# Properties

 Label 6.6.3359232.1 Degree $6$ Signature $[6, 0]$ Discriminant $3359232$ Root discriminant $12.24$ Ramified primes $2, 3$ Class number $1$ Class group trivial Galois group $C_6$ (as 6T1)

# Related objects

Show commands: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 12*x^4 + 36*x^2 - 8)

gp: K = bnfinit(x^6 - 12*x^4 + 36*x^2 - 8, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8, 0, 36, 0, -12, 0, 1]);

$$x^{6} - 12 x^{4} + 36 x^{2} - 8$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $6$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[6, 0]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$3359232$$$$\medspace = 2^{9}\cdot 3^{8}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $12.24$ 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: $2, 3$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Gal(K/\Q)|$: $6$ This field is Galois and abelian over $\Q$. Conductor: $$72=2^{3}\cdot 3^{2}$$ Dirichlet character group: $\lbrace$$\chi_{72}(1,·), \chi_{72}(37,·), \chi_{72}(49,·), \chi_{72}(25,·), \chi_{72}(13,·), \chi_{72}(61,·)$$\rbrace$ This is not a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$

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: $5$ 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: $$\frac{1}{4} a^{4} - 2 a^{2} + 2$$,  $$\frac{1}{2} a^{2} - 2$$,  $$\frac{1}{2} a^{3} - 3 a - 1$$,  $$\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - 3 a - 2$$,  $$\frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{3}{2} a^{2} + 3 a$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$15.053136738$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{6}\cdot(2\pi)^{0}\cdot 15.053136738 \cdot 1}{2\sqrt{3359232}}\approx 0.26281913742$

## Galois group

$C_6$ (as 6T1):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A cyclic group of order 6 The 6 conjugacy class representatives for $C_6$ Character table for $C_6$

## Intermediate fields

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

## Sibling algebras

 Twin sextic algebra: $$\Q(\zeta_{9})^+$$ $\times$ $$\Q(\sqrt{2})$$ $\times$ $$\Q$$

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type R R ${\href{/padicField/5.6.0.1}{6} }$ ${\href{/padicField/7.3.0.1}{3} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }$ ${\href{/padicField/13.6.0.1}{6} }$ ${\href{/padicField/17.1.0.1}{1} }^{6}$ ${\href{/padicField/19.2.0.1}{2} }^{3}$ ${\href{/padicField/23.3.0.1}{3} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }$ ${\href{/padicField/31.3.0.1}{3} }^{2}$ ${\href{/padicField/37.2.0.1}{2} }^{3}$ ${\href{/padicField/41.3.0.1}{3} }^{2}$ ${\href{/padicField/43.6.0.1}{6} }$ ${\href{/padicField/47.3.0.1}{3} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{3}$ ${\href{/padicField/59.6.0.1}{6} }$

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$2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3} 33.6.8.3x^{6} + 18 x^{2} + 9$$3$$2$$8$$C_6$$[2]^{2}$

## Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $$\Q$$ $C_1$ $1$ $1$
* 1.8.2t1.a.a$1$ $2^{3}$ $$\Q(\sqrt{2})$$ $C_2$ (as 2T1) $1$ $1$
* 1.9.3t1.a.a$1$ $3^{2}$ $$\Q(\zeta_{9})^+$$ $C_3$ (as 3T1) $0$ $1$
* 1.72.6t1.d.a$1$ $2^{3} \cdot 3^{2}$ 6.6.3359232.1 $C_6$ (as 6T1) $0$ $1$
* 1.9.3t1.a.b$1$ $3^{2}$ $$\Q(\zeta_{9})^+$$ $C_3$ (as 3T1) $0$ $1$
* 1.72.6t1.d.b$1$ $2^{3} \cdot 3^{2}$ 6.6.3359232.1 $C_6$ (as 6T1) $0$ $1$

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 *.