Properties

 Label 6.6.7930284509.2 Degree $6$ Signature $[6, 0]$ Discriminant $7930284509$ Root discriminant $$44.66$$ Ramified primes see page Class number $1$ Class group trivial Galois group $A_4\times C_2$ (as 6T6)

Related objects

Show commands: SageMath / Pari/GP / Magma

Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^5 - 120*x^4 + 106*x^3 + 4480*x^2 - 833*x - 50029)

gp: K = bnfinit(x^6 - 2*x^5 - 120*x^4 + 106*x^3 + 4480*x^2 - 833*x - 50029, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-50029, -833, 4480, 106, -120, -2, 1]);

$$x^{6} - 2x^{5} - 120x^{4} + 106x^{3} + 4480x^{2} - 833x - 50029$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

Invariants

 Degree: $6$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[6, 0]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$7930284509$$ 7930284509 $$\medspace = 7^{4}\cdot 251\cdot 13159$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $$44.66$$ 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: $$7$$, $$251$$, $$13159$$ 7, 251, 13159 sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $\card{ \Aut(K/\Q) }$: $2$ This field is not Galois over $\Q$. This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{1020089}a^{5}-\frac{42513}{1020089}a^{4}-\frac{327685}{1020089}a^{3}-\frac{118243}{1020089}a^{2}-\frac{52277}{145727}a+\frac{10678}{145727}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

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

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: $5$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$-1$$ -1  (order $2$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: $\frac{2136}{1020089}a^{5}-\frac{19847}{1020089}a^{4}-\frac{154106}{1020089}a^{3}+\frac{1435113}{1020089}a^{2}+\frac{400391}{145727}a-\frac{3568379}{145727}$, $\frac{2136}{1020089}a^{5}-\frac{19847}{1020089}a^{4}-\frac{154106}{1020089}a^{3}+\frac{1435113}{1020089}a^{2}+\frac{400391}{145727}a-\frac{3422652}{145727}$, $\frac{8640}{1020089}a^{5}-\frac{80280}{1020089}a^{4}-\frac{451425}{1020089}a^{3}+\frac{4590014}{1020089}a^{2}+\frac{517601}{145727}a-\frac{7127894}{145727}$, $\frac{22180}{145727}a^{5}-\frac{230377}{145727}a^{4}-\frac{1376445}{145727}a^{3}+\frac{16340503}{145727}a^{2}+\frac{18682755}{145727}a-\frac{280011366}{145727}$, $\frac{64242442}{1020089}a^{5}-\frac{547360033}{1020089}a^{4}-\frac{4140200475}{1020089}a^{3}+\frac{33804889501}{1020089}a^{2}+\frac{9627313863}{145727}a-\frac{70417820910}{145727}$ 2136/1020089*a^5 - 19847/1020089*a^4 - 154106/1020089*a^3 + 1435113/1020089*a^2 + 400391/145727*a - 3568379/145727, 2136/1020089*a^5 - 19847/1020089*a^4 - 154106/1020089*a^3 + 1435113/1020089*a^2 + 400391/145727*a - 3422652/145727, 8640/1020089*a^5 - 80280/1020089*a^4 - 451425/1020089*a^3 + 4590014/1020089*a^2 + 517601/145727*a - 7127894/145727, 22180/145727*a^5 - 230377/145727*a^4 - 1376445/145727*a^3 + 16340503/145727*a^2 + 18682755/145727*a - 280011366/145727, 64242442/1020089*a^5 - 547360033/1020089*a^4 - 4140200475/1020089*a^3 + 33804889501/1020089*a^2 + 9627313863/145727*a - 70417820910/145727 sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$860.494377619$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{6}\cdot(2\pi)^{0}\cdot 860.494377619 \cdot 1}{2\sqrt{7930284509}}\approx 0.309210071455$

Galois group

$C_2\times A_4$ (as 6T6):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A solvable group of order 24 The 8 conjugacy class representatives for $A_4\times C_2$ Character table for $A_4\times C_2$

Intermediate fields

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

Sibling algebras

 Galois closure: Deg 24 Twin sextic algebra: Deg 2 $\times$ Deg 4 Degree 8 sibling: Deg 8 Degree 12 siblings: Deg 12, some data not computed

Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type ${\href{/padicField/2.6.0.1}{6} }$ ${\href{/padicField/3.6.0.1}{6} }$ ${\href{/padicField/5.3.0.1}{3} }^{2}$ R ${\href{/padicField/11.3.0.1}{3} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.3.0.1}{3} }^{2}$ ${\href{/padicField/19.3.0.1}{3} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }$ ${\href{/padicField/29.2.0.1}{2} }^{3}$ ${\href{/padicField/31.3.0.1}{3} }^{2}$ ${\href{/padicField/37.3.0.1}{3} }^{2}$ ${\href{/padicField/41.1.0.1}{1} }^{6}$ ${\href{/padicField/43.1.0.1}{1} }^{6}$ ${\href{/padicField/47.6.0.1}{6} }$ ${\href{/padicField/53.6.0.1}{6} }$ ${\href{/padicField/59.6.0.1}{6} }$

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
$$7$$ 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}$
$$251$$ $\Q_{251}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{251}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{251}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{251}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2} $$13159$$ Deg 2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2} Deg 2$$2$$1$$1$$C_2$$[\ ]_{2}$