Properties

Label 6.2.33792250337.1
Degree $6$
Signature $[2, 2]$
Discriminant $33792250337$
Root discriminant $56.86$
Ramified primes $53, 61$
Class number $4$
Class group $[2, 2]$
Galois group $\PGL(2,5)$ (as 6T14)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 - 19*x^4 + 58*x^3 + 220*x^2 - 112*x - 741)
 
gp: K = bnfinit(x^6 - x^5 - 19*x^4 + 58*x^3 + 220*x^2 - 112*x - 741, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-741, -112, 220, 58, -19, -1, 1]);
 

\( x^{6} - x^{5} - 19 x^{4} + 58 x^{3} + 220 x^{2} - 112 x - 741 \)

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

Invariants

Degree:  $6$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[2, 2]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(33792250337\)\(\medspace = 53^{3}\cdot 61^{3}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $56.86$
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:  $53, 61$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{9057} a^{5} - \frac{363}{3019} a^{4} + \frac{455}{3019} a^{3} - \frac{911}{3019} a^{2} - \frac{3010}{9057} a + \frac{724}{3019}$

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

Class group and class number

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

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $3$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  \( \frac{122}{9057} a^{5} - \frac{22}{9057} a^{4} - \frac{2534}{9057} a^{3} + \frac{7721}{9057} a^{2} + \frac{34307}{9057} a - \frac{5261}{3019} \),  \( \frac{172}{9057} a^{5} - \frac{3149}{9057} a^{4} + \frac{5336}{9057} a^{3} + \frac{31078}{9057} a^{2} - \frac{113174}{9057} a - \frac{104916}{3019} \),  \( \frac{155}{3019} a^{5} + \frac{269}{3019} a^{4} - \frac{5793}{3019} a^{3} + \frac{11121}{3019} a^{2} + \frac{79889}{3019} a - \frac{158456}{3019} \)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 790.648107964 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{2}\cdot 790.648107964 \cdot 4}{2\sqrt{33792250337}}\approx 1.35838977168$

Galois group

$S_5$ (as 6T14):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 120
The 7 conjugacy class representatives for $\PGL(2,5)$
Character table for $\PGL(2,5)$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling algebras

Twin sextic algebra: \(\Q\) $\times$ 5.1.3233.1
Degree 5 sibling: 5.1.3233.1
Degree 10 siblings: data not computed
Degree 12 sibling: data not computed
Degree 15 sibling: data not computed
Degree 20 siblings: data not computed
Degree 24 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 sibling: data not computed

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.5.0.1}{5} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ R ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$53$53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$61$61.6.3.2$x^{6} - 3721 x^{2} + 2269810$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.3233.2t1.a.a$1$ $ 53 \cdot 61 $ $x^{2} - x - 808$ $C_2$ (as 2T1) $1$ $1$
4.33792250337.10t12.b.a$4$ $ 53^{3} \cdot 61^{3}$ $x^{6} - x^{5} - 19 x^{4} + 58 x^{3} + 220 x^{2} - 112 x - 741$ $\PGL(2,5)$ (as 6T14) $1$ $0$
4.3233.5t5.b.a$4$ $ 53 \cdot 61 $ $x^{6} - x^{5} - 19 x^{4} + 58 x^{3} + 220 x^{2} - 112 x - 741$ $\PGL(2,5)$ (as 6T14) $1$ $0$
5.10452289.10t13.b.a$5$ $ 53^{2} \cdot 61^{2}$ $x^{6} - x^{5} - 19 x^{4} + 58 x^{3} + 220 x^{2} - 112 x - 741$ $\PGL(2,5)$ (as 6T14) $1$ $1$
* 5.33792250337.6t14.b.a$5$ $ 53^{3} \cdot 61^{3}$ $x^{6} - x^{5} - 19 x^{4} + 58 x^{3} + 220 x^{2} - 112 x - 741$ $\PGL(2,5)$ (as 6T14) $1$ $1$
6.33792250337.20t30.b.a$6$ $ 53^{3} \cdot 61^{3}$ $x^{6} - x^{5} - 19 x^{4} + 58 x^{3} + 220 x^{2} - 112 x - 741$ $\PGL(2,5)$ (as 6T14) $1$ $-2$

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 *.