# Properties

 Label 6.6.330812181.2 Degree $6$ Signature $[6, 0]$ Discriminant $330812181$ Root discriminant $26.30$ Ramified primes $3, 7$ Class number $3$ Class group $[3]$ Galois group $C_6$ (as 6T1)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 21*x^4 - 14*x^3 + 63*x^2 + 84*x + 28)

gp: K = bnfinit(x^6 - 21*x^4 - 14*x^3 + 63*x^2 + 84*x + 28, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![28, 84, 63, -14, -21, 0, 1]);

$$x^{6} - 21 x^{4} - 14 x^{3} + 63 x^{2} + 84 x + 28$$

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: $$330812181$$$$\medspace = 3^{9}\cdot 7^{5}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $26.30$ 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: $3, 7$ 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: $$63=3^{2}\cdot 7$$ Dirichlet character group: $\lbrace$$\chi_{63}(1,·), \chi_{63}(5,·), \chi_{63}(38,·), \chi_{63}(25,·), \chi_{63}(58,·), \chi_{63}(62,·)$$\rbrace$ This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{8} a^{5} - \frac{1}{4} a^{4} - \frac{1}{8} a^{3} + \frac{3}{8} a + \frac{1}{4}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

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

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: 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: $$131.766082882$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{6}\cdot(2\pi)^{0}\cdot 131.766082882 \cdot 3}{2\sqrt{330812181}}\approx 0.695479162175$

## 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: 3.3.3969.1 $\times$ $$\Q(\sqrt{21})$$ $\times$ $$\Q$$

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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
$3$3.6.9.2$x^{6} + 3 x^{4} + 6$$6$$1$$9$$C_6$$[2]_{2} 77.6.5.6x^{6} + 224$$6$$1$$5$$C_6$$[\ ]_{6}$

## Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $$\Q$$ $C_1$ $1$ $1$
* 1.21.2t1.a.a$1$ $3 \cdot 7$ $$\Q(\sqrt{21})$$ $C_2$ (as 2T1) $1$ $1$
* 1.63.3t1.b.a$1$ $3^{2} \cdot 7$ 3.3.3969.1 $C_3$ (as 3T1) $0$ $1$
* 1.63.6t1.g.a$1$ $3^{2} \cdot 7$ 6.6.330812181.2 $C_6$ (as 6T1) $0$ $1$
* 1.63.3t1.b.b$1$ $3^{2} \cdot 7$ 3.3.3969.1 $C_3$ (as 3T1) $0$ $1$
* 1.63.6t1.g.b$1$ $3^{2} \cdot 7$ 6.6.330812181.2 $C_6$ (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 *.