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)

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Show commands: SageMath / Pari/GP / Magma

Normalized defining 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]);
 

\( x^{6} - 2x^{5} - 120x^{4} + 106x^{3} + 4480x^{2} - 833x - 50029 \) Copy content Toggle raw display

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:   \(7930284509\) \(\medspace = 7^{4}\cdot 251\cdot 13159\) Copy content Toggle raw display
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:   \(7\), \(251\), \(13159\) Copy content Toggle raw display
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 \(a\))

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

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:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:   $\frac{2136}{1020089}a^{5}-\frac{19847}{1020089}a^{4}-\frac{154106}{1020089}a^{3}+\frac{1435113}{1020089}a^{2}+\frac{400391}{145727}a-\frac{3568379}{145727}$, $\frac{2136}{1020089}a^{5}-\frac{19847}{1020089}a^{4}-\frac{154106}{1020089}a^{3}+\frac{1435113}{1020089}a^{2}+\frac{400391}{145727}a-\frac{3422652}{145727}$, $\frac{8640}{1020089}a^{5}-\frac{80280}{1020089}a^{4}-\frac{451425}{1020089}a^{3}+\frac{4590014}{1020089}a^{2}+\frac{517601}{145727}a-\frac{7127894}{145727}$, $\frac{22180}{145727}a^{5}-\frac{230377}{145727}a^{4}-\frac{1376445}{145727}a^{3}+\frac{16340503}{145727}a^{2}+\frac{18682755}{145727}a-\frac{280011366}{145727}$, $\frac{64242442}{1020089}a^{5}-\frac{547360033}{1020089}a^{4}-\frac{4140200475}{1020089}a^{3}+\frac{33804889501}{1020089}a^{2}+\frac{9627313863}{145727}a-\frac{70417820910}{145727}$ Copy content Toggle raw display
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

\(\Q(\zeta_{7})^+\)

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
\(7\) Copy content Toggle raw display 7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
\(251\) Copy content Toggle raw display $\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}$
\(13159\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$