Properties

Label 6.0.91794884831.2
Degree $6$
Signature $[0, 3]$
Discriminant $-91794884831$
Root discriminant $67.16$
Ramified primes $13, 347$
Class number $8$
Class group $[2, 4]$
Galois group $\PGL(2,5)$ (as 6T14)

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Show commands: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 + 27*x^3 + 339*x^2 + 631*x + 1044)
 
gp: K = bnfinit(x^6 - x^5 + 27*x^3 + 339*x^2 + 631*x + 1044, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1044, 631, 339, 27, 0, -1, 1]);
 

\(x^{6} - x^{5} + 27 x^{3} + 339 x^{2} + 631 x + 1044\)  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:  $[0, 3]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-91794884831\)\(\medspace = -\,13^{3}\cdot 347^{3}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $67.16$
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:  $13, 347$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{44053} a^{5} - \frac{16411}{44053} a^{4} + \frac{8521}{44053} a^{3} - \frac{5361}{44053} a^{2} + \frac{508}{44053} a - \frac{9632}{44053}$  Toggle raw display

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $2$
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:  \( \frac{4038}{44053} a^{5} - \frac{11906}{44053} a^{4} + \frac{2405}{44053} a^{3} + \frac{158517}{44053} a^{2} + \frac{1082138}{44053} a - \frac{523853}{44053} \),  \( \frac{2189247}{44053} a^{5} + \frac{5991212}{44053} a^{4} - \frac{31695694}{44053} a^{3} + \frac{93968089}{44053} a^{2} + \frac{1022489621}{44053} a + \frac{3276111857}{44053} \)  Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 339.170094186 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{3}\cdot 339.170094186 \cdot 8}{2\sqrt{91794884831}}\approx 1.11072848021$

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: \(\Q\) $\times$ 5.3.4511.1
Degree 5 sibling: 5.3.4511.1
Degree 10 siblings: 10.4.91794884831.1, Deg 10
Degree 12 sibling: Deg 12
Degree 15 sibling: Deg 15
Degree 20 siblings: Deg 20, Deg 20, Deg 20
Degree 24 sibling: Deg 24
Degree 30 siblings: Deg 30, Deg 30, Deg 30
Degree 40 sibling: Deg 40

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.5.0.1}{5} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ ${\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ ${\href{/padicField/5.5.0.1}{5} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ ${\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.6.0.1}{6} }$ R ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.3.0.1}{3} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.3.0.1}{3} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }$

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

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
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$347$Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$2$$2$$2$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.4511.2t1.a.a$1$ $ 13 \cdot 347 $ \(\Q(\sqrt{-4511}) \) $C_2$ (as 2T1) $1$ $-1$
4.91794884831.10t12.b.a$4$ $ 13^{3} \cdot 347^{3}$ 6.0.91794884831.2 $\PGL(2,5)$ (as 6T14) $1$ $-2$
4.4511.5t5.b.a$4$ $ 13 \cdot 347 $ 6.0.91794884831.2 $\PGL(2,5)$ (as 6T14) $1$ $2$
5.20349121.10t13.b.a$5$ $ 13^{2} \cdot 347^{2}$ 6.0.91794884831.2 $\PGL(2,5)$ (as 6T14) $1$ $1$
* 5.91794884831.6t14.b.a$5$ $ 13^{3} \cdot 347^{3}$ 6.0.91794884831.2 $\PGL(2,5)$ (as 6T14) $1$ $-1$
6.91794884831.20t30.b.a$6$ $ 13^{3} \cdot 347^{3}$ 6.0.91794884831.2 $\PGL(2,5)$ (as 6T14) $1$ $0$

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 *.