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Properties

Label 9.9.197309150940332809.1
Degree $9$
Signature $[9, 0]$
Discriminant $7^{6}\cdot 109^{6}$
Root discriminant $83.50$
Ramified primes $7, 109$
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![32089, 92067, -28900, -51103, 1679, 4231, -25, -116, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 116*x^7 - 25*x^6 + 4231*x^5 + 1679*x^4 - 51103*x^3 - 28900*x^2 + 92067*x + 32089)
 
gp: K = bnfinit(x^9 - 116*x^7 - 25*x^6 + 4231*x^5 + 1679*x^4 - 51103*x^3 - 28900*x^2 + 92067*x + 32089, 1)
 

Normalized defining polynomial

\( x^{9} - 116 x^{7} - 25 x^{6} + 4231 x^{5} + 1679 x^{4} - 51103 x^{3} - 28900 x^{2} + 92067 x + 32089 \)

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:  \(197309150940332809=7^{6}\cdot 109^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $83.50$
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:  $7, 109$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(763=7\cdot 109\)
Dirichlet character group:    $\lbrace$$\chi_{763}(1,·)$, $\chi_{763}(263,·)$, $\chi_{763}(172,·)$, $\chi_{763}(590,·)$, $\chi_{763}(655,·)$, $\chi_{763}(499,·)$, $\chi_{763}(372,·)$, $\chi_{763}(281,·)$, $\chi_{763}(219,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{12} a^{7} - \frac{1}{12} a^{6} + \frac{1}{12} a^{5} - \frac{1}{12} a^{4} + \frac{1}{12} a^{3} + \frac{5}{12} a^{2} + \frac{5}{12} a - \frac{1}{6}$, $\frac{1}{3473529379034868} a^{8} - \frac{30932796851843}{3473529379034868} a^{7} - \frac{47748909559417}{868382344758717} a^{6} - \frac{31272956976367}{1736764689517434} a^{5} - \frac{376203024548729}{1736764689517434} a^{4} + \frac{63559646039260}{868382344758717} a^{3} - \frac{156579593509217}{578921563172478} a^{2} + \frac{603443314749839}{3473529379034868} a - \frac{1570095438103621}{3473529379034868}$

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:  \( 147964.419082 \) (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

3.3.11881.1, 3.3.582169.2, 3.3.582169.1, \(\Q(\zeta_{7})^+\)

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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{3}$

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
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$109$109.9.6.1$x^{9} + 3270 x^{6} + 3552419 x^{3} + 1295029000$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$