↑

Properties

Label 9.3.1851804352.2
Degree $9$
Signature $[3, 3]$
Discriminant $-\,2^{6}\cdot 307^{3}$
Root discriminant $10.71$
Ramified primes $2, 307$
Class number $1$
Class group Trivial
Galois group $(C_3^2:C_3):C_2$ (as 9T12)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

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

Normalized defining polynomial

\( x^{9} - x^{7} - 3 x^{6} + x^{5} + 2 x^{4} - 4 x^{3} - x^{2} - x - 1 \)

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:  \(-1851804352=-\,2^{6}\cdot 307^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $10.71$
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:  $2, 307$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{53} a^{8} - \frac{3}{53} a^{7} + \frac{8}{53} a^{6} + \frac{26}{53} a^{5} - \frac{24}{53} a^{4} + \frac{21}{53} a^{3} - \frac{14}{53} a^{2} - \frac{12}{53} a - \frac{18}{53}$

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:  \( 12.6005120248 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2:S_3$ (as 9T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 10 conjugacy class representatives for $(C_3^2:C_3):C_2$
Character table for $(C_3^2:C_3):C_2$

Intermediate fields

3.1.1228.1

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

Sibling fields

Degree 9 siblings: data not computed
Degree 18 siblings: data not computed
Degree 27 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 ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
307Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.307.2t1.1c1$1$ $ 307 $ $x^{2} - x + 77$ $C_2$ (as 2T1) $1$ $-1$
* 2.2e2_307.3t2.1c1$2$ $ 2^{2} \cdot 307 $ $x^{3} - x^{2} + x - 7$ $S_3$ (as 3T2) $1$ $0$
2.2e2_307.3t2.2c1$2$ $ 2^{2} \cdot 307 $ $x^{3} + 4 x - 6$ $S_3$ (as 3T2) $1$ $0$
2.307.3t2.1c1$2$ $ 307 $ $x^{3} - x^{2} + 3 x + 2$ $S_3$ (as 3T2) $1$ $0$
2.2e2_307.3t2.3c1$2$ $ 2^{2} \cdot 307 $ $x^{3} - x^{2} + 7 x - 1$ $S_3$ (as 3T2) $1$ $0$
* 3.2e2_307.9t12.1c1$3$ $ 2^{2} \cdot 307 $ $x^{9} - x^{7} - 3 x^{6} + x^{5} + 2 x^{4} - 4 x^{3} - x^{2} - x - 1$ $(C_3^2:C_3):C_2$ (as 9T12) $0$ $1$
* 3.2e2_307.9t12.1c2$3$ $ 2^{2} \cdot 307 $ $x^{9} - x^{7} - 3 x^{6} + x^{5} + 2 x^{4} - 4 x^{3} - x^{2} - x - 1$ $(C_3^2:C_3):C_2$ (as 9T12) $0$ $1$
3.2e2_307e2.18t24.1c1$3$ $ 2^{2} \cdot 307^{2}$ $x^{9} - x^{7} - 3 x^{6} + x^{5} + 2 x^{4} - 4 x^{3} - x^{2} - x - 1$ $(C_3^2:C_3):C_2$ (as 9T12) $0$ $-1$
3.2e2_307e2.18t24.1c2$3$ $ 2^{2} \cdot 307^{2}$ $x^{9} - x^{7} - 3 x^{6} + x^{5} + 2 x^{4} - 4 x^{3} - x^{2} - x - 1$ $(C_3^2:C_3):C_2$ (as 9T12) $0$ $-1$

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 *.