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Properties

Label 6.0.666962499705...1375.1
Degree $6$
Signature $[0, 3]$
Discriminant $-\,3^{8}\cdot 5^{3}\cdot 7^{5}\cdot 13^{5}\cdot 19^{4}$
Root discriminant $2955.86$
Ramified primes $3, 5, 7, 13, 19$
Class number $35823060$ (GRH)
Class group $[3, 3, 3, 3, 3, 18, 8190]$ (GRH)
Galois group $C_6$ (as 6T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![22520129696, 1540417020, 27354072, -263493, -10029, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 3*x^5 - 10029*x^4 - 263493*x^3 + 27354072*x^2 + 1540417020*x + 22520129696)
 
gp: K = bnfinit(x^6 - 3*x^5 - 10029*x^4 - 263493*x^3 + 27354072*x^2 + 1540417020*x + 22520129696, 1)
 

Normalized defining polynomial

\( x^{6} - 3 x^{5} - 10029 x^{4} - 263493 x^{3} + 27354072 x^{2} + 1540417020 x + 22520129696 \)

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:  \(-666962499705409191375=-\,3^{8}\cdot 5^{3}\cdot 7^{5}\cdot 13^{5}\cdot 19^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2955.86$
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:  $3, 5, 7, 13, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(77805=3^{2}\cdot 5\cdot 7\cdot 13\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{77805}(1,·)$, $\chi_{77805}(4111,·)$, $\chi_{77805}(16636,·)$, $\chi_{77805}(41389,·)$, $\chi_{77805}(50959,·)$, $\chi_{77805}(68449,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{364} a^{3} - \frac{47}{364} a^{2} - \frac{11}{182} a - \frac{37}{91}$, $\frac{1}{1456} a^{4} + \frac{317}{1456} a^{2} + \frac{137}{728} a + \frac{43}{91}$, $\frac{1}{726430432} a^{5} - \frac{4025}{25943944} a^{4} + \frac{882361}{726430432} a^{3} + \frac{71567201}{363215216} a^{2} + \frac{1251869}{3242993} a + \frac{7908120}{22700951}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{18}\times C_{8190}$, which has order $35823060$ (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:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  \( \frac{93}{181607608} a^{5} - \frac{267}{12971972} a^{4} - \frac{761479}{181607608} a^{3} + \frac{469110}{22700951} a^{2} + \frac{36465109}{3492454} a + \frac{5170754675}{22700951} \),  \( \frac{201}{27939632} a^{5} - \frac{7570}{22700951} a^{4} - \frac{3051845}{51887888} a^{3} + \frac{134360035}{181607608} a^{2} + \frac{7780081885}{45401902} a + \frac{76122904861}{22700951} \) (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 212.1071224368399 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6$ (as 6T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 6
The 6 conjugacy class representatives for $C_6$
Character table for $C_6$

Intermediate fields

\(\Q(\sqrt{-455}) \), 3.3.242144721.4

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.1.0.1}{1} }^{6}$ R R R ${\href{/LocalNumberField/11.6.0.1}{6} }$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$3$3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.6.5.6$x^{6} + 224$$6$$1$$5$$C_6$$[\ ]_{6}$
$13$13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$
$19$19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$