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Properties

Label 9.3.293534171136.1
Degree $9$
Signature $[3, 3]$
Discriminant $-\,2^{27}\cdot 3^{7}$
Root discriminant $18.80$
Ramified primes $2, 3$
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![14, -23, -8, 20, -28, 14, -8, 4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 2*x^8 + 4*x^7 - 8*x^6 + 14*x^5 - 28*x^4 + 20*x^3 - 8*x^2 - 23*x + 14)
 
gp: K = bnfinit(x^9 - 2*x^8 + 4*x^7 - 8*x^6 + 14*x^5 - 28*x^4 + 20*x^3 - 8*x^2 - 23*x + 14, 1)
 

Normalized defining polynomial

\( x^{9} - 2 x^{8} + 4 x^{7} - 8 x^{6} + 14 x^{5} - 28 x^{4} + 20 x^{3} - 8 x^{2} - 23 x + 14 \)

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:  \(-293534171136=-\,2^{27}\cdot 3^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.80$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{5} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{4} a^{3} - \frac{3}{8} a^{2} - \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{1}{4} a^{4} - \frac{1}{8} a^{3} - \frac{1}{2} a^{2} + \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{16} a^{8} - \frac{1}{16} a^{7} - \frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{3}{16} a^{4} + \frac{1}{16} a^{3} + \frac{1}{16} a^{2} + \frac{1}{16} a + \frac{1}{8}$

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:  $5$
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:  \( 746.039765284 \)
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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.8.27.43$x^{8} + 8 x^{7} + 52 x^{4} + 40$$8$$1$$27$$QD_{16}$$[2, 3, 7/2, 9/2]$
$3$3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.6.7.3$x^{6} + 6 x^{3} + 6 x^{2} + 6$$6$$1$$7$$S_3\times C_3$$[3/2]_{2}^{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.2e3_3.2t1.2c1$1$ $ 2^{3} \cdot 3 $ $x^{2} + 6$ $C_2$ (as 2T1) $1$ $-1$
2.2e3_3e3.3t2.1c1$2$ $ 2^{3} \cdot 3^{3}$ $x^{3} + 3 x - 2$ $S_3$ (as 3T2) $1$ $0$
2.2e9_3e3.24t22.1c1$2$ $ 2^{9} \cdot 3^{3}$ $x^{8} - 4 x^{6} + 6 x^{4} - 6$ $\textrm{GL(2,3)}$ (as 8T23) $0$ $0$
2.2e9_3e3.24t22.1c2$2$ $ 2^{9} \cdot 3^{3}$ $x^{8} - 4 x^{6} + 6 x^{4} - 6$ $\textrm{GL(2,3)}$ (as 8T23) $0$ $0$
3.2e9_3e3.4t5.1c1$3$ $ 2^{9} \cdot 3^{3}$ $x^{4} - 4 x - 3$ $S_4$ (as 4T5) $1$ $1$
3.2e10_3e4.6t8.2c1$3$ $ 2^{10} \cdot 3^{4}$ $x^{4} - 4 x - 3$ $S_4$ (as 4T5) $1$ $-1$
4.2e18_3e4.8t23.1c1$4$ $ 2^{18} \cdot 3^{4}$ $x^{8} - 4 x^{6} + 6 x^{4} - 6$ $\textrm{GL(2,3)}$ (as 8T23) $1$ $0$
* 8.2e27_3e7.9t26.1c1$8$ $ 2^{27} \cdot 3^{7}$ $x^{9} - 2 x^{8} + 4 x^{7} - 8 x^{6} + 14 x^{5} - 28 x^{4} + 20 x^{3} - 8 x^{2} - 23 x + 14$ $((C_3^2:Q_8):C_3):C_2$ (as 9T26) $1$ $2$
8.2e31_3e9.18t157.1c1$8$ $ 2^{31} \cdot 3^{9}$ $x^{9} - 2 x^{8} + 4 x^{7} - 8 x^{6} + 14 x^{5} - 28 x^{4} + 20 x^{3} - 8 x^{2} - 23 x + 14$ $((C_3^2:Q_8):C_3):C_2$ (as 9T26) $1$ $-2$
16.2e58_3e20.24t1334.1c1$16$ $ 2^{58} \cdot 3^{20}$ $x^{9} - 2 x^{8} + 4 x^{7} - 8 x^{6} + 14 x^{5} - 28 x^{4} + 20 x^{3} - 8 x^{2} - 23 x + 14$ $((C_3^2:Q_8):C_3):C_2$ (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 *.