# Properties

 Label 6.0.6417228161287.1 Degree $6$ Signature $[0, 3]$ Discriminant $-6.417\times 10^{12}$ Root discriminant $136.32$ Ramified prime $18583$ Class number $8$ Class group $[8]$ Galois group $\PGL(2,5)$ (as 6T14)

# Related objects

Show commands: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 + 92*x^4 + 179*x^3 + 1121*x^2 + 1315*x + 22587)

gp: K = bnfinit(x^6 - x^5 + 92*x^4 + 179*x^3 + 1121*x^2 + 1315*x + 22587, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![22587, 1315, 1121, 179, 92, -1, 1]);

$$x^{6} - x^{5} + 92 x^{4} + 179 x^{3} + 1121 x^{2} + 1315 x + 22587$$

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: $$-6417228161287$$$$\medspace = -\,18583^{3}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $136.32$ 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: $18583$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{7} a^{4} - \frac{2}{7} a^{3} + \frac{1}{7} a^{2} - \frac{1}{7}$, $\frac{1}{3321073} a^{5} - \frac{111417}{3321073} a^{4} - \frac{76330}{474439} a^{3} + \frac{449614}{3321073} a^{2} + \frac{872829}{3321073} a + \frac{545037}{3321073}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

$C_{8}$, 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 $2$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: $$\frac{8552961591}{3321073} a^{5} - \frac{92288508993}{3321073} a^{4} - \frac{20807962621}{474439} a^{3} - \frac{976002877945}{3321073} a^{2} - \frac{471649830672}{3321073} a - \frac{28364787854161}{3321073}$$,  $$\frac{6148931112351}{3321073} a^{5} - \frac{242923371529764}{3321073} a^{4} - \frac{9845954436480}{67777} a^{3} - \frac{2586191877110454}{3321073} a^{2} - \frac{4590852063451212}{3321073} a - \frac{65695269732525758}{3321073}$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$4166.22435756$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{3}\cdot 4166.22435756 \cdot 8}{2\sqrt{6417228161287}}\approx 1.63180594013$

## 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: $$\Q$$ $\times$ 5.3.18583.1 Degree 5 sibling: 5.3.18583.1 Degree 10 siblings: 10.4.6417228161287.1, 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 ${\href{/padicField/2.5.0.1}{5} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ ${\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ ${\href{/padicField/5.6.0.1}{6} }$ ${\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.6.0.1}{6} }$ ${\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.3.0.1}{3} }^{2}$ ${\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.6.0.1}{6} }$ ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{3}$

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
$18583$Data not computed