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Properties

Label 9.1.6160309108009.1
Degree $9$
Signature $[1, 4]$
Discriminant $7^{4}\cdot 37^{6}$
Root discriminant $26.37$
Ramified primes $7, 37$
Class number $3$
Class group $[3]$
Galois group $C_3^2 : C_6$ (as 9T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8, 28, -8, -63, -28, 4, -6, 7, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 + 7*x^7 - 6*x^6 + 4*x^5 - 28*x^4 - 63*x^3 - 8*x^2 + 28*x - 8)
 
gp: K = bnfinit(x^9 + 7*x^7 - 6*x^6 + 4*x^5 - 28*x^4 - 63*x^3 - 8*x^2 + 28*x - 8, 1)
 

Normalized defining polynomial

\( x^{9} + 7 x^{7} - 6 x^{6} + 4 x^{5} - 28 x^{4} - 63 x^{3} - 8 x^{2} + 28 x - 8 \)

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

Invariants

Degree:  $9$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6160309108009=7^{4}\cdot 37^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.37$
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:  $7, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{6} a^{4} + \frac{1}{6} a^{3} + \frac{1}{3} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{6} a^{5} + \frac{1}{6} a^{3} + \frac{1}{6} a^{2} + \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{6} a^{6} - \frac{1}{6} a^{2} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{6} a^{7} - \frac{1}{6} a^{3} - \frac{1}{6} a^{2} + \frac{1}{3} a$, $\frac{1}{252} a^{8} - \frac{4}{63} a^{7} + \frac{11}{252} a^{6} - \frac{1}{18} a^{5} + \frac{1}{14} a^{4} - \frac{11}{126} a^{3} + \frac{121}{252} a^{2} + \frac{5}{42} a + \frac{13}{63}$

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

Class group and class number

$C_{3}$, which has order $3$

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:  \( 444.101181412 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$He_3:C_2$ (as 9T11):

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 $C_3^2 : C_6$
Character table for $C_3^2 : C_6$

Intermediate fields

3.1.9583.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ R ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$37$37.9.6.1$x^{9} + 222 x^{6} + 15059 x^{3} + 405224$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.7.2t1.1c1$1$ $ 7 $ $x^{2} - x + 2$ $C_2$ (as 2T1) $1$ $-1$
1.37.3t1.1c1$1$ $ 37 $ $x^{3} - x^{2} - 12 x - 11$ $C_3$ (as 3T1) $0$ $1$
1.7_37.6t1.4c1$1$ $ 7 \cdot 37 $ $x^{6} - x^{5} - 19 x^{4} + 43 x^{3} + 143 x^{2} - 581 x + 667$ $C_6$ (as 6T1) $0$ $-1$
1.7_37.6t1.4c2$1$ $ 7 \cdot 37 $ $x^{6} - x^{5} - 19 x^{4} + 43 x^{3} + 143 x^{2} - 581 x + 667$ $C_6$ (as 6T1) $0$ $-1$
1.37.3t1.1c2$1$ $ 37 $ $x^{3} - x^{2} - 12 x - 11$ $C_3$ (as 3T1) $0$ $1$
* 2.7_37e2.3t2.1c1$2$ $ 7 \cdot 37^{2}$ $x^{3} - x^{2} - 12 x + 63$ $S_3$ (as 3T2) $1$ $0$
2.7_37.6t5.1c1$2$ $ 7 \cdot 37 $ $x^{6} - x^{5} + 3 x^{4} + 2 x^{3} + 6 x^{2} + 3 x + 2$ $S_3\times C_3$ (as 6T5) $0$ $0$
2.7_37.6t5.1c2$2$ $ 7 \cdot 37 $ $x^{6} - x^{5} + 3 x^{4} + 2 x^{3} + 6 x^{2} + 3 x + 2$ $S_3\times C_3$ (as 6T5) $0$ $0$
* 6.7e3_37e4.9t13.1c1$6$ $ 7^{3} \cdot 37^{4}$ $x^{9} + 7 x^{7} - 6 x^{6} + 4 x^{5} - 28 x^{4} - 63 x^{3} - 8 x^{2} + 28 x - 8$ $C_3^2 : C_6$ (as 9T11) $1$ $0$

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 *.