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Properties

Label 9.9.2459374191553118401.1
Degree $9$
Signature $[9, 0]$
Discriminant $199^{8}$
Root discriminant $110.52$
Ramified prime $199$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_9$ (as 9T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-121, -3333, 7354, -1602, -3447, 775, 325, -88, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - x^8 - 88*x^7 + 325*x^6 + 775*x^5 - 3447*x^4 - 1602*x^3 + 7354*x^2 - 3333*x - 121)
 
gp: K = bnfinit(x^9 - x^8 - 88*x^7 + 325*x^6 + 775*x^5 - 3447*x^4 - 1602*x^3 + 7354*x^2 - 3333*x - 121, 1)
 

Normalized defining polynomial

\( x^{9} - x^{8} - 88 x^{7} + 325 x^{6} + 775 x^{5} - 3447 x^{4} - 1602 x^{3} + 7354 x^{2} - 3333 x - 121 \)

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:  \(2459374191553118401=199^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $110.52$
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:  $199$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(199\)
Dirichlet character group:    $\lbrace$$\chi_{199}(1,·)$, $\chi_{199}(162,·)$, $\chi_{199}(106,·)$, $\chi_{199}(43,·)$, $\chi_{199}(175,·)$, $\chi_{199}(178,·)$, $\chi_{199}(180,·)$, $\chi_{199}(58,·)$, $\chi_{199}(92,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{1133} a^{7} - \frac{520}{1133} a^{6} + \frac{292}{1133} a^{5} - \frac{171}{1133} a^{4} - \frac{335}{1133} a^{3} + \frac{193}{1133} a^{2} - \frac{404}{1133} a - \frac{22}{103}$, $\frac{1}{9606707} a^{8} + \frac{456}{9606707} a^{7} + \frac{3862753}{9606707} a^{6} - \frac{541136}{9606707} a^{5} + \frac{3230636}{9606707} a^{4} - \frac{3510497}{9606707} a^{3} - \frac{4167288}{9606707} a^{2} + \frac{1703770}{9606707} a + \frac{319252}{873337}$

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:  \( 2331050.1092 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_9$ (as 9T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 9
The 9 conjugacy class representatives for $C_9$
Character table for $C_9$

Intermediate fields

3.3.39601.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }$ ${\href{/LocalNumberField/3.9.0.1}{9} }$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/7.9.0.1}{9} }$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/13.9.0.1}{9} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/19.9.0.1}{9} }$ ${\href{/LocalNumberField/23.9.0.1}{9} }$ ${\href{/LocalNumberField/29.9.0.1}{9} }$ ${\href{/LocalNumberField/31.9.0.1}{9} }$ ${\href{/LocalNumberField/37.9.0.1}{9} }$ ${\href{/LocalNumberField/41.9.0.1}{9} }$ ${\href{/LocalNumberField/43.9.0.1}{9} }$ ${\href{/LocalNumberField/47.9.0.1}{9} }$ ${\href{/LocalNumberField/53.9.0.1}{9} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{3}$

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
$199$199.9.8.1$x^{9} - 199$$9$$1$$8$$C_9$$[\ ]_{9}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.199.9t1.1c1$1$ $ 199 $ $x^{9} - x^{8} - 88 x^{7} + 325 x^{6} + 775 x^{5} - 3447 x^{4} - 1602 x^{3} + 7354 x^{2} - 3333 x - 121$ $C_9$ (as 9T1) $0$ $1$
* 1.199.9t1.1c2$1$ $ 199 $ $x^{9} - x^{8} - 88 x^{7} + 325 x^{6} + 775 x^{5} - 3447 x^{4} - 1602 x^{3} + 7354 x^{2} - 3333 x - 121$ $C_9$ (as 9T1) $0$ $1$
* 1.199.3t1.1c1$1$ $ 199 $ $x^{3} - x^{2} - 66 x - 59$ $C_3$ (as 3T1) $0$ $1$
* 1.199.9t1.1c3$1$ $ 199 $ $x^{9} - x^{8} - 88 x^{7} + 325 x^{6} + 775 x^{5} - 3447 x^{4} - 1602 x^{3} + 7354 x^{2} - 3333 x - 121$ $C_9$ (as 9T1) $0$ $1$
* 1.199.9t1.1c4$1$ $ 199 $ $x^{9} - x^{8} - 88 x^{7} + 325 x^{6} + 775 x^{5} - 3447 x^{4} - 1602 x^{3} + 7354 x^{2} - 3333 x - 121$ $C_9$ (as 9T1) $0$ $1$
* 1.199.3t1.1c2$1$ $ 199 $ $x^{3} - x^{2} - 66 x - 59$ $C_3$ (as 3T1) $0$ $1$
* 1.199.9t1.1c5$1$ $ 199 $ $x^{9} - x^{8} - 88 x^{7} + 325 x^{6} + 775 x^{5} - 3447 x^{4} - 1602 x^{3} + 7354 x^{2} - 3333 x - 121$ $C_9$ (as 9T1) $0$ $1$
* 1.199.9t1.1c6$1$ $ 199 $ $x^{9} - x^{8} - 88 x^{7} + 325 x^{6} + 775 x^{5} - 3447 x^{4} - 1602 x^{3} + 7354 x^{2} - 3333 x - 121$ $C_9$ (as 9T1) $0$ $1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.