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Properties

Label 6.0.41836770602991.2
Degree $6$
Signature $[0, 3]$
Discriminant $-\,3^{3}\cdot 7^{3}\cdot 13^{5}\cdot 23^{3}$
Root discriminant $186.32$
Ramified primes $3, 7, 13, 23$
Class number $11144$ (GRH)
Class group $[2, 2, 2786]$ (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![23220625, 1570, 377526, -1569, 1568, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 + 1568*x^4 - 1569*x^3 + 377526*x^2 + 1570*x + 23220625)
 
gp: K = bnfinit(x^6 - x^5 + 1568*x^4 - 1569*x^3 + 377526*x^2 + 1570*x + 23220625, 1)
 

Normalized defining polynomial

\( x^{6} - x^{5} + 1568 x^{4} - 1569 x^{3} + 377526 x^{2} + 1570 x + 23220625 \)

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:  \(-41836770602991=-\,3^{3}\cdot 7^{3}\cdot 13^{5}\cdot 23^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $186.32$
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:  $3, 7, 13, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(6279=3\cdot 7\cdot 13\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{6279}(1,·)$, $\chi_{6279}(5795,·)$, $\chi_{6279}(484,·)$, $\chi_{6279}(6278,·)$, $\chi_{6279}(4346,·)$, $\chi_{6279}(1933,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{10} a^{4} - \frac{3}{10} a^{3} - \frac{1}{10} a^{2} - \frac{1}{5} a - \frac{1}{2}$, $\frac{1}{10165953270100} a^{5} + \frac{74331872098}{2541488317525} a^{4} - \frac{643783366754}{2541488317525} a^{3} - \frac{4137821249137}{10165953270100} a^{2} - \frac{101638234887}{406638130804} a - \frac{199646311221}{406638130804}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2786}$, which has order $11144$ (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:  \( \frac{3885661}{10165953270100} a^{5} + \frac{27731143}{2541488317525} a^{4} + \frac{1384868486}{2541488317525} a^{3} + \frac{127030440483}{10165953270100} a^{2} + \frac{127787664901}{2033190654020} a + \frac{315802197663}{406638130804} \),  \( \frac{5228289}{5082976635050} a^{5} + \frac{101767413}{5082976635050} a^{4} + \frac{7422024451}{5082976635050} a^{3} + \frac{77895820791}{2541488317525} a^{2} + \frac{33938352651}{203319065402} a + \frac{536363129766}{101659532701} \) (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 5.46019947038 \) (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

\(\Q(\sqrt{-6279}) \), 3.3.169.1

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

Sibling algebras

Twin sextic algebra: \(\Q\) $\times$ \(\Q(\sqrt{-6279}) \) $\times$ 3.3.169.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/5.1.0.1}{1} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }$ R ${\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }$ R ${\href{/LocalNumberField/29.6.0.1}{6} }$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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$$[\ ]_{2}^{3}$
$7$7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$13$13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$
$23$23.6.3.1$x^{6} - 46 x^{4} + 529 x^{2} - 194672$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$