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Properties

Label 9.1.540022788000000.1
Degree $9$
Signature $[1, 4]$
Discriminant $2^{8}\cdot 3^{9}\cdot 5^{6}\cdot 19^{3}$
Root discriminant $43.34$
Ramified primes $2, 3, 5, 19$
Class number $15$
Class group $[15]$
Galois group $C_3^2 : D_{6} $ (as 9T18)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-729, 729, -891, 437, -261, 135, -46, 12, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 3*x^8 + 12*x^7 - 46*x^6 + 135*x^5 - 261*x^4 + 437*x^3 - 891*x^2 + 729*x - 729)
 
gp: K = bnfinit(x^9 - 3*x^8 + 12*x^7 - 46*x^6 + 135*x^5 - 261*x^4 + 437*x^3 - 891*x^2 + 729*x - 729, 1)
 

Normalized defining polynomial

\( x^{9} - 3 x^{8} + 12 x^{7} - 46 x^{6} + 135 x^{5} - 261 x^{4} + 437 x^{3} - 891 x^{2} + 729 x - 729 \)

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:  \(540022788000000=2^{8}\cdot 3^{9}\cdot 5^{6}\cdot 19^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.34$
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:  $2, 3, 5, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{396} a^{7} - \frac{13}{132} a^{6} - \frac{8}{33} a^{5} + \frac{107}{396} a^{4} - \frac{15}{44} a^{3} - \frac{19}{44} a^{2} - \frac{29}{198} a + \frac{19}{44}$, $\frac{1}{431244} a^{8} + \frac{59}{143748} a^{7} + \frac{2656}{35937} a^{6} + \frac{22931}{431244} a^{5} + \frac{3433}{47916} a^{4} + \frac{18961}{47916} a^{3} - \frac{4439}{215622} a^{2} - \frac{9953}{47916} a - \frac{183}{1331}$

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

Class group and class number

$C_{15}$, which has order $15$

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:  \( 2230.11782747 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3.S_3^2$ (as 9T18):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 11 conjugacy class representatives for $C_3^2 : D_{6} $
Character table for $C_3^2 : D_{6} $

Intermediate fields

3.1.51300.1

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

Sibling fields

Degree 9 sibling: data not computed
Degree 18 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 R ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ R ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$0$Trivial$[\ ]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
$3$3.3.3.2$x^{3} + 3 x + 3$$3$$1$$3$$S_3$$[3/2]_{2}$
3.6.6.3$x^{6} + 3 x^{4} + 9$$3$$2$$6$$D_{6}$$[3/2]_{2}^{2}$
$5$5.9.6.1$x^{9} - 25 x^{3} + 250$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.2e2.2t1.1c1$1$ $ 2^{2}$ $x^{2} + 1$ $C_2$ (as 2T1) $1$ $-1$
1.2e2_3_19.2t1.1c1$1$ $ 2^{2} \cdot 3 \cdot 19 $ $x^{2} + 57$ $C_2$ (as 2T1) $1$ $-1$
1.3_19.2t1.1c1$1$ $ 3 \cdot 19 $ $x^{2} - x - 14$ $C_2$ (as 2T1) $1$ $1$
2.2e2_3e3_5e2_19.6t3.4c1$2$ $ 2^{2} \cdot 3^{3} \cdot 5^{2} \cdot 19 $ $x^{6} - 3 x^{5} + 24 x^{4} + 42 x^{3} - 447 x^{2} + 873 x - 1002$ $D_{6}$ (as 6T3) $1$ $0$
2.2e4_3_5e2_19.6t3.1c1$2$ $ 2^{4} \cdot 3 \cdot 5^{2} \cdot 19 $ $x^{6} + 17 x^{4} + 58 x^{2} + 9$ $D_{6}$ (as 6T3) $1$ $-2$
2.3_5e2_19.3t2.1c1$2$ $ 3 \cdot 5^{2} \cdot 19 $ $x^{3} - x^{2} - 8 x - 3$ $S_3$ (as 3T2) $1$ $2$
* 2.2e2_3e3_5e2_19.3t2.1c1$2$ $ 2^{2} \cdot 3^{3} \cdot 5^{2} \cdot 19 $ $x^{3} + 45 x - 130$ $S_3$ (as 3T2) $1$ $0$
4.2e4_3e6_5e2_19e2.6t9.3c1$4$ $ 2^{4} \cdot 3^{6} \cdot 5^{2} \cdot 19^{2}$ $x^{6} - 12 x^{4} - 44 x^{3} + 45 x^{2} + 270 x + 485$ $S_3^2$ (as 6T9) $1$ $0$
6.2e6_3e8_5e4_19e4.18t51.1c1$6$ $ 2^{6} \cdot 3^{8} \cdot 5^{4} \cdot 19^{4}$ $x^{9} - 3 x^{8} + 12 x^{7} - 46 x^{6} + 135 x^{5} - 261 x^{4} + 437 x^{3} - 891 x^{2} + 729 x - 729$ $C_3^2 : D_{6} $ (as 9T18) $1$ $0$
* 6.2e6_3e6_5e4_19e2.9t18.1c1$6$ $ 2^{6} \cdot 3^{6} \cdot 5^{4} \cdot 19^{2}$ $x^{9} - 3 x^{8} + 12 x^{7} - 46 x^{6} + 135 x^{5} - 261 x^{4} + 437 x^{3} - 891 x^{2} + 729 x - 729$ $C_3^2 : D_{6} $ (as 9T18) $1$ $0$

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 *.