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Properties

Label 9.1.165542400249498441.2
Degree $9$
Signature $[1, 4]$
Discriminant $3^{6}\cdot 13^{6}\cdot 19^{6}$
Root discriminant $81.89$
Ramified primes $3, 13, 19$
Class number $9$
Class group $[3, 3]$
Galois group $(C_3^2:Q_8):C_3$ (as 9T23)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![246, 397, 295, 337, 214, -5, 46, -8, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 2*x^8 - 8*x^7 + 46*x^6 - 5*x^5 + 214*x^4 + 337*x^3 + 295*x^2 + 397*x + 246)
 
gp: K = bnfinit(x^9 - 2*x^8 - 8*x^7 + 46*x^6 - 5*x^5 + 214*x^4 + 337*x^3 + 295*x^2 + 397*x + 246, 1)
 

Normalized defining polynomial

\( x^{9} - 2 x^{8} - 8 x^{7} + 46 x^{6} - 5 x^{5} + 214 x^{4} + 337 x^{3} + 295 x^{2} + 397 x + 246 \)

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:  \(165542400249498441=3^{6}\cdot 13^{6}\cdot 19^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.89$
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:  $3, 13, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{39} a^{6} + \frac{1}{39} a^{5} - \frac{8}{39} a^{4} - \frac{19}{39} a^{3} + \frac{2}{39} a^{2} - \frac{4}{39} a - \frac{4}{13}$, $\frac{1}{39} a^{7} + \frac{4}{39} a^{5} + \frac{5}{13} a^{4} - \frac{6}{13} a^{3} - \frac{19}{39} a^{2} + \frac{5}{39} a + \frac{4}{13}$, $\frac{1}{54483} a^{8} - \frac{505}{54483} a^{7} - \frac{19}{4191} a^{6} - \frac{46}{54483} a^{5} + \frac{706}{4953} a^{4} - \frac{5660}{54483} a^{3} - \frac{628}{1651} a^{2} - \frac{12520}{54483} a + \frac{1025}{18161}$

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

Class group and class number

$C_{3}\times C_{3}$, which has order $9$

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:  \( 48588.7452532 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$ASL(2,3)$ (as 9T23):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 216
The 10 conjugacy class representatives for $(C_3^2:Q_8):C_3$
Character table for $(C_3^2:Q_8):C_3$

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 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ R ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{3}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$13$13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$
$19$19.9.6.2$x^{9} + 228 x^{6} + 16967 x^{3} + 438976$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.13_19.3t1.1c1$1$ $ 13 \cdot 19 $ $x^{3} - x^{2} - 82 x + 64$ $C_3$ (as 3T1) $0$ $1$
1.13_19.3t1.1c2$1$ $ 13 \cdot 19 $ $x^{3} - x^{2} - 82 x + 64$ $C_3$ (as 3T1) $0$ $1$
2.3e2_13e2_19e2.24t7.4c1$2$ $ 3^{2} \cdot 13^{2} \cdot 19^{2}$ $x^{8} - x^{7} - 4 x^{6} - 13 x^{5} + 7 x^{4} + 127 x^{3} + 182 x^{2} + 67 x + 13$ $\SL(2,3)$ (as 8T12) $-1$ $-2$
2.3e2_13_19.8t12.2c1$2$ $ 3^{2} \cdot 13 \cdot 19 $ $x^{8} - x^{7} - 4 x^{6} - 13 x^{5} + 7 x^{4} + 127 x^{3} + 182 x^{2} + 67 x + 13$ $\SL(2,3)$ (as 8T12) $0$ $-2$
2.3e2_13_19.8t12.2c2$2$ $ 3^{2} \cdot 13 \cdot 19 $ $x^{8} - x^{7} - 4 x^{6} - 13 x^{5} + 7 x^{4} + 127 x^{3} + 182 x^{2} + 67 x + 13$ $\SL(2,3)$ (as 8T12) $0$ $-2$
3.3e2_13e2_19e2.4t4.1c1$3$ $ 3^{2} \cdot 13^{2} \cdot 19^{2}$ $x^{4} - x^{3} - 13 x^{2} + 10 x + 4$ $A_4$ (as 4T4) $1$ $3$
* 8.3e6_13e6_19e6.9t23.1c1$8$ $ 3^{6} \cdot 13^{6} \cdot 19^{6}$ $x^{9} - 2 x^{8} - 8 x^{7} + 46 x^{6} - 5 x^{5} + 214 x^{4} + 337 x^{3} + 295 x^{2} + 397 x + 246$ $(C_3^2:Q_8):C_3$ (as 9T23) $1$ $0$
8.3e6_13e5_19e5.24t569.1c1$8$ $ 3^{6} \cdot 13^{5} \cdot 19^{5}$ $x^{9} - 2 x^{8} - 8 x^{7} + 46 x^{6} - 5 x^{5} + 214 x^{4} + 337 x^{3} + 295 x^{2} + 397 x + 246$ $(C_3^2:Q_8):C_3$ (as 9T23) $0$ $0$
8.3e6_13e5_19e5.24t569.1c2$8$ $ 3^{6} \cdot 13^{5} \cdot 19^{5}$ $x^{9} - 2 x^{8} - 8 x^{7} + 46 x^{6} - 5 x^{5} + 214 x^{4} + 337 x^{3} + 295 x^{2} + 397 x + 246$ $(C_3^2:Q_8):C_3$ (as 9T23) $0$ $0$

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 *.