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Properties

Label 9.9.178322008965...0000.1
Degree $9$
Signature $[9, 0]$
Discriminant $2^{31}\cdot 3^{12}\cdot 5^{6}$
Root discriminant $137.73$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
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![-20938, 6366, 12696, -4152, -2232, 744, 144, -48, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 3*x^8 - 48*x^7 + 144*x^6 + 744*x^5 - 2232*x^4 - 4152*x^3 + 12696*x^2 + 6366*x - 20938)
 
gp: K = bnfinit(x^9 - 3*x^8 - 48*x^7 + 144*x^6 + 744*x^5 - 2232*x^4 - 4152*x^3 + 12696*x^2 + 6366*x - 20938, 1)
 

Normalized defining polynomial

\( x^{9} - 3 x^{8} - 48 x^{7} + 144 x^{6} + 744 x^{5} - 2232 x^{4} - 4152 x^{3} + 12696 x^{2} + 6366 x - 20938 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $9$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[9, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(17832200896512000000=2^{31}\cdot 3^{12}\cdot 5^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $137.73$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7} a^{7} - \frac{2}{7} a^{6} + \frac{2}{7} a^{4} - \frac{3}{7} a^{3} + \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{25565687} a^{8} + \frac{67647}{25565687} a^{7} - \frac{10895194}{25565687} a^{6} - \frac{4769987}{25565687} a^{5} + \frac{481508}{25565687} a^{4} + \frac{6980971}{25565687} a^{3} - \frac{4947471}{25565687} a^{2} + \frac{9573750}{25565687} a + \frac{1340613}{25565687}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $8$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 33755161.1223 \) (assuming GRH)
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 R R R ${\href{/LocalNumberField/7.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.8.31.4$x^{8} + 24 x^{4} + 8 x^{2} + 16 x + 46$$8$$1$$31$$C_8$$[3, 4, 5]$
$3$3.9.12.26$x^{9} + 3 x^{4} + 3 x^{3} + 3$$9$$1$$12$$C_3^2:C_8$$[3/2, 3/2]_{2}^{4}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.8.6.4$x^{8} - 5 x^{4} + 50$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.2e3.2t1.1c1$1$ $ 2^{3}$ $x^{2} - 2$ $C_2$ (as 2T1) $1$ $1$
1.2e4_5.4t1.3c1$1$ $ 2^{4} \cdot 5 $ $x^{4} - 20 x^{2} + 50$ $C_4$ (as 4T1) $0$ $1$
1.2e4_5.4t1.3c2$1$ $ 2^{4} \cdot 5 $ $x^{4} - 20 x^{2} + 50$ $C_4$ (as 4T1) $0$ $1$
1.2e5_3_5.8t1.1c1$1$ $ 2^{5} \cdot 3 \cdot 5 $ $x^{8} - 120 x^{6} + 4500 x^{4} - 54000 x^{2} + 4050$ $C_8$ (as 8T1) $0$ $1$
1.2e5_3_5.8t1.1c2$1$ $ 2^{5} \cdot 3 \cdot 5 $ $x^{8} - 120 x^{6} + 4500 x^{4} - 54000 x^{2} + 4050$ $C_8$ (as 8T1) $0$ $1$
1.2e5_3_5.8t1.1c3$1$ $ 2^{5} \cdot 3 \cdot 5 $ $x^{8} - 120 x^{6} + 4500 x^{4} - 54000 x^{2} + 4050$ $C_8$ (as 8T1) $0$ $1$
1.2e5_3_5.8t1.1c4$1$ $ 2^{5} \cdot 3 \cdot 5 $ $x^{8} - 120 x^{6} + 4500 x^{4} - 54000 x^{2} + 4050$ $C_8$ (as 8T1) $0$ $1$
* 8.2e31_3e12_5e6.9t15.1c1$8$ $ 2^{31} \cdot 3^{12} \cdot 5^{6}$ $x^{9} - 3 x^{8} - 48 x^{7} + 144 x^{6} + 744 x^{5} - 2232 x^{4} - 4152 x^{3} + 12696 x^{2} + 6366 x - 20938$ $C_3^2:C_8$ (as 9T15) $1$ $8$

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 *.