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Properties

Label 9.9.140479058634340321.1
Degree $9$
Signature $[9, 0]$
Discriminant $7^{6}\cdot 103^{6}$
Root discriminant $80.41$
Ramified primes $7, 103$
Class number $1$ (GRH)
Class group Trivial (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![-10688, -138256, -151552, -35773, 9892, 3793, -148, -110, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 110*x^7 - 148*x^6 + 3793*x^5 + 9892*x^4 - 35773*x^3 - 151552*x^2 - 138256*x - 10688)
 
gp: K = bnfinit(x^9 - 110*x^7 - 148*x^6 + 3793*x^5 + 9892*x^4 - 35773*x^3 - 151552*x^2 - 138256*x - 10688, 1)
 

Normalized defining polynomial

\( x^{9} - 110 x^{7} - 148 x^{6} + 3793 x^{5} + 9892 x^{4} - 35773 x^{3} - 151552 x^{2} - 138256 x - 10688 \)

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:  \(140479058634340321=7^{6}\cdot 103^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $80.41$
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:  $7, 103$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(721=7\cdot 103\)
Dirichlet character group:    $\lbrace$$\chi_{721}(1,·)$, $\chi_{721}(674,·)$, $\chi_{721}(365,·)$, $\chi_{721}(46,·)$, $\chi_{721}(207,·)$, $\chi_{721}(561,·)$, $\chi_{721}(149,·)$, $\chi_{721}(310,·)$, $\chi_{721}(571,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{6} + \frac{1}{6} a^{5} - \frac{1}{3} a^{3} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{156} a^{7} - \frac{5}{78} a^{6} - \frac{2}{39} a^{5} + \frac{4}{39} a^{4} - \frac{23}{156} a^{3} - \frac{16}{39} a^{2} + \frac{25}{156} a - \frac{16}{39}$, $\frac{1}{511458136176} a^{8} + \frac{68320640}{31966133511} a^{7} + \frac{6477861925}{255729068088} a^{6} - \frac{21245239903}{127864534044} a^{5} + \frac{15947523217}{511458136176} a^{4} + \frac{1432435499}{3278577796} a^{3} + \frac{112456628755}{511458136176} a^{2} - \frac{18046253539}{63932267022} a + \frac{14172806021}{31966133511}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 644577.805246 \) (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

3.3.519841.2, 3.3.519841.1, 3.3.10609.1, \(\Q(\zeta_{7})^+\)

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 ${\href{/LocalNumberField/11.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{3}$

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
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$103$103.9.6.1$x^{9} + 8961 x^{6} + 26755898 x^{3} + 26650518803$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$