# Properties

 Label 6.2.916733952.1 Degree $6$ Signature $[2, 2]$ Discriminant $916733952$ Root discriminant $31.17$ Ramified primes $2, 3, 6217$ Class number $1$ Class group trivial Galois group $C_3^2:D_4$ (as 6T13)

# Related objects

Show commands: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^6 + 6*x^4 - 8*x^3 - 18*x^2 - 48*x - 124)

gp: K = bnfinit(x^6 + 6*x^4 - 8*x^3 - 18*x^2 - 48*x - 124, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-124, -48, -18, -8, 6, 0, 1]);

$$x^{6} + 6 x^{4} - 8 x^{3} - 18 x^{2} - 48 x - 124$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $6$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[2, 2]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$916733952$$$$\medspace = 2^{14}\cdot 3^{2}\cdot 6217$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $31.17$ 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: $2, 3, 6217$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $\card{ \Aut(K/\Q) }$: $1$ This field is not Galois over $\Q$. This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{6} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{42} a^{5} + \frac{1}{21} a^{4} - \frac{2}{21} a^{3} - \frac{1}{21} a^{2} - \frac{4}{21} a + \frac{1}{7}$

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: $3$ 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: $$\frac{1}{42} a^{5} + \frac{1}{21} a^{4} - \frac{2}{21} a^{3} - \frac{1}{21} a^{2} - \frac{25}{21} a + \frac{1}{7}$$,  $$\frac{1}{7} a^{5} - \frac{8}{21} a^{4} + \frac{2}{21} a^{3} - \frac{34}{21} a^{2} + \frac{4}{21} a + \frac{193}{21}$$,  $$\frac{564758743}{42} a^{5} + \frac{407570572}{21} a^{4} - \frac{251215208}{21} a^{3} + \frac{140680565}{21} a^{2} - \frac{9624091657}{21} a - \frac{5562477757}{7}$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$513.175678584$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{2}\cdot 513.175678584 \cdot 1}{2\sqrt{916733952}}\approx 1.33824040919$

## Galois group

$S_3\wr C_2$ (as 6T13):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A solvable group of order 72 The 9 conjugacy class representatives for $C_3^2:D_4$ Character table for $C_3^2:D_4$

## Intermediate fields

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

## Sibling algebras

 Twin sextic algebra: Deg 6 Degree 6 sibling: Deg 6 Degree 9 sibling: Deg 9 Degree 12 siblings: Deg 12, Deg 12, Deg 12, Deg 12, Deg 12, Deg 12 Degree 18 siblings: Deg 18, Deg 18, Deg 18 Degree 24 siblings: Deg 24, Deg 24 Degree 36 siblings: Deg 36, Deg 36, Deg 36

## 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 ${\href{/padicField/5.6.0.1}{6} }$ ${\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }$ ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ ${\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.2.0.1}{2} }^{3}$ ${\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }$

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$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3] 2.4.11.1x^{4} + 12 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2} 3.4.2.2x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$6217$$\Q_{6217}$$x$$1$$1$$0Trivial[\ ] Deg 2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$