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Properties

Label 9.1.949294310003793.1
Degree $9$
Signature $[1, 4]$
Discriminant $3^{4}\cdot 13^{4}\cdot 17^{7}$
Root discriminant $46.15$
Ramified primes $3, 13, 17$
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![-1905, 1318, 489, -319, -212, 102, 31, -15, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 2*x^8 - 15*x^7 + 31*x^6 + 102*x^5 - 212*x^4 - 319*x^3 + 489*x^2 + 1318*x - 1905)
 
gp: K = bnfinit(x^9 - 2*x^8 - 15*x^7 + 31*x^6 + 102*x^5 - 212*x^4 - 319*x^3 + 489*x^2 + 1318*x - 1905, 1)
 

Normalized defining polynomial

\( x^{9} - 2 x^{8} - 15 x^{7} + 31 x^{6} + 102 x^{5} - 212 x^{4} - 319 x^{3} + 489 x^{2} + 1318 x - 1905 \)

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:  \(949294310003793=3^{4}\cdot 13^{4}\cdot 17^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.15$
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:  $3, 13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{147860051} a^{8} - \frac{44007579}{147860051} a^{7} - \frac{34350555}{147860051} a^{6} + \frac{54163863}{147860051} a^{5} + \frac{16390288}{147860051} a^{4} - \frac{46022148}{147860051} a^{3} - \frac{29100942}{147860051} a^{2} + \frac{59651162}{147860051} a - \frac{47340635}{147860051}$

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:  \( 10616.296287 \)
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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ R ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.17.2t1.1c1$1$ $ 17 $ $x^{2} - x - 4$ $C_2$ (as 2T1) $1$ $1$
1.17.4t1.1c1$1$ $ 17 $ $x^{4} - x^{3} - 6 x^{2} + x + 1$ $C_4$ (as 4T1) $0$ $1$
1.17.4t1.1c2$1$ $ 17 $ $x^{4} - x^{3} - 6 x^{2} + x + 1$ $C_4$ (as 4T1) $0$ $1$
1.3_13_17.8t1.1c1$1$ $ 3 \cdot 13 \cdot 17 $ $x^{8} - x^{7} + 163 x^{6} - 164 x^{5} + 6135 x^{4} - 11230 x^{3} + 78700 x^{2} - 168806 x + 300901$ $C_8$ (as 8T1) $0$ $-1$
1.3_13_17.8t1.1c2$1$ $ 3 \cdot 13 \cdot 17 $ $x^{8} - x^{7} + 163 x^{6} - 164 x^{5} + 6135 x^{4} - 11230 x^{3} + 78700 x^{2} - 168806 x + 300901$ $C_8$ (as 8T1) $0$ $-1$
1.3_13_17.8t1.1c3$1$ $ 3 \cdot 13 \cdot 17 $ $x^{8} - x^{7} + 163 x^{6} - 164 x^{5} + 6135 x^{4} - 11230 x^{3} + 78700 x^{2} - 168806 x + 300901$ $C_8$ (as 8T1) $0$ $-1$
1.3_13_17.8t1.1c4$1$ $ 3 \cdot 13 \cdot 17 $ $x^{8} - x^{7} + 163 x^{6} - 164 x^{5} + 6135 x^{4} - 11230 x^{3} + 78700 x^{2} - 168806 x + 300901$ $C_8$ (as 8T1) $0$ $-1$
* 8.3e4_13e4_17e7.9t15.1c1$8$ $ 3^{4} \cdot 13^{4} \cdot 17^{7}$ $x^{9} - 2 x^{8} - 15 x^{7} + 31 x^{6} + 102 x^{5} - 212 x^{4} - 319 x^{3} + 489 x^{2} + 1318 x - 1905$ $C_3^2:C_8$ (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 *.