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Properties

Label 9.1.108811522174...9184.1
Degree $9$
Signature $[1, 4]$
Discriminant $2^{6}\cdot 3^{18}\cdot 7^{7}\cdot 127^{7}$
Root discriminant $2808.79$
Ramified primes $2, 3, 7, 127$
Class number $734832$ (GRH)
Class group $[9, 18, 18, 252]$ (GRH)
Galois group $S_3^2$ (as 9T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-513536239999, -23042879280, -86389200, 192600057, 1440900, 10800, -23997, 180, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 + 180*x^7 - 23997*x^6 + 10800*x^5 + 1440900*x^4 + 192600057*x^3 - 86389200*x^2 - 23042879280*x - 513536239999)
 
gp: K = bnfinit(x^9 + 180*x^7 - 23997*x^6 + 10800*x^5 + 1440900*x^4 + 192600057*x^3 - 86389200*x^2 - 23042879280*x - 513536239999, 1)
 

Normalized defining polynomial

\( x^{9} + 180 x^{7} - 23997 x^{6} + 10800 x^{5} + 1440900 x^{4} + 192600057 x^{3} - 86389200 x^{2} - 23042879280 x - 513536239999 \)

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

Invariants

Degree:  $9$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(10881152217470823144247586309184=2^{6}\cdot 3^{18}\cdot 7^{7}\cdot 127^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2808.79$
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:  $2, 3, 7, 127$
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 \(a\))

$1$, $a$, $\frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{9} a^{3} - \frac{1}{3} a + \frac{2}{9}$, $\frac{1}{27} a^{4} - \frac{1}{27} a^{3} - \frac{1}{9} a^{2} + \frac{5}{27} a - \frac{2}{27}$, $\frac{1}{27} a^{5} - \frac{1}{27} a^{3} + \frac{2}{27} a^{2} - \frac{2}{9} a + \frac{4}{27}$, $\frac{1}{72009} a^{6} + \frac{400}{24003} a^{5} + \frac{20}{24003} a^{4} - \frac{11}{72009} a^{3} + \frac{800}{24003} a^{2} - \frac{80}{24003} a - \frac{8}{72009}$, $\frac{1}{216027} a^{7} - \frac{1}{216027} a^{6} - \frac{320}{72009} a^{5} - \frac{62}{216027} a^{4} - \frac{8392}{216027} a^{3} - \frac{760}{72009} a^{2} + \frac{10315}{30861} a - \frac{38398}{216027}$, $\frac{1}{292662968422411511757} a^{8} + \frac{222358814345191}{292662968422411511757} a^{7} + \frac{127209207266611}{292662968422411511757} a^{6} + \frac{100274889491940682}{41808995488915930251} a^{5} + \frac{603472788777181780}{292662968422411511757} a^{4} - \frac{1800102100981968797}{292662968422411511757} a^{3} - \frac{1090079114099367149}{292662968422411511757} a^{2} + \frac{12778792822064350168}{41808995488915930251} a + \frac{67751231442328171771}{292662968422411511757}$

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

Class group and class number

$C_{9}\times C_{18}\times C_{18}\times C_{252}$, which has order $734832$ (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:  $4$
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:  \( 4185072.3103982336 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3^2$ (as 9T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 9 conjugacy class representatives for $S_3^2$
Character table for $S_3^2$

Intermediate fields

3.1.96012.2, 3.1.192048003.1

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

Sibling fields

Galois closure: data not computed
Degree 6 sibling: data not computed
Degree 12 sibling: data not computed
Degree 18 siblings: 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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$3$3.9.18.2$x^{9} + 3 x^{3} + 9 x^{2} + 9 x + 3$$9$$1$$18$$C_3^2:C_2$$[3/2, 5/2]_{2}$
$7$7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.3$x^{6} - 112$$6$$1$$5$$C_6$$[\ ]_{6}$
$127$127.3.2.1$x^{3} - 127$$3$$1$$2$$C_3$$[\ ]_{3}$
127.6.5.5$x^{6} + 92583$$6$$1$$5$$C_6$$[\ ]_{6}$