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Properties

Label 9.9.5154052352809441.1
Degree $9$
Signature $[9, 0]$
Discriminant $37^{4}\cdot 229^{4}$
Root discriminant $55.69$
Ramified primes $37, 229$
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![-999, 765, 2226, -1454, -714, 394, 68, -35, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 2*x^8 - 35*x^7 + 68*x^6 + 394*x^5 - 714*x^4 - 1454*x^3 + 2226*x^2 + 765*x - 999)
 
gp: K = bnfinit(x^9 - 2*x^8 - 35*x^7 + 68*x^6 + 394*x^5 - 714*x^4 - 1454*x^3 + 2226*x^2 + 765*x - 999, 1)
 

Normalized defining polynomial

\( x^{9} - 2 x^{8} - 35 x^{7} + 68 x^{6} + 394 x^{5} - 714 x^{4} - 1454 x^{3} + 2226 x^{2} + 765 x - 999 \)

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:  \(5154052352809441=37^{4}\cdot 229^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.69$
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:  $37, 229$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{15} a^{7} - \frac{1}{15} a^{5} + \frac{1}{15} a^{4} + \frac{7}{15} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{225} a^{8} + \frac{4}{225} a^{7} - \frac{11}{225} a^{6} + \frac{2}{225} a^{5} + \frac{31}{225} a^{4} + \frac{8}{25} a^{3} + \frac{103}{225} a^{2} - \frac{2}{75} a + \frac{6}{25}$

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:  $8$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 65796.3764102 \)
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

3.3.229.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ R ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$37$$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
229Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.229.2t1.1c1$1$ $ 229 $ $x^{2} - x - 57$ $C_2$ (as 2T1) $1$ $1$
1.37.3t1.1c1$1$ $ 37 $ $x^{3} - x^{2} - 12 x - 11$ $C_3$ (as 3T1) $0$ $1$
1.37_229.6t1.1c1$1$ $ 37 \cdot 229 $ $x^{6} - x^{5} - 196 x^{4} + 161 x^{3} + 9937 x^{2} + 1484 x - 115799$ $C_6$ (as 6T1) $0$ $1$
1.37_229.6t1.1c2$1$ $ 37 \cdot 229 $ $x^{6} - x^{5} - 196 x^{4} + 161 x^{3} + 9937 x^{2} + 1484 x - 115799$ $C_6$ (as 6T1) $0$ $1$
1.37.3t1.1c2$1$ $ 37 $ $x^{3} - x^{2} - 12 x - 11$ $C_3$ (as 3T1) $0$ $1$
* 2.229.3t2.1c1$2$ $ 229 $ $x^{3} - 4 x - 1$ $S_3$ (as 3T2) $1$ $2$
2.37e2_229.6t5.1c1$2$ $ 37^{2} \cdot 229 $ $x^{6} - 3 x^{5} - 114 x^{4} + 196 x^{3} + 709 x^{2} + 321 x - 45$ $S_3\times C_3$ (as 6T5) $0$ $2$
2.37e2_229.6t5.1c2$2$ $ 37^{2} \cdot 229 $ $x^{6} - 3 x^{5} - 114 x^{4} + 196 x^{3} + 709 x^{2} + 321 x - 45$ $S_3\times C_3$ (as 6T5) $0$ $2$
* 6.37e4_229e3.9t13.1c1$6$ $ 37^{4} \cdot 229^{3}$ $x^{9} - 2 x^{8} - 35 x^{7} + 68 x^{6} + 394 x^{5} - 714 x^{4} - 1454 x^{3} + 2226 x^{2} + 765 x - 999$ $C_3^2 : C_6$ (as 9T11) $1$ $6$

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 *.