Properties

Label 6.6.48737056617.2
Degree $6$
Signature $[6, 0]$
Discriminant $48737056617$
Root discriminant $60.44$
Ramified primes $3, 19$
Class number $3$
Class group $[3]$
Galois group $C_6$ (as 6T1)

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Show commands: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 57*x^4 - 76*x^3 + 684*x^2 + 1824*x + 1216)
 
gp: K = bnfinit(x^6 - 57*x^4 - 76*x^3 + 684*x^2 + 1824*x + 1216, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1216, 1824, 684, -76, -57, 0, 1]);
 

\(x^{6} - 57 x^{4} - 76 x^{3} + 684 x^{2} + 1824 x + 1216\)  Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $6$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[6, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(48737056617\)\(\medspace = 3^{9}\cdot 19^{5}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $60.44$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $3, 19$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $6$
This field is Galois and abelian over $\Q$.
Conductor:  \(171=3^{2}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{171}(1,·)$, $\chi_{171}(164,·)$, $\chi_{171}(49,·)$, $\chi_{171}(7,·)$, $\chi_{171}(170,·)$, $\chi_{171}(122,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{4} - \frac{1}{8} a^{3} + \frac{3}{16} a^{2} - \frac{1}{8} a - \frac{1}{2}$, $\frac{1}{64} a^{5} - \frac{1}{64} a^{3} + \frac{1}{16} a^{2} + \frac{5}{16} a - \frac{1}{4}$  Toggle raw display

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

Class group and class number

$C_{3}$, which has order $3$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $5$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 33830.112687 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{6}\cdot(2\pi)^{0}\cdot 33830.112687 \cdot 3}{2\sqrt{48737056617}}\approx 14.711095975$

Galois group

$C_6$ (as 6T1):

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

Intermediate fields

\(\Q(\sqrt{57}) \), 3.3.29241.1

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

Sibling algebras

Twin sextic algebra: 3.3.29241.1 $\times$ \(\Q(\sqrt{57}) \) $\times$ \(\Q\)

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.1.0.1}{1} }^{6}$ R ${\href{/padicField/5.6.0.1}{6} }$ ${\href{/padicField/7.3.0.1}{3} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }$ ${\href{/padicField/13.2.0.1}{2} }^{3}$ ${\href{/padicField/17.6.0.1}{6} }$ R ${\href{/padicField/23.2.0.1}{2} }^{3}$ ${\href{/padicField/29.3.0.1}{3} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }$ ${\href{/padicField/37.2.0.1}{2} }^{3}$ ${\href{/padicField/41.3.0.1}{3} }^{2}$ ${\href{/padicField/43.1.0.1}{1} }^{6}$ ${\href{/padicField/47.6.0.1}{6} }$ ${\href{/padicField/53.3.0.1}{3} }^{2}$ ${\href{/padicField/59.3.0.1}{3} }^{2}$

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

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.6.9.1$x^{6} + 3 x^{4} + 15$$6$$1$$9$$C_6$$[2]_{2}$
$19$19.6.5.5$x^{6} + 1216$$6$$1$$5$$C_6$$[\ ]_{6}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.57.2t1.a.a$1$ $ 3 \cdot 19 $ \(\Q(\sqrt{57}) \) $C_2$ (as 2T1) $1$ $1$
* 1.171.3t1.a.a$1$ $ 3^{2} \cdot 19 $ 3.3.29241.1 $C_3$ (as 3T1) $0$ $1$
* 1.171.6t1.e.a$1$ $ 3^{2} \cdot 19 $ 6.6.48737056617.2 $C_6$ (as 6T1) $0$ $1$
* 1.171.3t1.a.b$1$ $ 3^{2} \cdot 19 $ 3.3.29241.1 $C_3$ (as 3T1) $0$ $1$
* 1.171.6t1.e.b$1$ $ 3^{2} \cdot 19 $ 6.6.48737056617.2 $C_6$ (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 *.