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Properties

Label 9.3.359209479168.1
Degree $9$
Signature $[3, 3]$
Discriminant $-\,2^{10}\cdot 3^{4}\cdot 163^{3}$
Root discriminant $19.23$
Ramified primes $2, 3, 163$
Class number $1$
Class group Trivial
Galois group $((C_3^2:Q_8):C_3):C_2$ (as 9T26)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -6, -4, 2, -10, 8, -8, 2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 2*x^8 + 2*x^7 - 8*x^6 + 8*x^5 - 10*x^4 + 2*x^3 - 4*x^2 - 6*x + 4)
 
gp: K = bnfinit(x^9 - 2*x^8 + 2*x^7 - 8*x^6 + 8*x^5 - 10*x^4 + 2*x^3 - 4*x^2 - 6*x + 4, 1)
 

Normalized defining polynomial

\( x^{9} - 2 x^{8} + 2 x^{7} - 8 x^{6} + 8 x^{5} - 10 x^{4} + 2 x^{3} - 4 x^{2} - 6 x + 4 \)

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

Invariants

Degree:  $9$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-359209479168=-\,2^{10}\cdot 3^{4}\cdot 163^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.23$
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:  $2, 3, 163$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{926} a^{8} - \frac{95}{463} a^{7} - \frac{196}{463} a^{6} - \frac{196}{463} a^{5} - \frac{188}{463} a^{4} + \frac{151}{463} a^{3} - \frac{144}{463} a^{2} + \frac{216}{463} a + \frac{133}{463}$

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:  $5$
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:  \( 815.150141791 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$AGL(2,3)$ (as 9T26):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 432
The 11 conjugacy class representatives for $((C_3^2:Q_8):C_3):C_2$
Character table for $((C_3^2:Q_8):C_3):C_2$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 12 sibling: data not computed
Degree 18 sibling: data not computed
Degree 24 siblings: data not computed
Degree 27 sibling: data not computed
Degree 36 siblings: 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 R ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.8.10.2$x^{8} + 20 x^{2} + 20$$8$$1$$10$$\textrm{GL(2,3)}$$[4/3, 4/3, 3/2]_{3}^{2}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
163Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.163.2t1.1c1$1$ $ 163 $ $x^{2} - x + 41$ $C_2$ (as 2T1) $1$ $-1$
2.2e2_163.3t2.1c1$2$ $ 2^{2} \cdot 163 $ $x^{3} - x^{2} + 7 x + 5$ $S_3$ (as 3T2) $1$ $0$
2.2e3_3e2_163.24t22.1c1$2$ $ 2^{3} \cdot 3^{2} \cdot 163 $ $x^{8} - 4 x^{7} - 6 x^{6} + 32 x^{5} + 54 x^{4} + 74 x^{3} + 158 x^{2} + 36 x - 110$ $\textrm{GL(2,3)}$ (as 8T23) $0$ $0$
2.2e3_3e2_163.24t22.1c2$2$ $ 2^{3} \cdot 3^{2} \cdot 163 $ $x^{8} - 4 x^{7} - 6 x^{6} + 32 x^{5} + 54 x^{4} + 74 x^{3} + 158 x^{2} + 36 x - 110$ $\textrm{GL(2,3)}$ (as 8T23) $0$ $0$
3.2e4_163.4t5.1c1$3$ $ 2^{4} \cdot 163 $ $x^{4} - 2 x^{2} - 2 x - 2$ $S_4$ (as 4T5) $1$ $1$
3.2e4_163e2.6t8.1c1$3$ $ 2^{4} \cdot 163^{2}$ $x^{4} - 2 x^{2} - 2 x - 2$ $S_4$ (as 4T5) $1$ $-1$
4.2e6_3e4_163e2.8t23.1c1$4$ $ 2^{6} \cdot 3^{4} \cdot 163^{2}$ $x^{8} - 4 x^{7} - 6 x^{6} + 32 x^{5} + 54 x^{4} + 74 x^{3} + 158 x^{2} + 36 x - 110$ $\textrm{GL(2,3)}$ (as 8T23) $1$ $0$
* 8.2e10_3e4_163e3.9t26.1c1$8$ $ 2^{10} \cdot 3^{4} \cdot 163^{3}$ $x^{9} - 2 x^{8} + 2 x^{7} - 8 x^{6} + 8 x^{5} - 10 x^{4} + 2 x^{3} - 4 x^{2} - 6 x + 4$ $((C_3^2:Q_8):C_3):C_2$ (as 9T26) $1$ $2$
8.2e10_3e4_163e5.18t157.1c1$8$ $ 2^{10} \cdot 3^{4} \cdot 163^{5}$ $x^{9} - 2 x^{8} + 2 x^{7} - 8 x^{6} + 8 x^{5} - 10 x^{4} + 2 x^{3} - 4 x^{2} - 6 x + 4$ $((C_3^2:Q_8):C_3):C_2$ (as 9T26) $1$ $-2$
16.2e22_3e8_163e8.24t1334.1c1$16$ $ 2^{22} \cdot 3^{8} \cdot 163^{8}$ $x^{9} - 2 x^{8} + 2 x^{7} - 8 x^{6} + 8 x^{5} - 10 x^{4} + 2 x^{3} - 4 x^{2} - 6 x + 4$ $((C_3^2:Q_8):C_3):C_2$ (as 9T26) $1$ $0$

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 *.