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Properties

Label 9.1.148718980881.2
Degree $9$
Signature $[1, 4]$
Discriminant $3^{12}\cdot 23^{4}$
Root discriminant $17.43$
Ramified primes $3, 23$
Class number $3$
Class group $[3]$
Galois group $D_{9}$ (as 9T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-37, 60, -45, -11, -3, 27, -5, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 6*x^7 - 5*x^6 + 27*x^5 - 3*x^4 - 11*x^3 - 45*x^2 + 60*x - 37)
 
gp: K = bnfinit(x^9 - 6*x^7 - 5*x^6 + 27*x^5 - 3*x^4 - 11*x^3 - 45*x^2 + 60*x - 37, 1)
 

Normalized defining polynomial

\( x^{9} - 6 x^{7} - 5 x^{6} + 27 x^{5} - 3 x^{4} - 11 x^{3} - 45 x^{2} + 60 x - 37 \)

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:  \(148718980881=3^{12}\cdot 23^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.43$
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:  $3, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{7} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a^{2} - \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{32935} a^{8} - \frac{1423}{32935} a^{7} + \frac{2714}{32935} a^{6} - \frac{8632}{32935} a^{5} + \frac{1039}{6587} a^{4} - \frac{8461}{32935} a^{3} - \frac{14218}{32935} a^{2} - \frac{9682}{32935} a - \frac{1809}{6587}$

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:  \( 33.411581888 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_9$ (as 9T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 6 conjugacy class representatives for $D_{9}$
Character table for $D_{9}$

Intermediate fields

3.1.23.1

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

Sibling fields

Galois closure: 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 ${\href{/LocalNumberField/2.9.0.1}{9} }$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.9.0.1}{9} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ R ${\href{/LocalNumberField/29.9.0.1}{9} }$ ${\href{/LocalNumberField/31.9.0.1}{9} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.9.0.1}{9} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.9.0.1}{9} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{3}$

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
$3$3.9.12.3$x^{9} + 6 x^{8} + 3 x^{6} + 9 x^{3} + 135$$3$$3$$12$$C_9$$[2]^{3}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.23.2t1.1c1$1$ $ 23 $ $x^{2} - x + 6$ $C_2$ (as 2T1) $1$ $-1$
* 2.23.3t2.1c1$2$ $ 23 $ $x^{3} - x^{2} + 1$ $S_3$ (as 3T2) $1$ $0$
* 2.3e4_23.9t3.1c1$2$ $ 3^{4} \cdot 23 $ $x^{9} - 6 x^{7} - 5 x^{6} + 27 x^{5} - 3 x^{4} - 11 x^{3} - 45 x^{2} + 60 x - 37$ $D_{9}$ (as 9T3) $1$ $0$
* 2.3e4_23.9t3.1c2$2$ $ 3^{4} \cdot 23 $ $x^{9} - 6 x^{7} - 5 x^{6} + 27 x^{5} - 3 x^{4} - 11 x^{3} - 45 x^{2} + 60 x - 37$ $D_{9}$ (as 9T3) $1$ $0$
* 2.3e4_23.9t3.1c3$2$ $ 3^{4} \cdot 23 $ $x^{9} - 6 x^{7} - 5 x^{6} + 27 x^{5} - 3 x^{4} - 11 x^{3} - 45 x^{2} + 60 x - 37$ $D_{9}$ (as 9T3) $1$ $0$

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 *.