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Properties

Label 9.3.98673100992.1
Degree $9$
Signature $[3, 3]$
Discriminant $-\,2^{6}\cdot 3^{7}\cdot 89^{3}$
Root discriminant $16.66$
Ramified primes $2, 3, 89$
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![1, 0, 9, -21, 3, 12, -6, 3, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 3*x^8 + 3*x^7 - 6*x^6 + 12*x^5 + 3*x^4 - 21*x^3 + 9*x^2 + 1)
 
gp: K = bnfinit(x^9 - 3*x^8 + 3*x^7 - 6*x^6 + 12*x^5 + 3*x^4 - 21*x^3 + 9*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{9} - 3 x^{8} + 3 x^{7} - 6 x^{6} + 12 x^{5} + 3 x^{4} - 21 x^{3} + 9 x^{2} + 1 \)

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:  \(-98673100992=-\,2^{6}\cdot 3^{7}\cdot 89^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.66$
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:  $2, 3, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{123} a^{8} + \frac{50}{123} a^{7} - \frac{53}{123} a^{6} + \frac{14}{123} a^{5} + \frac{16}{123} a^{4} - \frac{10}{123} a^{3} - \frac{59}{123} a^{2} - \frac{43}{123} a + \frac{58}{123}$

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:  \( 212.318104201 \)
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 ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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
$2$2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.8.7.1$x^{8} + 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
89Data 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_89.2t1.1c1$1$ $ 3 \cdot 89 $ $x^{2} - x + 67$ $C_2$ (as 2T1) $1$ $-1$
2.2e2_3_89.3t2.1c1$2$ $ 2^{2} \cdot 3 \cdot 89 $ $x^{3} - x^{2} - x + 13$ $S_3$ (as 3T2) $1$ $0$
2.2e2_3e2_89.24t22.4c1$2$ $ 2^{2} \cdot 3^{2} \cdot 89 $ $x^{8} - x^{7} + 10 x^{6} - 7 x^{5} + 16 x^{4} - 13 x^{3} - 23 x^{2} + 74 x + 16$ $\textrm{GL(2,3)}$ (as 8T23) $0$ $0$
2.2e2_3e2_89.24t22.4c2$2$ $ 2^{2} \cdot 3^{2} \cdot 89 $ $x^{8} - x^{7} + 10 x^{6} - 7 x^{5} + 16 x^{4} - 13 x^{3} - 23 x^{2} + 74 x + 16$ $\textrm{GL(2,3)}$ (as 8T23) $0$ $0$
3.2e2_3e3_89.4t5.1c1$3$ $ 2^{2} \cdot 3^{3} \cdot 89 $ $x^{4} - x^{3} - 3 x^{2} - x - 2$ $S_4$ (as 4T5) $1$ $1$
3.2e2_3e2_89e2.6t8.2c1$3$ $ 2^{2} \cdot 3^{2} \cdot 89^{2}$ $x^{4} - x^{3} - 3 x^{2} - x - 2$ $S_4$ (as 4T5) $1$ $-1$
4.2e2_3e4_89e2.8t23.4c1$4$ $ 2^{2} \cdot 3^{4} \cdot 89^{2}$ $x^{8} - x^{7} + 10 x^{6} - 7 x^{5} + 16 x^{4} - 13 x^{3} - 23 x^{2} + 74 x + 16$ $\textrm{GL(2,3)}$ (as 8T23) $1$ $0$
* 8.2e6_3e7_89e3.9t26.1c1$8$ $ 2^{6} \cdot 3^{7} \cdot 89^{3}$ $x^{9} - 3 x^{8} + 3 x^{7} - 6 x^{6} + 12 x^{5} + 3 x^{4} - 21 x^{3} + 9 x^{2} + 1$ $((C_3^2:Q_8):C_3):C_2$ (as 9T26) $1$ $2$
8.2e6_3e7_89e5.18t157.1c1$8$ $ 2^{6} \cdot 3^{7} \cdot 89^{5}$ $x^{9} - 3 x^{8} + 3 x^{7} - 6 x^{6} + 12 x^{5} + 3 x^{4} - 21 x^{3} + 9 x^{2} + 1$ $((C_3^2:Q_8):C_3):C_2$ (as 9T26) $1$ $-2$
16.2e10_3e14_89e8.24t1334.1c1$16$ $ 2^{10} \cdot 3^{14} \cdot 89^{8}$ $x^{9} - 3 x^{8} + 3 x^{7} - 6 x^{6} + 12 x^{5} + 3 x^{4} - 21 x^{3} + 9 x^{2} + 1$ $((C_3^2:Q_8):C_3):C_2$ (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 *.