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Properties

Label 9.9.9025761726072081.2
Degree $9$
Signature $[9, 0]$
Discriminant $3^{12}\cdot 19^{8}$
Root discriminant $59.27$
Ramified primes $3, 19$
Class number $3$
Class group $[3]$
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![-703, -4275, -7524, -3192, 1254, 855, -38, -57, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 57*x^7 - 38*x^6 + 855*x^5 + 1254*x^4 - 3192*x^3 - 7524*x^2 - 4275*x - 703)
 
gp: K = bnfinit(x^9 - 57*x^7 - 38*x^6 + 855*x^5 + 1254*x^4 - 3192*x^3 - 7524*x^2 - 4275*x - 703, 1)
 

Normalized defining polynomial

\( x^{9} - 57 x^{7} - 38 x^{6} + 855 x^{5} + 1254 x^{4} - 3192 x^{3} - 7524 x^{2} - 4275 x - 703 \)

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:  \(9025761726072081=3^{12}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.27$
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:  $3, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(171=3^{2}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{171}(64,·)$, $\chi_{171}(1,·)$, $\chi_{171}(43,·)$, $\chi_{171}(4,·)$, $\chi_{171}(169,·)$, $\chi_{171}(139,·)$, $\chi_{171}(16,·)$, $\chi_{171}(163,·)$, $\chi_{171}(85,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{7} a^{5} + \frac{1}{7} a^{4} + \frac{1}{7} a^{3} + \frac{1}{7} a^{2} + \frac{1}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{6} - \frac{1}{7}$, $\frac{1}{49} a^{7} - \frac{3}{49} a^{6} - \frac{1}{49} a^{5} + \frac{20}{49} a^{4} + \frac{13}{49} a^{3} - \frac{1}{49} a^{2} + \frac{19}{49} a + \frac{16}{49}$, $\frac{1}{7627291} a^{8} + \frac{59856}{7627291} a^{7} + \frac{35542}{1089613} a^{6} - \frac{394222}{7627291} a^{5} + \frac{1041905}{7627291} a^{4} - \frac{2068006}{7627291} a^{3} + \frac{1669265}{7627291} a^{2} - \frac{3239056}{7627291} a + \frac{23200}{206143}$

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

Class group and class number

$C_{3}$, which has order $3$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 86043.1062668 \)
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

3.3.361.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }$ R ${\href{/LocalNumberField/5.9.0.1}{9} }$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }$ ${\href{/LocalNumberField/17.9.0.1}{9} }$ R ${\href{/LocalNumberField/23.9.0.1}{9} }$ ${\href{/LocalNumberField/29.9.0.1}{9} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }$ ${\href{/LocalNumberField/43.9.0.1}{9} }$ ${\href{/LocalNumberField/47.9.0.1}{9} }$ ${\href{/LocalNumberField/53.9.0.1}{9} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }$

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$3.9.12.2$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 54$$3$$3$$12$$C_9$$[2]^{3}$
$19$19.9.8.8$x^{9} - 19$$9$$1$$8$$C_9$$[\ ]_{9}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.3e2_19.9t1.1c1$1$ $ 3^{2} \cdot 19 $ $x^{9} - 57 x^{7} - 38 x^{6} + 855 x^{5} + 1254 x^{4} - 3192 x^{3} - 7524 x^{2} - 4275 x - 703$ $C_9$ (as 9T1) $0$ $1$
* 1.3e2_19.9t1.1c2$1$ $ 3^{2} \cdot 19 $ $x^{9} - 57 x^{7} - 38 x^{6} + 855 x^{5} + 1254 x^{4} - 3192 x^{3} - 7524 x^{2} - 4275 x - 703$ $C_9$ (as 9T1) $0$ $1$
* 1.19.3t1.1c1$1$ $ 19 $ $x^{3} - x^{2} - 6 x + 7$ $C_3$ (as 3T1) $0$ $1$
* 1.3e2_19.9t1.1c3$1$ $ 3^{2} \cdot 19 $ $x^{9} - 57 x^{7} - 38 x^{6} + 855 x^{5} + 1254 x^{4} - 3192 x^{3} - 7524 x^{2} - 4275 x - 703$ $C_9$ (as 9T1) $0$ $1$
* 1.3e2_19.9t1.1c4$1$ $ 3^{2} \cdot 19 $ $x^{9} - 57 x^{7} - 38 x^{6} + 855 x^{5} + 1254 x^{4} - 3192 x^{3} - 7524 x^{2} - 4275 x - 703$ $C_9$ (as 9T1) $0$ $1$
* 1.19.3t1.1c2$1$ $ 19 $ $x^{3} - x^{2} - 6 x + 7$ $C_3$ (as 3T1) $0$ $1$
* 1.3e2_19.9t1.1c5$1$ $ 3^{2} \cdot 19 $ $x^{9} - 57 x^{7} - 38 x^{6} + 855 x^{5} + 1254 x^{4} - 3192 x^{3} - 7524 x^{2} - 4275 x - 703$ $C_9$ (as 9T1) $0$ $1$
* 1.3e2_19.9t1.1c6$1$ $ 3^{2} \cdot 19 $ $x^{9} - 57 x^{7} - 38 x^{6} + 855 x^{5} + 1254 x^{4} - 3192 x^{3} - 7524 x^{2} - 4275 x - 703$ $C_9$ (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 *.