↑

Properties

Label 9.1.1320495160384.1
Degree $9$
Signature $[1, 4]$
Discriminant $2^{6}\cdot 379^{4}$
Root discriminant $22.22$
Ramified primes $2, 379$
Class number $1$
Class group Trivial
Galois group $D_{9}$ (as 9T3)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![27, -18, -108, 163, -46, -32, 15, 4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 4*x^8 + 4*x^7 + 15*x^6 - 32*x^5 - 46*x^4 + 163*x^3 - 108*x^2 - 18*x + 27)
 
gp: K = bnfinit(x^9 - 4*x^8 + 4*x^7 + 15*x^6 - 32*x^5 - 46*x^4 + 163*x^3 - 108*x^2 - 18*x + 27, 1)
 

Normalized defining polynomial

\( x^{9} - 4 x^{8} + 4 x^{7} + 15 x^{6} - 32 x^{5} - 46 x^{4} + 163 x^{3} - 108 x^{2} - 18 x + 27 \)

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

Invariants

Degree:  $9$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1320495160384=2^{6}\cdot 379^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.22$
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:  $2, 379$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{1}{3} a^{4} - \frac{2}{9} a^{3} - \frac{2}{9} a^{2}$, $\frac{1}{9009} a^{8} + \frac{1}{99} a^{7} - \frac{40}{1001} a^{6} + \frac{617}{3003} a^{5} - \frac{397}{819} a^{4} - \frac{71}{1287} a^{3} - \frac{223}{1001} a^{2} - \frac{16}{91} a + \frac{295}{1001}$

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

Class group and class number

Trivial group, which has order $1$

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $4$
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:  \( 1049.83879169 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_9$ (as 9T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 6 conjugacy class representatives for $D_{9}$
Character table for $D_{9}$

Intermediate fields

3.1.379.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.9.0.1}{9} }$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.9.0.1}{9} }$ ${\href{/LocalNumberField/23.9.0.1}{9} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.9.0.1}{9} }$ ${\href{/LocalNumberField/41.9.0.1}{9} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$2$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
379Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.379.2t1.1c1$1$ $ 379 $ $x^{2} - x + 95$ $C_2$ (as 2T1) $1$ $-1$
* 2.379.3t2.1c1$2$ $ 379 $ $x^{3} - x^{2} + x - 4$ $S_3$ (as 3T2) $1$ $0$
* 2.2e2_379.9t3.1c1$2$ $ 2^{2} \cdot 379 $ $x^{9} - 4 x^{8} + 4 x^{7} + 15 x^{6} - 32 x^{5} - 46 x^{4} + 163 x^{3} - 108 x^{2} - 18 x + 27$ $D_{9}$ (as 9T3) $1$ $0$
* 2.2e2_379.9t3.1c2$2$ $ 2^{2} \cdot 379 $ $x^{9} - 4 x^{8} + 4 x^{7} + 15 x^{6} - 32 x^{5} - 46 x^{4} + 163 x^{3} - 108 x^{2} - 18 x + 27$ $D_{9}$ (as 9T3) $1$ $0$
* 2.2e2_379.9t3.1c3$2$ $ 2^{2} \cdot 379 $ $x^{9} - 4 x^{8} + 4 x^{7} + 15 x^{6} - 32 x^{5} - 46 x^{4} + 163 x^{3} - 108 x^{2} - 18 x + 27$ $D_{9}$ (as 9T3) $1$ $0$

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 *.