Properties

 Label 6.2.643264.2 Degree $6$ Signature $[2, 2]$ Discriminant $643264$ Root discriminant $9.29$ Ramified primes $2, 19, 23$ Class number $1$ Class group trivial Galois group $S_4\times C_2$ (as 6T11)

Related objects

Show commands: SageMath / Pari/GP / Magma

Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 5*x^4 + 15*x^2 - 19)

gp: K = bnfinit(x^6 - 5*x^4 + 15*x^2 - 19, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-19, 0, 15, 0, -5, 0, 1]);

$$x^{6} - 5 x^{4} + 15 x^{2} - 19$$

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: $$643264$$$$\medspace = 2^{6}\cdot 19\cdot 23^{2}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $9.29$ 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: $2, 19, 23$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Aut(K/\Q)|$: $2$ This field is not Galois over $\Q$. This is not a CM field.

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

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

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: $$\frac{1}{4} a^{4} - \frac{1}{2} a^{2} + \frac{5}{4}$$,  $$\frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - a^{2} + \frac{1}{4} a - \frac{1}{4}$$,  $$\frac{1}{4} a^{5} - \frac{1}{4} a^{4} - a^{3} + a^{2} + \frac{11}{4} a - \frac{15}{4}$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$3.93740172863$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{2}\cdot 3.93740172863 \cdot 1}{2\sqrt{643264}}\approx 0.387618803391$

Galois group

$C_2\times S_4$ (as 6T11):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A solvable group of order 48 The 10 conjugacy class representatives for $S_4\times C_2$ Character table for $S_4\times C_2$

Intermediate fields

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

Sibling algebras

 Twin sextic algebra: $$\Q(\sqrt{-437})$$ $\times$ 4.2.531392.1 Degree 6 sibling: 6.0.14795072.2 Degree 8 siblings: 8.4.1129509830656.3, 8.0.597510700417024.6 Degree 12 siblings: Deg 12, Deg 12, Deg 12, Deg 12, Deg 12, Deg 12 Degree 16 sibling: Deg 16 Degree 24 siblings: Deg 24, Deg 24, Deg 24, Deg 24

Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type R ${\href{/padicField/3.3.0.1}{3} }^{2}$ ${\href{/padicField/5.2.0.1}{2} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ ${\href{/padicField/7.2.0.1}{2} }^{3}$ ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.6.0.1}{6} }$ ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ R R ${\href{/padicField/29.6.0.1}{6} }$ ${\href{/padicField/31.3.0.1}{3} }^{2}$ ${\href{/padicField/37.2.0.1}{2} }^{3}$ ${\href{/padicField/41.6.0.1}{6} }$ ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }$ ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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$2.6.6.6$x^{6} - 13 x^{4} + 7 x^{2} - 3$$2$$3$$6$$A_4\times C_2$$[2, 2, 2]^{3} 1919.2.1.2x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4} 2323.2.0.1x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$