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Properties

Label 9.1.121721421175625.1
Degree $9$
Signature $[1, 4]$
Discriminant $5^{4}\cdot 41^{7}$
Root discriminant $36.73$
Ramified primes $5, 41$
Class number $3$
Class group $[3]$
Galois group $C_3^2:C_8$ (as 9T15)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-166, 215, -129, 84, -39, 3, 9, -4, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - x^8 - 4*x^7 + 9*x^6 + 3*x^5 - 39*x^4 + 84*x^3 - 129*x^2 + 215*x - 166)
 
gp: K = bnfinit(x^9 - x^8 - 4*x^7 + 9*x^6 + 3*x^5 - 39*x^4 + 84*x^3 - 129*x^2 + 215*x - 166, 1)
 

Normalized defining polynomial

\( x^{9} - x^{8} - 4 x^{7} + 9 x^{6} + 3 x^{5} - 39 x^{4} + 84 x^{3} - 129 x^{2} + 215 x - 166 \)

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:  \(121721421175625=5^{4}\cdot 41^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.73$
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:  $5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{1504537} a^{8} - \frac{749717}{1504537} a^{7} - \frac{633851}{1504537} a^{6} + \frac{224875}{1504537} a^{5} + \frac{12575}{1504537} a^{4} - \frac{249897}{1504537} a^{3} - \frac{690589}{1504537} a^{2} - \frac{163456}{1504537} a - \frac{464476}{1504537}$

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

Class group and class number

$C_{3}$, which has order $3$

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:  \( 2724.92791059 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_9$ (as 9T15):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 9 conjugacy class representatives for $C_3^2:C_8$
Character table for $C_3^2:C_8$

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 sibling: data not computed
Degree 36 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{3}$

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
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.41.2t1.1c1$1$ $ 41 $ $x^{2} - x - 10$ $C_2$ (as 2T1) $1$ $1$
1.41.4t1.1c1$1$ $ 41 $ $x^{4} - x^{3} - 15 x^{2} - 18 x - 4$ $C_4$ (as 4T1) $0$ $1$
1.41.4t1.1c2$1$ $ 41 $ $x^{4} - x^{3} - 15 x^{2} - 18 x - 4$ $C_4$ (as 4T1) $0$ $1$
1.5_41.8t1.3c1$1$ $ 5 \cdot 41 $ $x^{8} - x^{7} + 44 x^{6} - 52 x^{5} + 823 x^{4} + 94 x^{3} + 6016 x^{2} + 6843 x + 4651$ $C_8$ (as 8T1) $0$ $-1$
1.5_41.8t1.3c2$1$ $ 5 \cdot 41 $ $x^{8} - x^{7} + 44 x^{6} - 52 x^{5} + 823 x^{4} + 94 x^{3} + 6016 x^{2} + 6843 x + 4651$ $C_8$ (as 8T1) $0$ $-1$
1.5_41.8t1.3c3$1$ $ 5 \cdot 41 $ $x^{8} - x^{7} + 44 x^{6} - 52 x^{5} + 823 x^{4} + 94 x^{3} + 6016 x^{2} + 6843 x + 4651$ $C_8$ (as 8T1) $0$ $-1$
1.5_41.8t1.3c4$1$ $ 5 \cdot 41 $ $x^{8} - x^{7} + 44 x^{6} - 52 x^{5} + 823 x^{4} + 94 x^{3} + 6016 x^{2} + 6843 x + 4651$ $C_8$ (as 8T1) $0$ $-1$
* 8.5e4_41e7.9t15.1c1$8$ $ 5^{4} \cdot 41^{7}$ $x^{9} - x^{8} - 4 x^{7} + 9 x^{6} + 3 x^{5} - 39 x^{4} + 84 x^{3} - 129 x^{2} + 215 x - 166$ $C_3^2:C_8$ (as 9T15) $1$ $0$

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 *.