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Properties

Label 9.3.528604308547...1408.2
Degree $9$
Signature $[3, 3]$
Discriminant $-\,2^{6}\cdot 3^{13}\cdot 13^{6}\cdot 181^{4}$
Root discriminant $432.42$
Ramified primes $2, 3, 13, 181$
Class number $1701$ (GRH)
Class group $[3, 9, 63]$ (GRH)
Galois group $(C_3^2:C_3):C_2$ (as 9T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-13890664, -10614564, -5307282, -830247, -58644, -14661, -786, -81, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 81*x^7 - 786*x^6 - 14661*x^5 - 58644*x^4 - 830247*x^3 - 5307282*x^2 - 10614564*x - 13890664)
 
gp: K = bnfinit(x^9 - 81*x^7 - 786*x^6 - 14661*x^5 - 58644*x^4 - 830247*x^3 - 5307282*x^2 - 10614564*x - 13890664, 1)
 

Normalized defining polynomial

\( x^{9} - 81 x^{7} - 786 x^{6} - 14661 x^{5} - 58644 x^{4} - 830247 x^{3} - 5307282 x^{2} - 10614564 x - 13890664 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $9$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-528604308547604359881408=-\,2^{6}\cdot 3^{13}\cdot 13^{6}\cdot 181^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $432.42$
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:  $2, 3, 13, 181$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{3} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{12} a^{4} - \frac{1}{4} a^{2} + \frac{1}{6} a$, $\frac{1}{36} a^{5} + \frac{1}{36} a^{4} + \frac{1}{36} a^{3} + \frac{5}{36} a^{2} - \frac{1}{9} a - \frac{1}{9}$, $\frac{1}{1016496} a^{6} - \frac{23}{1872} a^{5} - \frac{12697}{338832} a^{4} - \frac{60335}{1016496} a^{3} + \frac{205}{936} a^{2} - \frac{17}{468} a - \frac{25}{702}$, $\frac{1}{4065984} a^{7} - \frac{1}{2032992} a^{6} - \frac{2095}{677664} a^{5} - \frac{9749}{1016496} a^{4} - \frac{13451}{4065984} a^{3} - \frac{419}{3744} a^{2} - \frac{2669}{5616} a + \frac{371}{2808}$, $\frac{1}{422862336} a^{8} - \frac{1}{140954112} a^{7} - \frac{1}{5873088} a^{6} + \frac{2807}{1168128} a^{5} - \frac{5641}{778752} a^{4} - \frac{4830533}{140954112} a^{3} + \frac{5651}{89856} a^{2} + \frac{41945}{194688} a - \frac{5873}{97344}$

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

Class group and class number

$C_{3}\times C_{9}\times C_{63}$, which has order $1701$ (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:  $5$
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:  \( 2645850.04215605 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2:S_3$ (as 9T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 10 conjugacy class representatives for $(C_3^2:C_3):C_2$
Character table for $(C_3^2:C_3):C_2$

Intermediate fields

3.1.2028.1

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

Sibling fields

Degree 9 siblings: data not computed
Degree 18 siblings: data not computed
Degree 27 sibling: 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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$2$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
$3$3.3.4.1$x^{3} - 3 x^{2} + 21$$3$$1$$4$$C_3$$[2]$
3.6.9.9$x^{6} + 6 x^{4} + 21$$6$$1$$9$$C_6$$[2]_{2}$
$13$13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$
$181$$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
181.3.2.1$x^{3} - 181$$3$$1$$2$$C_3$$[\ ]_{3}$
181.3.2.1$x^{3} - 181$$3$$1$$2$$C_3$$[\ ]_{3}$