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Properties

Label 9.3.334110914019.5
Degree $9$
Signature $[3, 3]$
Discriminant $-\,3^{9}\cdot 257^{3}$
Root discriminant $19.07$
Ramified primes $3, 257$
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![2, 3, -9, 13, -33, 39, -20, 6, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 3*x^8 + 6*x^7 - 20*x^6 + 39*x^5 - 33*x^4 + 13*x^3 - 9*x^2 + 3*x + 2)
 
gp: K = bnfinit(x^9 - 3*x^8 + 6*x^7 - 20*x^6 + 39*x^5 - 33*x^4 + 13*x^3 - 9*x^2 + 3*x + 2, 1)
 

Normalized defining polynomial

\( x^{9} - 3 x^{8} + 6 x^{7} - 20 x^{6} + 39 x^{5} - 33 x^{4} + 13 x^{3} - 9 x^{2} + 3 x + 2 \)

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:  \(-334110914019=-\,3^{9}\cdot 257^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.07$
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:  $3, 257$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{509} a^{8} - \frac{194}{509} a^{7} - \frac{97}{509} a^{6} + \frac{183}{509} a^{5} + \frac{207}{509} a^{4} + \frac{132}{509} a^{3} + \frac{251}{509} a^{2} - \frac{104}{509} a + \frac{16}{509}$

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:  \( 332.03116888 \)
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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ R ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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$3.9.9.1$x^{9} + 54 x^{5} + 27 x^{3} + 189$$3$$3$$9$$S_3\times C_3$$[3/2]_{2}^{3}$
257Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.3_257.2t1.1c1$1$ $ 3 \cdot 257 $ $x^{2} - x + 193$ $C_2$ (as 2T1) $1$ $-1$
2.3e3_257.3t2.3c1$2$ $ 3^{3} \cdot 257 $ $x^{3} + 12 x - 1$ $S_3$ (as 3T2) $1$ $0$
2.3e3_257.24t22.1c1$2$ $ 3^{3} \cdot 257 $ $x^{8} - 4 x^{7} + 7 x^{6} + 6 x^{5} - 24 x^{4} + 21 x^{3} + 9 x^{2} - 27$ $\textrm{GL(2,3)}$ (as 8T23) $0$ $0$
2.3e3_257.24t22.1c2$2$ $ 3^{3} \cdot 257 $ $x^{8} - 4 x^{7} + 7 x^{6} + 6 x^{5} - 24 x^{4} + 21 x^{3} + 9 x^{2} - 27$ $\textrm{GL(2,3)}$ (as 8T23) $0$ $0$
3.3e3_257.4t5.1c1$3$ $ 3^{3} \cdot 257 $ $x^{4} - x - 3$ $S_4$ (as 4T5) $1$ $1$
3.3e4_257e2.6t8.1c1$3$ $ 3^{4} \cdot 257^{2}$ $x^{4} - x - 3$ $S_4$ (as 4T5) $1$ $-1$
4.3e4_257e2.8t23.1c1$4$ $ 3^{4} \cdot 257^{2}$ $x^{8} - 4 x^{7} + 7 x^{6} + 6 x^{5} - 24 x^{4} + 21 x^{3} + 9 x^{2} - 27$ $\textrm{GL(2,3)}$ (as 8T23) $1$ $0$
* 8.3e9_257e3.9t26.1c1$8$ $ 3^{9} \cdot 257^{3}$ $x^{9} - 3 x^{8} + 6 x^{7} - 20 x^{6} + 39 x^{5} - 33 x^{4} + 13 x^{3} - 9 x^{2} + 3 x + 2$ $((C_3^2:Q_8):C_3):C_2$ (as 9T26) $1$ $2$
8.3e11_257e5.18t157.1c1$8$ $ 3^{11} \cdot 257^{5}$ $x^{9} - 3 x^{8} + 6 x^{7} - 20 x^{6} + 39 x^{5} - 33 x^{4} + 13 x^{3} - 9 x^{2} + 3 x + 2$ $((C_3^2:Q_8):C_3):C_2$ (as 9T26) $1$ $-2$
16.3e18_257e8.24t1334.1c1$16$ $ 3^{18} \cdot 257^{8}$ $x^{9} - 3 x^{8} + 6 x^{7} - 20 x^{6} + 39 x^{5} - 33 x^{4} + 13 x^{3} - 9 x^{2} + 3 x + 2$ $((C_3^2:Q_8):C_3):C_2$ (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 *.