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Properties

Label 9.3.345646515675...1568.1
Degree $9$
Signature $[3, 3]$
Discriminant $-\,2^{9}\cdot 7^{6}\cdot 13^{3}\cdot 31^{7}\cdot 37^{7}$
Root discriminant $4124.81$
Ramified primes $2, 7, 13, 31, 37$
Class number $2337336$ (GRH)
Class group $[3, 6, 6, 21642]$ (GRH)
Galois group $S_3\times C_3$ (as 9T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-24496392693493, -6510682901835, -613128224240, -16652946295, 463533973, 21555397, -86135, -8054, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 8054*x^7 - 86135*x^6 + 21555397*x^5 + 463533973*x^4 - 16652946295*x^3 - 613128224240*x^2 - 6510682901835*x - 24496392693493)
 
gp: K = bnfinit(x^9 - 8054*x^7 - 86135*x^6 + 21555397*x^5 + 463533973*x^4 - 16652946295*x^3 - 613128224240*x^2 - 6510682901835*x - 24496392693493, 1)
 

Normalized defining polynomial

\( x^{9} - 8054 x^{7} - 86135 x^{6} + 21555397 x^{5} + 463533973 x^{4} - 16652946295 x^{3} - 613128224240 x^{2} - 6510682901835 x - 24496392693493 \)

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:  \(-345646515675090906115747805101568=-\,2^{9}\cdot 7^{6}\cdot 13^{3}\cdot 31^{7}\cdot 37^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $4124.81$
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:  $2, 7, 13, 31, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{66526} a^{6} - \frac{6443}{66526} a^{5} - \frac{5361}{66526} a^{4} - \frac{16621}{66526} a^{3} + \frac{24813}{66526} a^{2} + \frac{4869}{66526} a - \frac{10463}{66526}$, $\frac{1}{4590294} a^{7} - \frac{1}{199578} a^{6} - \frac{1387295}{4590294} a^{5} - \frac{7537}{199578} a^{4} + \frac{2154529}{4590294} a^{3} - \frac{73417}{199578} a^{2} + \frac{580031}{4590294} a + \frac{9086}{99789}$, $\frac{1}{110020383565438216947319933398} a^{8} + \frac{136627695641928029882}{2391747468813874281463476813} a^{7} + \frac{8915673451471247248652}{1896903164921348568057240231} a^{6} + \frac{365499362248308430905896711}{2391747468813874281463476813} a^{5} - \frac{23427799851332024246076552776}{55010191782719108473659966699} a^{4} + \frac{867598365262659380768199983}{2391747468813874281463476813} a^{3} - \frac{3991479012979176646560376079}{18336730594239702824553322233} a^{2} - \frac{822245830711405752012600961}{4783494937627748562926953626} a + \frac{28661847183771366864895177}{103989020383211925281020731}$

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

Class group and class number

$C_{3}\times C_{6}\times C_{6}\times C_{21642}$, which has order $2337336$ (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:  \( 3103842.3227068773 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 9T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3\times C_3$
Character table for $S_3\times C_3$

Intermediate fields

3.3.64464841.3, 3.1.119288.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ R R ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$13$13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.6.3.2$x^{6} - 338 x^{2} + 13182$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$31$31.3.2.3$x^{3} - 1519$$3$$1$$2$$C_3$$[\ ]_{3}$
31.6.5.6$x^{6} + 521017$$6$$1$$5$$C_6$$[\ ]_{6}$
$37$37.3.2.3$x^{3} - 148$$3$$1$$2$$C_3$$[\ ]_{3}$
37.6.5.6$x^{6} + 1184$$6$$1$$5$$C_6$$[\ ]_{6}$