# Properties

 Label 6.2.4439449.1 Degree $6$ Signature $[2, 2]$ Discriminant $7^{4}\cdot 43^{2}$ Root discriminant $12.82$ Ramified primes $7, 43$ Class number $2$ Class group $[2]$ Galois group $A_4$ (as 6T4)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-56, 28, -14, 1, -1, -2, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^5 - x^4 + x^3 - 14*x^2 + 28*x - 56)

gp: K = bnfinit(x^6 - 2*x^5 - x^4 + x^3 - 14*x^2 + 28*x - 56, 1)

## Normalizeddefining polynomial

$$x^{6} - 2 x^{5} - x^{4} + x^{3} - 14 x^{2} + 28 x - 56$$

magma: DefiningPolynomial(K);

sage: K.defining_polynomial()

gp: K.pol

## Invariants

 Degree: $6$ magma: Degree(K);  sage: K.degree()  gp: poldegree(K.pol) Signature: $[2, 2]$ magma: Signature(K);  sage: K.signature()  gp: K.sign Discriminant: $$4439449=7^{4}\cdot 43^{2}$$ magma: Discriminant(Integers(K));  sage: K.disc()  gp: K.disc Root discriminant: $12.82$ magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));  sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $7, 43$ magma: PrimeDivisors(Discriminant(Integers(K)));  sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~ $|\Aut(K/\Q)|$: $2$ This field is not Galois over $\Q$. This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{688} a^{5} - \frac{5}{43} a^{4} + \frac{47}{688} a^{3} - \frac{225}{688} a^{2} + \frac{21}{43} a - \frac{9}{172}$

magma: IntegralBasis(K);

sage: K.integral_basis()

gp: K.zk

## Class group and class number

$C_{2}$, which has order $2$

magma: ClassGroup(K);

sage: K.class_group().invariants()

gp: K.clgp

## Unit group

magma: UK, f := UnitGroup(K);

sage: UK = K.unit_group()

 Rank: $3$ 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: $$\frac{7}{688} a^{5} - \frac{11}{172} a^{4} - \frac{15}{688} a^{3} - \frac{27}{688} a^{2} + \frac{29}{172} a + \frac{195}{172}$$,  $$\frac{13}{688} a^{5} - \frac{1}{86} a^{4} - \frac{77}{688} a^{3} + \frac{171}{688} a^{2} - \frac{13}{86} a + \frac{55}{172}$$,  $$\frac{31}{688} a^{5} - \frac{9}{86} a^{4} + \frac{81}{688} a^{3} - \frac{439}{688} a^{2} + \frac{55}{86} a - \frac{107}{172}$$ magma: [K!f(g): g in Generators(UK)];  sage: UK.fundamental_units()  gp: K.fu Regulator: $$5.08829060986$$ magma: Regulator(K);  sage: K.regulator()  gp: K.reg

## Galois group

$A_4$ (as 6T4):

magma: GaloisGroup(K);

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

 A solvable group of order 12 The 4 conjugacy class representatives for $A_4$ Character table for $A_4$

## Intermediate fields

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

## Sibling algebras

 Galois closure: Deg 12 Twin sextic algebra: $$\Q$$ $\times$ $$\Q$$ $\times$ 4.0.90601.1 Degree 4 sibling: 4.0.90601.1

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 ${\href{/LocalNumberField/2.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3} 7.3.2.2x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
$43$$\Q_{43}$$x + 9$$1$$1$$0Trivial[\ ] \Q_{43}$$x + 9$$1$$1$$0Trivial[\ ] 43.2.1.2x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$