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Properties

Label 9.9.165247690404...9401.2
Degree $9$
Signature $[9, 0]$
Discriminant $19^{6}\cdot 37^{8}$
Root discriminant $176.38$
Ramified primes $19, 37$
Class number $3$ (GRH)
Class group $[3]$ (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![2605243, 1831729, -1055566, -365670, 49437, 17789, -285, -238, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - x^8 - 238*x^7 - 285*x^6 + 17789*x^5 + 49437*x^4 - 365670*x^3 - 1055566*x^2 + 1831729*x + 2605243)
 
gp: K = bnfinit(x^9 - x^8 - 238*x^7 - 285*x^6 + 17789*x^5 + 49437*x^4 - 365670*x^3 - 1055566*x^2 + 1831729*x + 2605243, 1)
 

Normalized defining polynomial

\( x^{9} - x^{8} - 238 x^{7} - 285 x^{6} + 17789 x^{5} + 49437 x^{4} - 365670 x^{3} - 1055566 x^{2} + 1831729 x + 2605243 \)

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:  \(165247690404112349401=19^{6}\cdot 37^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $176.38$
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:  $19, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(703=19\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{703}(1,·)$, $\chi_{703}(197,·)$, $\chi_{703}(144,·)$, $\chi_{703}(562,·)$, $\chi_{703}(83,·)$, $\chi_{703}(182,·)$, $\chi_{703}(343,·)$, $\chi_{703}(248,·)$, $\chi_{703}(349,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{43554768209529120877} a^{8} - \frac{4837909454597857459}{43554768209529120877} a^{7} - \frac{15984292010579457114}{43554768209529120877} a^{6} - \frac{21076552508513992063}{43554768209529120877} a^{5} + \frac{7789569489271201440}{43554768209529120877} a^{4} + \frac{7363482438679798196}{43554768209529120877} a^{3} + \frac{3254224686140233540}{43554768209529120877} a^{2} + \frac{21376704166277936527}{43554768209529120877} a + \frac{10769910343957673700}{43554768209529120877}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 4298056.78822 \) (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

3.3.1369.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }$ ${\href{/LocalNumberField/3.9.0.1}{9} }$ ${\href{/LocalNumberField/5.9.0.1}{9} }$ ${\href{/LocalNumberField/7.9.0.1}{9} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }$ ${\href{/LocalNumberField/17.9.0.1}{9} }$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{9}$ R ${\href{/LocalNumberField/41.9.0.1}{9} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/53.9.0.1}{9} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }$

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
$19$19.9.6.3$x^{9} - 361 x^{3} + 27436$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$
$37$37.9.8.1$x^{9} - 37$$9$$1$$8$$C_9$$[\ ]_{9}$