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Properties

Label 9.1.33860761.1
Degree $9$
Signature $[1, 4]$
Discriminant $11^{2}\cdot 23^{4}$
Root discriminant $6.86$
Ramified primes $11, 23$
Class number $1$
Class group Trivial
Galois group $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 0, 0, 2, -2, 0, 2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 2*x^8 + 2*x^7 - 2*x^5 + 2*x^4 - x + 1)
 
gp: K = bnfinit(x^9 - 2*x^8 + 2*x^7 - 2*x^5 + 2*x^4 - x + 1, 1)
 

Normalized defining polynomial

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

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

Invariants

Degree:  $9$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(33860761=11^{2}\cdot 23^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $6.86$
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:  $11, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$

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:  $4$
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:  \( a \),  \( a^{8} - a^{7} + a^{6} - a^{4} + a - 1 \),  \( a^{8} - a^{7} + a^{6} - a^{4} + a^{3} + a \),  \( a - 1 \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 0.744153660637 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 648
The 14 conjugacy class representatives for $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$
Character table for $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$

Intermediate fields

3.1.23.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 12 siblings: data not computed
Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 27 siblings: 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 ${\href{/LocalNumberField/2.9.0.1}{9} }$ ${\href{/LocalNumberField/3.9.0.1}{9} }$ ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ R ${\href{/LocalNumberField/13.9.0.1}{9} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ R ${\href{/LocalNumberField/29.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/31.9.0.1}{9} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.9.0.1}{9} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.23.2t1.a.a$1$ $ 23 $ $x^{2} - x + 6$ $C_2$ (as 2T1) $1$ $-1$
* 2.23.3t2.b.a$2$ $ 23 $ $x^{3} - x^{2} + 1$ $S_3$ (as 3T2) $1$ $0$
3.64009.6t8.a.a$3$ $ 11^{2} \cdot 23^{2}$ $x^{4} - x^{3} + 9 x^{2} - 36 x + 31$ $S_4$ (as 4T5) $1$ $-1$
3.1472207.4t5.a.a$3$ $ 11^{2} \cdot 23^{3}$ $x^{4} - x^{3} + 9 x^{2} - 36 x + 31$ $S_4$ (as 4T5) $1$ $1$
6.778797503.18t217.a.a$6$ $ 11^{2} \cdot 23^{5}$ $x^{9} - 2 x^{8} + 2 x^{7} - 2 x^{5} + 2 x^{4} - x + 1$ $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $0$
* 6.1472207.9t30.a.a$6$ $ 11^{2} \cdot 23^{3}$ $x^{9} - 2 x^{8} + 2 x^{7} - 2 x^{5} + 2 x^{4} - x + 1$ $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $0$
6.94234497863.36t1121.a.a$6$ $ 11^{4} \cdot 23^{5}$ $x^{9} - 2 x^{8} + 2 x^{7} - 2 x^{5} + 2 x^{4} - x + 1$ $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $0$
6.94234497863.36t1121.a.b$6$ $ 11^{4} \cdot 23^{5}$ $x^{9} - 2 x^{8} + 2 x^{7} - 2 x^{5} + 2 x^{4} - x + 1$ $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $0$
8.2167393450849.12t177.a.a$8$ $ 11^{4} \cdot 23^{6}$ $x^{9} - 2 x^{8} + 2 x^{7} - 2 x^{5} + 2 x^{4} - x + 1$ $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $0$
8.2167393450849.12t177.b.a$8$ $ 11^{4} \cdot 23^{6}$ $x^{9} - 2 x^{8} + 2 x^{7} - 2 x^{5} + 2 x^{4} - x + 1$ $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $0$
8.2167393450849.12t178.a.a$8$ $ 11^{4} \cdot 23^{6}$ $x^{9} - 2 x^{8} + 2 x^{7} - 2 x^{5} + 2 x^{4} - x + 1$ $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $0$
12.3190851810094053743.36t1123.a.a$12$ $ 11^{6} \cdot 23^{9}$ $x^{9} - 2 x^{8} + 2 x^{7} - 2 x^{5} + 2 x^{4} - x + 1$ $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $-2$
12.3190851810094053743.18t218.a.a$12$ $ 11^{6} \cdot 23^{9}$ $x^{9} - 2 x^{8} + 2 x^{7} - 2 x^{5} + 2 x^{4} - x + 1$ $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $2$

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 *.