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Properties

Label 9.1.11198680244449489.1
Degree $9$
Signature $[1, 4]$
Discriminant $11^{6}\cdot 43^{6}$
Root discriminant $60.71$
Ramified primes $11, 43$
Class number $9$
Class group $[9]$
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![-1368, 651, 1160, -386, -350, 17, 0, -3, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - x^8 - 3*x^7 + 17*x^5 - 350*x^4 - 386*x^3 + 1160*x^2 + 651*x - 1368)
 
gp: K = bnfinit(x^9 - x^8 - 3*x^7 + 17*x^5 - 350*x^4 - 386*x^3 + 1160*x^2 + 651*x - 1368, 1)
 

Normalized defining polynomial

\( x^{9} - x^{8} - 3 x^{7} + 17 x^{5} - 350 x^{4} - 386 x^{3} + 1160 x^{2} + 651 x - 1368 \)

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:  \(11198680244449489=11^{6}\cdot 43^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.71$
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:  $11, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{43} a^{6} + \frac{5}{43} a^{5} + \frac{15}{43} a^{4} - \frac{5}{43} a^{3} + \frac{19}{43} a^{2} - \frac{13}{43} a - \frac{1}{43}$, $\frac{1}{43} a^{7} - \frac{10}{43} a^{5} + \frac{6}{43} a^{4} + \frac{1}{43} a^{3} + \frac{21}{43} a^{2} + \frac{21}{43} a + \frac{5}{43}$, $\frac{1}{23587521} a^{8} + \frac{101240}{23587521} a^{7} + \frac{12714}{7862507} a^{6} + \frac{103963}{7862507} a^{5} - \frac{11027635}{23587521} a^{4} + \frac{3428245}{23587521} a^{3} + \frac{11070571}{23587521} a^{2} - \frac{487552}{23587521} a + \frac{1530081}{7862507}$

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

Class group and class number

$C_{9}$, 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:  \( 9954.39768125 \)
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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ R ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.8.6.1$x^{8} + 143 x^{4} + 5929$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$43$43.3.2.1$x^{3} - 43$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.2.1$x^{3} - 43$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.2.1$x^{3} - 43$$3$$1$$2$$C_3$$[\ ]_{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.43.3t1.1c1$1$ $ 43 $ $x^{3} - x^{2} - 14 x - 8$ $C_3$ (as 3T1) $0$ $1$
1.43.3t1.1c2$1$ $ 43 $ $x^{3} - x^{2} - 14 x - 8$ $C_3$ (as 3T1) $0$ $1$
2.11e2_43e2.24t7.3c1$2$ $ 11^{2} \cdot 43^{2}$ $x^{8} - 3 x^{7} + 7 x^{6} + 18 x^{5} - 59 x^{4} - 148 x^{3} + 68 x^{2} + 468 x + 421$ $\SL(2,3)$ (as 8T12) $-1$ $-2$
2.11e2_43.8t12.2c1$2$ $ 11^{2} \cdot 43 $ $x^{8} - 3 x^{7} + 7 x^{6} + 18 x^{5} - 59 x^{4} - 148 x^{3} + 68 x^{2} + 468 x + 421$ $\SL(2,3)$ (as 8T12) $0$ $-2$
2.11e2_43.8t12.2c2$2$ $ 11^{2} \cdot 43 $ $x^{8} - 3 x^{7} + 7 x^{6} + 18 x^{5} - 59 x^{4} - 148 x^{3} + 68 x^{2} + 468 x + 421$ $\SL(2,3)$ (as 8T12) $0$ $-2$
3.11e2_43e2.4t4.1c1$3$ $ 11^{2} \cdot 43^{2}$ $x^{4} - x^{3} - 16 x^{2} - 7 x + 27$ $A_4$ (as 4T4) $1$ $3$
* 8.11e6_43e6.9t23.1c1$8$ $ 11^{6} \cdot 43^{6}$ $x^{9} - x^{8} - 3 x^{7} + 17 x^{5} - 350 x^{4} - 386 x^{3} + 1160 x^{2} + 651 x - 1368$ $(C_3^2:Q_8):C_3$ (as 9T23) $1$ $0$
8.11e6_43e5.24t569.1c1$8$ $ 11^{6} \cdot 43^{5}$ $x^{9} - x^{8} - 3 x^{7} + 17 x^{5} - 350 x^{4} - 386 x^{3} + 1160 x^{2} + 651 x - 1368$ $(C_3^2:Q_8):C_3$ (as 9T23) $0$ $0$
8.11e6_43e5.24t569.1c2$8$ $ 11^{6} \cdot 43^{5}$ $x^{9} - x^{8} - 3 x^{7} + 17 x^{5} - 350 x^{4} - 386 x^{3} + 1160 x^{2} + 651 x - 1368$ $(C_3^2:Q_8):C_3$ (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 *.