# Properties

 Label 6.6.14414517.1 Degree $6$ Signature $[6, 0]$ Discriminant $3^{8}\cdot 13^{3}$ Root discriminant $15.60$ Ramified primes $3, 13$ Class number $1$ Class group Trivial Galois Group $C_6$ (as 6T1)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![17, -48, 21, 27, -12, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 3*x^5 - 12*x^4 + 27*x^3 + 21*x^2 - 48*x + 17)
gp: K = bnfinit(x^6 - 3*x^5 - 12*x^4 + 27*x^3 + 21*x^2 - 48*x + 17, 1)

## Normalizeddefining polynomial

$$x^{6}$$ $$\mathstrut -\mathstrut 3 x^{5}$$ $$\mathstrut -\mathstrut 12 x^{4}$$ $$\mathstrut +\mathstrut 27 x^{3}$$ $$\mathstrut +\mathstrut 21 x^{2}$$ $$\mathstrut -\mathstrut 48 x$$ $$\mathstrut +\mathstrut 17$$

magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol

## Invariants

 Degree: $6$ magma: Degree(K); sage: K.degree() gp: poldegree(K.pol) Signature: $[6, 0]$ magma: Signature(K); sage: K.signature() gp: K.sign Discriminant: $$14414517=3^{8}\cdot 13^{3}$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $15.60$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $3, 13$ magma: PrimeDivisors(Discriminant(K)); sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ This field is Galois and abelian over $\Q$. Conductor: $$117=3^{2}\cdot 13$$ Dirichlet character group: $\lbrace$$\chi_{117}(64,·), \chi_{117}(1,·), \chi_{117}(103,·), \chi_{117}(40,·), \chi_{117}(25,·), \chi_{117}(79,·)$$\rbrace$ This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{127} a^{5} + \frac{4}{127} a^{4} + \frac{16}{127} a^{3} + \frac{12}{127} a^{2} - \frac{22}{127} a + \frac{52}{127}$

magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk

## Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp

## Unit group

magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
 Rank: $5$ 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: $$54.5168595864$$ magma: Regulator(K); sage: K.regulator() gp: K.reg

## Galois group

$C_6$ (as 6T1):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
 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: $$\Q(\zeta_{9})^+$$ $\times$ $$\Q(\sqrt{13})$$ $\times$ $$\Q$$

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 ${\href{/LocalNumberField/2.6.0.1}{6} }$ R ${\href{/LocalNumberField/5.6.0.1}{6} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }$ R ${\href{/LocalNumberField/17.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }$

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

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$3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2] 3.3.4.2x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
$13$13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.13.2t1.1c1$1$ $13$ $x^{2} - x - 3$ $C_2$ (as 2T1) $1$ $1$
* 1.3e2.3t1.1c1$1$ $3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
* 1.3e2_13.6t1.2c1$1$ $3^{2} \cdot 13$ $x^{6} - 3 x^{5} - 12 x^{4} + 27 x^{3} + 21 x^{2} - 48 x + 17$ $C_6$ (as 6T1) $0$ $1$
* 1.3e2.3t1.1c2$1$ $3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
* 1.3e2_13.6t1.2c2$1$ $3^{2} \cdot 13$ $x^{6} - 3 x^{5} - 12 x^{4} + 27 x^{3} + 21 x^{2} - 48 x + 17$ $C_6$ (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 *.