# Properties

 Label 9.1.2125764000000.5 Degree $9$ Signature $[1, 4]$ Discriminant $2^{8}\cdot 3^{12}\cdot 5^{6}$ Root discriminant $23.43$ Ramified primes $2, 3, 5$ Class number $1$ Class group Trivial Galois group $C_3^2:C_4$ (as 9T9)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 3*x^8 + 3*x^7 - 15*x^6 + 33*x^5 - 3*x^4 + 24*x^3 + 6*x^2 - 4)

gp: K = bnfinit(x^9 - 3*x^8 + 3*x^7 - 15*x^6 + 33*x^5 - 3*x^4 + 24*x^3 + 6*x^2 - 4, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, 0, 6, 24, -3, 33, -15, 3, -3, 1]);

## Normalizeddefining polynomial

$$x^{9} - 3 x^{8} + 3 x^{7} - 15 x^{6} + 33 x^{5} - 3 x^{4} + 24 x^{3} + 6 x^{2} - 4$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $9$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[1, 4]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$2125764000000=2^{8}\cdot 3^{12}\cdot 5^{6}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $23.43$ sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol))  magma: Abs(Discriminant(Integers(K)))^(1/Degree(K)); Ramified primes: $2, 3, 5$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Aut(K/\Q)|$: $1$ This field is not Galois over $\Q$. This is not a CM field.

## Integral basis (with respect to field generator $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{6} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{50} a^{7} + \frac{3}{50} a^{6} - \frac{3}{50} a^{5} - \frac{1}{2} a^{4} + \frac{3}{10} a^{3} + \frac{7}{50} a^{2} + \frac{8}{25} a + \frac{7}{25}$, $\frac{1}{50} a^{8} - \frac{1}{25} a^{6} + \frac{2}{25} a^{5} - \frac{2}{5} a^{4} - \frac{9}{25} a^{3} + \frac{1}{10} a^{2} + \frac{8}{25} a - \frac{6}{25}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

## Unit group

sage: UK = K.unit_group()

magma: UK, f := UnitGroup(K);

 Rank: $4$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$-1$$ (order $2$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); 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 sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$1978.70362296$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Galois group

$C_3:S_3.C_2$ (as 9T9):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A solvable group of order 36 The 6 conjugacy class representatives for $C_3^2:C_4$ Character table for $C_3^2:C_4$

## Intermediate fields

 The extension is primitive: there are no intermediate fields between this field and $\Q$.

## Sibling fields

 Galois closure: data not computed Degree 6 siblings: 6.2.7290000.1, 6.2.7290000.2 Degree 12 siblings: Deg 12, Deg 12 Degree 18 sibling: Deg 18

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])

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];

## Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$$\Q_{2}$$x + 1$$1$$1$$0Trivial[\ ] 2.4.4.2x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2} 33.9.12.21x^{9} + 3 x^{4} + 6$$9$$1$$12$$C_3^2:C_4$$[3/2, 3/2]_{2}^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0Trivial[\ ] 5.4.3.1x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$