# Properties

 Label 6.0.3518667.1 Degree $6$ Signature $[0, 3]$ Discriminant $-3518667$ Root discriminant $$12.33$$ Ramified primes see page Class number $3$ Class group $[3]$ 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 - x^5 + 7*x^4 - 8*x^3 + 43*x^2 - 42*x + 49)

gp: K = bnfinit(x^6 - x^5 + 7*x^4 - 8*x^3 + 43*x^2 - 42*x + 49, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![49, -42, 43, -8, 7, -1, 1]);

$$x^{6} - x^{5} + 7x^{4} - 8x^{3} + 43x^{2} - 42x + 49$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $6$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[0, 3]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$-3518667$$ -3518667 $$\medspace = -\,3^{3}\cdot 19^{4}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $$12.33$$ 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: $$3$$, $$19$$ 3, 19 sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $\card{ \Gal(K/\Q) }$: $6$ This field is Galois and abelian over $\Q$. Conductor: $$57=3\cdot 19$$ Dirichlet character group: $\lbrace$$\chi_{57}(1,·), \chi_{57}(20,·), \chi_{57}(49,·), \chi_{57}(7,·), \chi_{57}(26,·), \chi_{57}(11,·)$$\rbrace$ This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{7}a^{4}-\frac{1}{7}a$, $\frac{1}{259}a^{5}-\frac{1}{37}a^{4}+\frac{7}{37}a^{3}-\frac{43}{259}a^{2}+\frac{6}{37}a-\frac{5}{37}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

 Monogenic: Not computed Index: $1$ Inessential primes: None

## 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: $2$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$\frac{6}{259} a^{5} - \frac{5}{259} a^{4} + \frac{5}{37} a^{3} + \frac{1}{259} a^{2} + \frac{215}{259} a + \frac{7}{37}$$ (6)/(259)*a^(5) - (5)/(259)*a^(4) + (5)/(37)*a^(3) + (1)/(259)*a^(2) + (215)/(259)*a + (7)/(37)  (order $6$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: $\frac{30}{259}a^{5}-\frac{25}{259}a^{4}+\frac{25}{37}a^{3}-\frac{254}{259}a^{2}+\frac{1075}{259}a-\frac{150}{37}$, $\frac{5}{259}a^{5}-\frac{5}{37}a^{4}-\frac{2}{37}a^{3}-\frac{215}{259}a^{2}+\frac{30}{37}a-\frac{99}{37}$ 30/259*a^5 - 25/259*a^4 + 25/37*a^3 - 254/259*a^2 + 1075/259*a - 150/37, 5/259*a^5 - 5/37*a^4 - 2/37*a^3 - 215/259*a^2 + 30/37*a - 99/37 sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$7.80862678603$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{3}\cdot 7.80862678603 \cdot 3}{6\sqrt{3518667}}\approx 0.516291757423$

## 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: 3.3.361.1 $\times$ $$\Q(\sqrt{-3})$$ $\times$ $$\Q$$

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type ${\href{/padicField/2.6.0.1}{6} }$ R ${\href{/padicField/5.6.0.1}{6} }$ ${\href{/padicField/7.1.0.1}{1} }^{6}$ ${\href{/padicField/11.2.0.1}{2} }^{3}$ ${\href{/padicField/13.3.0.1}{3} }^{2}$ ${\href{/padicField/17.6.0.1}{6} }$ R ${\href{/padicField/23.6.0.1}{6} }$ ${\href{/padicField/29.6.0.1}{6} }$ ${\href{/padicField/31.1.0.1}{1} }^{6}$ ${\href{/padicField/37.1.0.1}{1} }^{6}$ ${\href{/padicField/41.6.0.1}{6} }$ ${\href{/padicField/43.3.0.1}{3} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }$ ${\href{/padicField/53.6.0.1}{6} }$ ${\href{/padicField/59.6.0.1}{6} }$

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
$$3$$ 3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3} $$19$$ 19.3.2.2x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$

## Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $$\Q$$ $C_1$ $1$ $1$
* 1.3.2t1.a.a$1$ $3$ $$\Q(\sqrt{-3})$$ $C_2$ (as 2T1) $1$ $-1$
* 1.19.3t1.a.a$1$ $19$ 3.3.361.1 $C_3$ (as 3T1) $0$ $1$
* 1.57.6t1.a.a$1$ $3 \cdot 19$ 6.0.3518667.1 $C_6$ (as 6T1) $0$ $-1$
* 1.19.3t1.a.b$1$ $19$ 3.3.361.1 $C_3$ (as 3T1) $0$ $1$
* 1.57.6t1.a.b$1$ $3 \cdot 19$ 6.0.3518667.1 $C_6$ (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 *.