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Properties

Label 6.0.33141820926375.3
Degree $6$
Signature $[0, 3]$
Discriminant $-\,3^{3}\cdot 5^{3}\cdot 7^{3}\cdot 31^{5}$
Root discriminant $179.22$
Ramified primes $3, 5, 7, 31$
Class number $12520$
Class group $[2, 2, 3130]$
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![9143296, -172520, 173334, -1623, 809, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 + 809*x^4 - 1623*x^3 + 173334*x^2 - 172520*x + 9143296)
 
gp: K = bnfinit(x^6 - x^5 + 809*x^4 - 1623*x^3 + 173334*x^2 - 172520*x + 9143296, 1)
 

Normalized defining polynomial

\( x^{6} - x^{5} + 809 x^{4} - 1623 x^{3} + 173334 x^{2} - 172520 x + 9143296 \)

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:  \(-33141820926375=-\,3^{3}\cdot 5^{3}\cdot 7^{3}\cdot 31^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $179.22$
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:  $3, 5, 7, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3255=3\cdot 5\cdot 7\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{3255}(1,·)$, $\chi_{3255}(211,·)$, $\chi_{3255}(3044,·)$, $\chi_{3255}(3254,·)$, $\chi_{3255}(1049,·)$, $\chi_{3255}(2206,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{52553581112} a^{5} + \frac{1416431373}{26276790556} a^{4} + \frac{7948836035}{52553581112} a^{3} - \frac{2380928129}{26276790556} a^{2} - \frac{561217117}{13138395278} a - \frac{1556552194}{6569197639}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{3130}$, which has order $12520$

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:  \( \frac{32807}{13138395278} a^{5} + \frac{9020539}{26276790556} a^{4} + \frac{12562431}{6569197639} a^{3} + \frac{6310791345}{26276790556} a^{2} - \frac{125098847}{13138395278} a + \frac{229321990199}{6569197639} \),  \( \frac{33745}{26276790556} a^{5} - \frac{5339479}{13138395278} a^{4} - \frac{5994573}{26276790556} a^{3} - \frac{3132588135}{13138395278} a^{2} + \frac{725180072}{6569197639} a - \frac{108775290035}{6569197639} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 48.7831276503 \)
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

\(\Q(\sqrt{-3255}) \), 3.3.961.1

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

Sibling algebras

Twin sextic algebra: \(\Q\) $\times$ \(\Q(\sqrt{-3255}) \) $\times$ 3.3.961.1

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 ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ R ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$31$31.6.5.1$x^{6} - 31$$6$$1$$5$$C_6$$[\ ]_{6}$