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Properties

Label 9.3.1940970550347.1
Degree $9$
Signature $[3, 3]$
Discriminant $-\,3^{7}\cdot 31^{6}$
Root discriminant $23.19$
Ramified primes $3, 31$
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![3, 18, 15, -69, -21, 33, 3, -12, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - x^8 - 12*x^7 + 3*x^6 + 33*x^5 - 21*x^4 - 69*x^3 + 15*x^2 + 18*x + 3)
 
gp: K = bnfinit(x^9 - x^8 - 12*x^7 + 3*x^6 + 33*x^5 - 21*x^4 - 69*x^3 + 15*x^2 + 18*x + 3, 1)
 

Normalized defining polynomial

\( x^{9} - x^{8} - 12 x^{7} + 3 x^{6} + 33 x^{5} - 21 x^{4} - 69 x^{3} + 15 x^{2} + 18 x + 3 \)

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:  \(-1940970550347=-\,3^{7}\cdot 31^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.19$
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:  $3, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7} a^{7} + \frac{1}{7} a^{6} - \frac{2}{7} a^{5} + \frac{3}{7} a^{3} - \frac{1}{7} a^{2} + \frac{2}{7} a - \frac{3}{7}$, $\frac{1}{12887} a^{8} + \frac{396}{12887} a^{7} + \frac{715}{12887} a^{6} - \frac{1497}{12887} a^{5} + \frac{2208}{12887} a^{4} + \frac{239}{12887} a^{3} - \frac{918}{12887} a^{2} - \frac{1754}{12887} a - \frac{4104}{12887}$

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:  \( 1728.76981926 \)
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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ R ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$0$Trivial$[\ ]$
3.8.7.2$x^{8} - 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.8.6.2$x^{8} + 713 x^{4} + 138384$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.3_31.2t1.1c1$1$ $ 3 \cdot 31 $ $x^{2} - x - 23$ $C_2$ (as 2T1) $1$ $1$
1.31.2t1.1c1$1$ $ 31 $ $x^{2} - x + 8$ $C_2$ (as 2T1) $1$ $-1$
1.3.2t1.1c1$1$ $ 3 $ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
2.3e2_31.4t3.1c1$2$ $ 3^{2} \cdot 31 $ $x^{4} + 9 x^{2} - 3$ $D_{4}$ (as 4T3) $1$ $0$
2.3e2_31e2.8t8.1c1$2$ $ 3^{2} \cdot 31^{2}$ $x^{8} - 2 x^{7} + x^{6} - 23 x^{5} + 55 x^{4} - 32 x^{3} + 109 x^{2} - 275 x + 163$ $QD_{16}$ (as 8T8) $0$ $0$
2.3e2_31e2.8t8.1c2$2$ $ 3^{2} \cdot 31^{2}$ $x^{8} - 2 x^{7} + x^{6} - 23 x^{5} + 55 x^{4} - 32 x^{3} + 109 x^{2} - 275 x + 163$ $QD_{16}$ (as 8T8) $0$ $0$
8.3e7_31e6.18t68.1c1$8$ $ 3^{7} \cdot 31^{6}$ $x^{9} - x^{8} - 12 x^{7} + 3 x^{6} + 33 x^{5} - 21 x^{4} - 69 x^{3} + 15 x^{2} + 18 x + 3$ $(C_3^2:C_8):C_2$ (as 9T19) $1$ $-2$
* 8.3e7_31e6.9t19.1c1$8$ $ 3^{7} \cdot 31^{6}$ $x^{9} - x^{8} - 12 x^{7} + 3 x^{6} + 33 x^{5} - 21 x^{4} - 69 x^{3} + 15 x^{2} + 18 x + 3$ $(C_3^2:C_8):C_2$ (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 *.