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Properties

Label 6.0.373837707.1
Degree $6$
Signature $[0, 3]$
Discriminant $-\,3^{3}\cdot 61^{4}$
Root discriminant $26.84$
Ramified primes $3, 61$
Class number $4$
Class group $[2, 2]$
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![81, -180, 409, 2, 21, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 + 21*x^4 + 2*x^3 + 409*x^2 - 180*x + 81)
 
gp: K = bnfinit(x^6 - x^5 + 21*x^4 + 2*x^3 + 409*x^2 - 180*x + 81, 1)
 

Normalized defining polynomial

\( x^{6} - x^{5} + 21 x^{4} + 2 x^{3} + 409 x^{2} - 180 x + 81 \)

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:  \(-373837707=-\,3^{3}\cdot 61^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.84$
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:  $3, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(183=3\cdot 61\)
Dirichlet character group:    $\lbrace$$\chi_{183}(1,·)$, $\chi_{183}(169,·)$, $\chi_{183}(74,·)$, $\chi_{183}(13,·)$, $\chi_{183}(62,·)$, $\chi_{183}(47,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{4} - \frac{1}{9} a$, $\frac{1}{25227} a^{5} - \frac{7}{8409} a^{4} + \frac{49}{2803} a^{3} + \frac{8000}{25227} a^{2} + \frac{2863}{8409} a - \frac{420}{2803}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

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:  \( \frac{140}{25227} a^{5} - \frac{137}{25227} a^{4} + \frac{959}{8409} a^{3} + \frac{1603}{25227} a^{2} + \frac{56033}{25227} a + \frac{63}{2803} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  \( \frac{125}{25227} a^{5} + \frac{178}{25227} a^{4} - \frac{1246}{8409} a^{3} + \frac{32965}{25227} a^{2} - \frac{72802}{25227} a + \frac{3560}{2803} \),  \( \frac{5603}{25227} a^{5} - \frac{5543}{25227} a^{4} + \frac{38801}{8409} a^{3} + \frac{12439}{25227} a^{2} + \frac{2267087}{25227} a - \frac{110860}{2803} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 54.8341682926 \)
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

\(\Q(\sqrt{-3}) \), 3.3.3721.1

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

Sibling algebras

Twin sextic algebra: 3.3.3721.1 $\times$ \(\Q(\sqrt{-3}) \) $\times$ \(\Q\)

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }$ R ${\href{/LocalNumberField/5.6.0.1}{6} }$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }$

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$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$61$61.3.2.1$x^{3} - 61$$3$$1$$2$$C_3$$[\ ]_{3}$
61.3.2.1$x^{3} - 61$$3$$1$$2$$C_3$$[\ ]_{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.3.2t1.1c1$1$ $ 3 $ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
* 1.61.3t1.1c1$1$ $ 61 $ $x^{3} - x^{2} - 20 x + 9$ $C_3$ (as 3T1) $0$ $1$
* 1.3_61.6t1.1c1$1$ $ 3 \cdot 61 $ $x^{6} - x^{5} + 21 x^{4} + 2 x^{3} + 409 x^{2} - 180 x + 81$ $C_6$ (as 6T1) $0$ $-1$
* 1.61.3t1.1c2$1$ $ 61 $ $x^{3} - x^{2} - 20 x + 9$ $C_3$ (as 3T1) $0$ $1$
* 1.3_61.6t1.1c2$1$ $ 3 \cdot 61 $ $x^{6} - x^{5} + 21 x^{4} + 2 x^{3} + 409 x^{2} - 180 x + 81$ $C_6$ (as 6T1) $0$ $-1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.