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Properties

Label 9.9.129186640449535761.1
Degree $9$
Signature $[9, 0]$
Discriminant $3^{12}\cdot 79^{6}$
Root discriminant $79.66$
Ramified primes $3, 79$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_3^2$ (as 9T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-9197, -53817, -64641, -21345, 3303, 2445, 50, -84, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 3*x^8 - 84*x^7 + 50*x^6 + 2445*x^5 + 3303*x^4 - 21345*x^3 - 64641*x^2 - 53817*x - 9197)
 
gp: K = bnfinit(x^9 - 3*x^8 - 84*x^7 + 50*x^6 + 2445*x^5 + 3303*x^4 - 21345*x^3 - 64641*x^2 - 53817*x - 9197, 1)
 

Normalized defining polynomial

\( x^{9} - 3 x^{8} - 84 x^{7} + 50 x^{6} + 2445 x^{5} + 3303 x^{4} - 21345 x^{3} - 64641 x^{2} - 53817 x - 9197 \)

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

Invariants

Degree:  $9$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[9, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(129186640449535761=3^{12}\cdot 79^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.66$
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:  $3, 79$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(711=3^{2}\cdot 79\)
Dirichlet character group:    $\lbrace$$\chi_{711}(1,·)$, $\chi_{711}(418,·)$, $\chi_{711}(292,·)$, $\chi_{711}(238,·)$, $\chi_{711}(655,·)$, $\chi_{711}(529,·)$, $\chi_{711}(181,·)$, $\chi_{711}(55,·)$, $\chi_{711}(475,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{14} a^{6} - \frac{1}{7} a^{5} + \frac{3}{14} a^{4} + \frac{5}{14} a^{3} - \frac{1}{2} a^{2} - \frac{3}{7} a - \frac{1}{7}$, $\frac{1}{1190} a^{7} - \frac{29}{1190} a^{6} - \frac{69}{1190} a^{5} + \frac{31}{238} a^{4} + \frac{223}{595} a^{3} + \frac{291}{595} a^{2} + \frac{279}{1190} a + \frac{11}{70}$, $\frac{1}{32356100} a^{8} - \frac{2893}{16178050} a^{7} + \frac{38151}{8089025} a^{6} - \frac{1101958}{8089025} a^{5} - \frac{7471399}{32356100} a^{4} - \frac{157049}{462230} a^{3} - \frac{2173981}{6471220} a^{2} + \frac{1266281}{8089025} a + \frac{550923}{1903300}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $8$
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:  \( 131790.629862 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2$ (as 9T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 9
The 9 conjugacy class representatives for $C_3^2$
Character table for $C_3^2$

Intermediate fields

\(\Q(\zeta_{9})^+\), 3.3.6241.1, 3.3.505521.2, 3.3.505521.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.3.0.1}{3} }^{3}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{3}$

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$$[2]^{3}$
$79$79.9.6.1$x^{9} + 948 x^{6} + 293327 x^{3} + 31554496$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$