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Properties

Label 9.3.834966881766...8419.1
Degree $9$
Signature $[3, 3]$
Discriminant $-\,3^{12}\cdot 7^{6}\cdot 17^{3}\cdot 43^{7}$
Root discriminant $758.90$
Ramified primes $3, 7, 17, 43$
Class number $13608$ (GRH)
Class group $[3, 3, 6, 6, 42]$ (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![-223609972474, -14286781089, 1666695996, 5278035, 1011360, -530964, -15050, -903, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 903*x^7 - 15050*x^6 - 530964*x^5 + 1011360*x^4 + 5278035*x^3 + 1666695996*x^2 - 14286781089*x - 223609972474)
 
gp: K = bnfinit(x^9 - 903*x^7 - 15050*x^6 - 530964*x^5 + 1011360*x^4 + 5278035*x^3 + 1666695996*x^2 - 14286781089*x - 223609972474, 1)
 

Normalized defining polynomial

\( x^{9} - 903 x^{7} - 15050 x^{6} - 530964 x^{5} + 1011360 x^{4} + 5278035 x^{3} + 1666695996 x^{2} - 14286781089 x - 223609972474 \)

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:  \(-83496688176695495276938419=-\,3^{12}\cdot 7^{6}\cdot 17^{3}\cdot 43^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $758.90$
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:  $3, 7, 17, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{7} a^{3}$, $\frac{1}{14} a^{4} - \frac{1}{2} a$, $\frac{1}{14} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4214} a^{6} - \frac{1}{14} a^{3}$, $\frac{1}{8428} a^{7} - \frac{1}{14} a^{3} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{23704343703597020903815682992} a^{8} - \frac{135594675465706211560655}{3386334814799574414830811856} a^{7} + \frac{5237640876861980192891}{1693167407399787207415405928} a^{6} - \frac{12932147682759860157081}{2812570444185693035573764} a^{5} + \frac{2844256498999093064835}{200897888870406645398126} a^{4} - \frac{2201381329328596752189}{703142611046423258893441} a^{3} - \frac{4741582124737288650809487}{11250281776742772142295056} a^{2} + \frac{634806229146523980354813}{1607183110963253163185008} a - \frac{946621281211653594677}{27710053637297468330776}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{6}\times C_{6}\times C_{42}$, which has order $13608$ (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:  \( 6365194.594055798 \) (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

3.3.7338681.1, 3.1.731.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{3}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{3}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ R ${\href{/LocalNumberField/47.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{3}$

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
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
$43$43.3.2.3$x^{3} - 3483$$3$$1$$2$$C_3$$[\ ]_{3}$
43.6.5.6$x^{6} + 2539107$$6$$1$$5$$C_6$$[\ ]_{6}$