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Properties

Label 9.3.806215680000.1
Degree $9$
Signature $[3, 3]$
Discriminant $-\,2^{16}\cdot 3^{9}\cdot 5^{4}$
Root discriminant $21.03$
Ramified primes $2, 3, 5$
Class number $1$
Class group Trivial
Galois group $(C_3^2:C_8):C_2$ (as 9T19)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, 30, 60, 84, 48, -12, -12, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 6*x^7 - 12*x^6 - 12*x^5 + 48*x^4 + 84*x^3 + 60*x^2 + 30*x + 8)
 
gp: K = bnfinit(x^9 - 6*x^7 - 12*x^6 - 12*x^5 + 48*x^4 + 84*x^3 + 60*x^2 + 30*x + 8, 1)
 

Normalized defining polynomial

\( x^{9} - 6 x^{7} - 12 x^{6} - 12 x^{5} + 48 x^{4} + 84 x^{3} + 60 x^{2} + 30 x + 8 \)

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:  \(-806215680000=-\,2^{16}\cdot 3^{9}\cdot 5^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.03$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{24238} a^{8} + \frac{1667}{12119} a^{7} - \frac{4846}{12119} a^{6} - \frac{1943}{12119} a^{5} + \frac{5697}{12119} a^{4} + \frac{3349}{12119} a^{3} + \frac{4009}{12119} a^{2} - \frac{1221}{12119} a + \frac{1185}{12119}$

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:  \( 1461.68516613 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$PSU(3,2):C_2$ (as 9T19):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 144
The 9 conjugacy class representatives for $(C_3^2:C_8):C_2$
Character table for $(C_3^2:C_8):C_2$

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 siblings: data not computed
Degree 24 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 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.8.16.66$x^{8} + 8 x^{4} + 336$$8$$1$$16$$QD_{16}$$[2, 2, 5/2]^{2}$
$3$3.9.9.12$x^{9} + 6 x + 6$$9$$1$$9$$(C_3^2:C_8):C_2$$[9/8, 9/8]_{8}^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.8.4.2$x^{8} + 25 x^{4} - 250 x^{2} + 1250$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$

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_3.2t1.1c1$1$ $ 2^{2} \cdot 3 $ $x^{2} - 3$ $C_2$ (as 2T1) $1$ $1$
1.2e2.2t1.1c1$1$ $ 2^{2}$ $x^{2} + 1$ $C_2$ (as 2T1) $1$ $-1$
1.3.2t1.1c1$1$ $ 3 $ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
2.2e4_3e2.4t3.1c1$2$ $ 2^{4} \cdot 3^{2}$ $x^{4} - 2 x^{3} - 2 x + 1$ $D_{4}$ (as 4T3) $1$ $0$
2.2e5_3e2_5e2.8t8.3c1$2$ $ 2^{5} \cdot 3^{2} \cdot 5^{2}$ $x^{8} - 4 x^{7} + 10 x^{6} - 16 x^{5} - 20 x^{4} + 32 x^{3} - 212 x^{2} - 196 x - 74$ $QD_{16}$ (as 8T8) $0$ $0$
2.2e5_3e2_5e2.8t8.3c2$2$ $ 2^{5} \cdot 3^{2} \cdot 5^{2}$ $x^{8} - 4 x^{7} + 10 x^{6} - 16 x^{5} - 20 x^{4} + 32 x^{3} - 212 x^{2} - 196 x - 74$ $QD_{16}$ (as 8T8) $0$ $0$
8.2e16_3e9_5e4.18t68.1c1$8$ $ 2^{16} \cdot 3^{9} \cdot 5^{4}$ $x^{9} - 6 x^{7} - 12 x^{6} - 12 x^{5} + 48 x^{4} + 84 x^{3} + 60 x^{2} + 30 x + 8$ $(C_3^2:C_8):C_2$ (as 9T19) $1$ $-2$
* 8.2e16_3e9_5e4.9t19.1c1$8$ $ 2^{16} \cdot 3^{9} \cdot 5^{4}$ $x^{9} - 6 x^{7} - 12 x^{6} - 12 x^{5} + 48 x^{4} + 84 x^{3} + 60 x^{2} + 30 x + 8$ $(C_3^2:C_8):C_2$ (as 9T19) $1$ $2$

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 *.