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Properties

Label 9.1.464465622145...3136.1
Degree $9$
Signature $[1, 4]$
Discriminant $2^{6}\cdot 3^{3}\cdot 13^{7}\cdot 1231^{7}$
Root discriminant $4262.47$
Ramified primes $2, 3, 13, 1231$
Class number $154548$ (GRH)
Class group $[3, 3, 3, 6, 954]$ (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![4090575971282, -34558675410, -752932776, 778261881, -1014429, -94164, 47771, 72, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 3*x^8 + 72*x^7 + 47771*x^6 - 94164*x^5 - 1014429*x^4 + 778261881*x^3 - 752932776*x^2 - 34558675410*x + 4090575971282)
 
gp: K = bnfinit(x^9 - 3*x^8 + 72*x^7 + 47771*x^6 - 94164*x^5 - 1014429*x^4 + 778261881*x^3 - 752932776*x^2 - 34558675410*x + 4090575971282, 1)
 

Normalized defining polynomial

\( x^{9} - 3 x^{8} + 72 x^{7} + 47771 x^{6} - 94164 x^{5} - 1014429 x^{4} + 778261881 x^{3} - 752932776 x^{2} - 34558675410 x + 4090575971282 \)

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:  \(464465622145778726630024995523136=2^{6}\cdot 3^{3}\cdot 13^{7}\cdot 1231^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $4262.47$
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:  $2, 3, 13, 1231$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{48009} a^{6} + \frac{13888}{48009} a^{5} + \frac{19549}{48009} a^{4} - \frac{18242}{48009} a^{3} + \frac{6335}{48009} a^{2} - \frac{6241}{16003} a + \frac{22075}{48009}$, $\frac{1}{48009} a^{7} - \frac{1614}{16003} a^{5} - \frac{7953}{16003} a^{4} + \frac{7738}{48009} a^{3} + \frac{1294}{48009} a^{2} - \frac{1358}{3693} a + \frac{7874}{48009}$, $\frac{1}{61968075792452732506343415} a^{8} - \frac{93696012077349654659}{20656025264150910835447805} a^{7} - \frac{36193524988840296128}{12393615158490546501268683} a^{6} + \frac{5215251647194678009178951}{61968075792452732506343415} a^{5} - \frac{7816045781722798772390061}{20656025264150910835447805} a^{4} + \frac{8350051842933586345905611}{20656025264150910835447805} a^{3} - \frac{15611006898795579721052101}{61968075792452732506343415} a^{2} - \frac{12509905131001114845896782}{61968075792452732506343415} a + \frac{1390469442069885280499981}{4766775060957902500487955}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{6}\times C_{954}$, which has order $154548$ (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:  \( 387447255.3792636 \) (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

3.1.768288027.1, 3.1.64012.3

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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.3.2.3$x^{3} - 52$$3$$1$$2$$C_3$$[\ ]_{3}$
13.6.5.1$x^{6} - 52$$6$$1$$5$$C_6$$[\ ]_{6}$
1231Data not computed