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Properties

Label 6.0.32291634881943.2
Degree $6$
Signature $[0, 3]$
Discriminant $-\,3^{3}\cdot 31^{4}\cdot 109^{3}$
Root discriminant $178.45$
Ramified primes $3, 31, 109$
Class number $13104$
Class group $[2, 2, 3276]$
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![693376, -4376, 20198, -159, 225, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 + 225*x^4 - 159*x^3 + 20198*x^2 - 4376*x + 693376)
 
gp: K = bnfinit(x^6 - x^5 + 225*x^4 - 159*x^3 + 20198*x^2 - 4376*x + 693376, 1)
 

Normalized defining polynomial

\( x^{6} - x^{5} + 225 x^{4} - 159 x^{3} + 20198 x^{2} - 4376 x + 693376 \)

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:  \(-32291634881943=-\,3^{3}\cdot 31^{4}\cdot 109^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $178.45$
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:  $3, 31, 109$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(10137=3\cdot 31\cdot 109\)
Dirichlet character group:    $\lbrace$$\chi_{10137}(2288,·)$, $\chi_{10137}(1,·)$, $\chi_{10137}(5233,·)$, $\chi_{10137}(7193,·)$, $\chi_{10137}(1307,·)$, $\chi_{10137}(4252,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{5239184} a^{5} - \frac{159195}{2619592} a^{4} + \frac{212515}{5239184} a^{3} - \frac{498363}{2619592} a^{2} - \frac{159277}{327449} a - \frac{145875}{327449}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{3276}$, which has order $13104$

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:  \( \frac{145}{1309796} a^{5} + \frac{3759}{1309796} a^{4} + \frac{34469}{1309796} a^{3} + \frac{534637}{1309796} a^{2} + \frac{902263}{654898} a + \frac{5440975}{327449} \),  \( \frac{179}{2619592} a^{5} - \frac{3921}{654898} a^{4} + \frac{56101}{2619592} a^{3} - \frac{280799}{327449} a^{2} + \frac{892267}{654898} a - \frac{15876411}{327449} \)
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

\(\Q(\sqrt{-327}) \), 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: \(\Q\) $\times$ \(\Q(\sqrt{-327}) \) $\times$ 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 ${\href{/LocalNumberField/5.6.0.1}{6} }$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/37.6.0.1}{6} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$31$31.3.2.1$x^{3} - 31$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.1$x^{3} - 31$$3$$1$$2$$C_3$$[\ ]_{3}$
$109$109.2.1.1$x^{2} - 109$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.1$x^{2} - 109$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.1$x^{2} - 109$$2$$1$$1$$C_2$$[\ ]_{2}$