Properties

Label 9.9.3691950281939241.1
Degree $9$
Signature $[9, 0]$
Discriminant $3.692\times 10^{15}$
Root discriminant $53.67$
Ramified primes $3, 7$
Class number $3$
Class group $[3]$
Galois group $C_9$ (as 9T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 63*x^7 + 1323*x^5 - 10290*x^3 + 21609*x - 12691)
 
gp: K = bnfinit(x^9 - 63*x^7 + 1323*x^5 - 10290*x^3 + 21609*x - 12691, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-12691, 21609, 0, -10290, 0, 1323, 0, -63, 0, 1]);
 

\(x^{9} - 63 x^{7} + 1323 x^{5} - 10290 x^{3} + 21609 x - 12691\)  Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $9$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[9, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(3691950281939241\)\(\medspace = 3^{22}\cdot 7^{6}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $53.67$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $3, 7$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $9$
This field is Galois and abelian over $\Q$.
Conductor:  \(189=3^{3}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{189}(64,·)$, $\chi_{189}(1,·)$, $\chi_{189}(130,·)$, $\chi_{189}(67,·)$, $\chi_{189}(4,·)$, $\chi_{189}(142,·)$, $\chi_{189}(79,·)$, $\chi_{189}(16,·)$, $\chi_{189}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{7} a^{3}$, $\frac{1}{7} a^{4}$, $\frac{1}{7} a^{5}$, $\frac{1}{49} a^{6}$, $\frac{1}{49} a^{7}$, $\frac{1}{49} a^{8}$  Toggle raw display

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

Class group and class number

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

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $8$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 17156.8683251 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{9}\cdot(2\pi)^{0}\cdot 17156.8683251 \cdot 3}{2\sqrt{3691950281939241}}\approx 0.216855936823$

Galois group

$C_9$ (as 9T1):

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

Intermediate fields

\(\Q(\zeta_{9})^+\)

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 ${\href{/padicField/2.9.0.1}{9} }$ R ${\href{/padicField/5.9.0.1}{9} }$ R ${\href{/padicField/11.9.0.1}{9} }$ ${\href{/padicField/13.9.0.1}{9} }$ ${\href{/padicField/17.3.0.1}{3} }^{3}$ ${\href{/padicField/19.3.0.1}{3} }^{3}$ ${\href{/padicField/23.9.0.1}{9} }$ ${\href{/padicField/29.9.0.1}{9} }$ ${\href{/padicField/31.9.0.1}{9} }$ ${\href{/padicField/37.1.0.1}{1} }^{9}$ ${\href{/padicField/41.9.0.1}{9} }$ ${\href{/padicField/43.9.0.1}{9} }$ ${\href{/padicField/47.9.0.1}{9} }$ ${\href{/padicField/53.3.0.1}{3} }^{3}$ ${\href{/padicField/59.9.0.1}{9} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.9.22.2$x^{9} + 9 x^{7} + 3 x^{6} + 18 x^{5} + 51$$9$$1$$22$$C_9$$[2, 3]$
$7$7.9.6.2$x^{9} - 49 x^{3} + 686$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.189.9t1.a.a$1$ $ 3^{3} \cdot 7 $ 9.9.3691950281939241.1 $C_9$ (as 9T1) $0$ $1$
* 1.189.9t1.a.b$1$ $ 3^{3} \cdot 7 $ 9.9.3691950281939241.1 $C_9$ (as 9T1) $0$ $1$
* 1.9.3t1.a.a$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
* 1.189.9t1.a.c$1$ $ 3^{3} \cdot 7 $ 9.9.3691950281939241.1 $C_9$ (as 9T1) $0$ $1$
* 1.189.9t1.a.d$1$ $ 3^{3} \cdot 7 $ 9.9.3691950281939241.1 $C_9$ (as 9T1) $0$ $1$
* 1.9.3t1.a.b$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
* 1.189.9t1.a.e$1$ $ 3^{3} \cdot 7 $ 9.9.3691950281939241.1 $C_9$ (as 9T1) $0$ $1$
* 1.189.9t1.a.f$1$ $ 3^{3} \cdot 7 $ 9.9.3691950281939241.1 $C_9$ (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 *.