frac{2}{19} a + \frac{3}{19}$, $\frac{1}{2178929404943} a^{8} - \frac{611665}{2178929404943} a^{7} - \frac{2522966605}{2178929404943} a^{6} + \frac{291623954761}{2178929404943} a^{5} + \frac{104008254144}{2178929404943} a^{4} + \frac{755848297815}{2178929404943} a^{3} - \frac{8821120489}{19990178027} a^{2} + \frac{2774558506}{19990178027} a - \frac{497300107}{19990178027}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 3015185.10042 \) (assuming GRH)
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

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

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 ${\href{/LocalNumberField/5.9.0.1}{9} }$ ${\href{/LocalNumberField/7.9.0.1}{9} }$ ${\href{/LocalNumberField/11.9.0.1}{9} }$ ${\href{/LocalNumberField/13.9.0.1}{9} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/23.9.0.1}{9} }$ ${\href{/LocalNumberField/29.9.0.1}{9} }$ ${\href{/LocalNumberField/31.9.0.1}{9} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.9.0.1}{9} }$ ${\href{/LocalNumberField/43.9.0.1}{9} }$ ${\href{/LocalNumberField/47.9.0.1}{9} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.9.0.1}{9} }$

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$3.9.22.11$x^{9} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 3$$9$$1$$22$$C_9:C_3$$[2, 3]^{3}$
$109$109.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
109.3.2.2$x^{3} + 654$$3$$1$$2$$C_3$$[\ ]_{3}$
109.3.2.3$x^{3} - 3924$$3$$1$$2$$C_3$$[\ ]_{3}$