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Properties

Label 6.0.317992113503...0000.1
Degree $6$
Signature $[0, 3]$
Discriminant $-\,2^{4}\cdot 3^{7}\cdot 5^{4}\cdot 79^{4}\cdot 139^{4}$
Root discriminant $8261.70$
Ramified primes $2, 3, 5, 79, 139$
Class number $20155392$ (GRH)
Class group $[3, 3, 3, 3, 3, 288, 288]$ (GRH)
Galois group $S_3$ (as 6T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![48233383641, -658863, -658854, 439233, 6, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 3*x^5 + 6*x^4 + 439233*x^3 - 658854*x^2 - 658863*x + 48233383641)
 
gp: K = bnfinit(x^6 - 3*x^5 + 6*x^4 + 439233*x^3 - 658854*x^2 - 658863*x + 48233383641, 1)
 

Normalized defining polynomial

\( x^{6} - 3 x^{5} + 6 x^{4} + 439233 x^{3} - 658854 x^{2} - 658863 x + 48233383641 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $6$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-317992113503391600270000=-\,2^{4}\cdot 3^{7}\cdot 5^{4}\cdot 79^{4}\cdot 139^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $8261.70$
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:  $2, 3, 5, 79, 139$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{18} a^{4} + \frac{1}{6} a^{2} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{289397666394} a^{5} - \frac{8038714259}{289397666394} a^{4} - \frac{73205}{96465888798} a^{3} - \frac{32155003441}{96465888798} a^{2} - \frac{24116728421}{96465888798} a - \frac{73207}{439238}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{288}\times C_{288}$, which has order $20155392$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $2$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{2}{144698833197} a^{5} - \frac{73205}{48232944399} a^{4} + \frac{146410}{48232944399} a^{3} - \frac{585650}{48232944399} a^{2} - \frac{5359045230}{16077648133} a + \frac{146414}{219619} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  \( \frac{659219}{144698833197} a^{5} + \frac{12591248}{48232944399} a^{4} - \frac{25182496}{48232944399} a^{3} + \frac{48336970478}{48232944399} a^{2} + \frac{921754901088}{16077648133} a - \frac{27159791}{219619} \),  \( \frac{368}{144698833197} a^{5} + \frac{13469720}{48232944399} a^{4} - \frac{26939440}{48232944399} a^{3} + \frac{107759600}{48232944399} a^{2} + \frac{1034297266719}{16077648133} a - \frac{26940176}{219619} \) (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 396.44689226796163 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3$ (as 6T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 6
The 3 conjugacy class representatives for $S_3$
Character table for $S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.325572374700.1 x3

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

Sibling algebras

Twin sextic algebra: data not computed
Degree 3 sibling: 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 R R R ${\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$2$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$3$3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
$5$5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$79$79.3.2.1$x^{3} - 79$$3$$1$$2$$C_3$$[\ ]_{3}$
79.3.2.1$x^{3} - 79$$3$$1$$2$$C_3$$[\ ]_{3}$
$139$139.3.2.3$x^{3} - 2224$$3$$1$$2$$C_3$$[\ ]_{3}$
139.3.2.3$x^{3} - 2224$$3$$1$$2$$C_3$$[\ ]_{3}$