Properties

Label 6.2.18927429625.2
Degree $6$
Signature $[2, 2]$
Discriminant $18927429625$
Root discriminant $51.62$
Ramified primes $5, 13, 41$
Class number $8$
Class group $[2, 2, 2]$
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 - 17*x^4 + 117*x^3 - 188*x^2 + 23*x + 594)
 
gp: K = bnfinit(x^6 - x^5 - 17*x^4 + 117*x^3 - 188*x^2 + 23*x + 594, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![594, 23, -188, 117, -17, -1, 1]);
 

\(x^{6} - x^{5} - 17 x^{4} + 117 x^{3} - 188 x^{2} + 23 x + 594\)  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:  $[2, 2]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(18927429625\)\(\medspace = 5^{3}\cdot 13^{3}\cdot 41^{3}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $51.62$
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:  $5, 13, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{4577} a^{5} + \frac{80}{199} a^{4} + \frac{443}{4577} a^{3} + \frac{974}{4577} a^{2} - \frac{1238}{4577} a + \frac{211}{4577}$  Toggle raw display

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, 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:  $3$
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:  \( \frac{30}{4577} a^{5} + \frac{12}{199} a^{4} - \frac{441}{4577} a^{3} - \frac{2819}{4577} a^{2} + \frac{22361}{4577} a - \frac{25709}{4577} \),  \( \frac{147}{4577} a^{5} + \frac{19}{199} a^{4} - \frac{3534}{4577} a^{3} + \frac{1291}{4577} a^{2} + \frac{28556}{4577} a - \frac{69677}{4577} \),  \( \frac{68}{4577} a^{5} + \frac{67}{199} a^{4} + \frac{2662}{4577} a^{3} - \frac{16154}{4577} a^{2} + \frac{66857}{4577} a + \frac{101311}{4577} \)  Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 215.331906731 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{2}\cdot 215.331906731 \cdot 8}{2\sqrt{18927429625}}\approx 0.988649350436$

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: \(\Q\) $\times$ 5.1.2665.1
Degree 5 sibling: 5.1.2665.1
Degree 10 siblings: Deg 10, 10.2.18927429625.1
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 ${\href{/padicField/2.5.0.1}{5} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ ${\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ R ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ ${\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ R ${\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.2.0.1}{2} }^{3}$ ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.3.0.1}{3} }^{2}$ R ${\href{/padicField/43.6.0.1}{6} }$ ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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
$5$5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$13$13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$41$41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.2665.2t1.a.a$1$ $ 5 \cdot 13 \cdot 41 $ \(\Q(\sqrt{2665}) \) $C_2$ (as 2T1) $1$ $1$
4.18927429625.10t12.a.a$4$ $ 5^{3} \cdot 13^{3} \cdot 41^{3}$ 6.2.18927429625.2 $\PGL(2,5)$ (as 6T14) $1$ $0$
4.2665.5t5.a.a$4$ $ 5 \cdot 13 \cdot 41 $ 6.2.18927429625.2 $\PGL(2,5)$ (as 6T14) $1$ $0$
5.7102225.10t13.a.a$5$ $ 5^{2} \cdot 13^{2} \cdot 41^{2}$ 6.2.18927429625.2 $\PGL(2,5)$ (as 6T14) $1$ $1$
* 5.18927429625.6t14.a.a$5$ $ 5^{3} \cdot 13^{3} \cdot 41^{3}$ 6.2.18927429625.2 $\PGL(2,5)$ (as 6T14) $1$ $1$
6.18927429625.20t30.a.a$6$ $ 5^{3} \cdot 13^{3} \cdot 41^{3}$ 6.2.18927429625.2 $\PGL(2,5)$ (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 *.