# Properties

 Label 6.2.5611284433.1 Degree $6$ Signature $[2, 2]$ Discriminant $5611284433$ Root discriminant $42.15$ Ramified prime $1777$ Class number $2$ Class group $[2]$ 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 - 3*x^5 + 8*x^4 + 45*x^3 - 211*x^2 + 293*x - 121)

gp: K = bnfinit(x^6 - 3*x^5 + 8*x^4 + 45*x^3 - 211*x^2 + 293*x - 121, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-121, 293, -211, 45, 8, -3, 1]);

$$x^{6} - 3 x^{5} + 8 x^{4} + 45 x^{3} - 211 x^{2} + 293 x - 121$$

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: $$5611284433$$$$\medspace = 1777^{3}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $42.15$ 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: $1777$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $\card{ \Aut(K/\Q) }$: $1$ This field is not Galois over $\Q$. This is not a CM field.

## Integral basis (with respect to field generator $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2147} a^{5} + \frac{13}{113} a^{4} - \frac{505}{2147} a^{3} + \frac{468}{2147} a^{2} + \frac{851}{2147} a + \frac{490}{2147}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

$C_{2}$, which has order $2$

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: $$\frac{3374}{2147} a^{5} - \frac{321}{113} a^{4} + \frac{20171}{2147} a^{3} + \frac{174894}{2147} a^{2} - \frac{501663}{2147} a + \frac{412294}{2147}$$,  $$\frac{34}{2147} a^{5} - \frac{10}{113} a^{4} + \frac{6}{2147} a^{3} + \frac{883}{2147} a^{2} - \frac{11859}{2147} a + \frac{5925}{2147}$$,  $$\frac{717}{2147} a^{5} - \frac{58}{113} a^{4} + \frac{2905}{2147} a^{3} + \frac{41417}{2147} a^{2} - \frac{115519}{2147} a + \frac{61485}{2147}$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$513.537962715$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{2}\cdot 513.537962715 \cdot 2}{2\sqrt{5611284433}}\approx 1.08258284342$

## 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: $$\Q$$ $\times$ 5.1.1777.1 Degree 5 sibling: 5.1.1777.1 Degree 10 siblings: 10.2.17718915581332657.1, 10.2.5611284433.1 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 ${\href{/padicField/2.5.0.1}{5} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ ${\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ ${\href{/padicField/5.6.0.1}{6} }$ ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.3.0.1}{3} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{3}$ ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.6.0.1}{6} }$ ${\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
$1777$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$1$ $1$ $$\Q$$ $C_1$ $1$ $1$
1.1777.2t1.a.a$1$ $1777$ $$\Q(\sqrt{1777})$$ $C_2$ (as 2T1) $1$ $1$
4.5611284433.10t12.b.a$4$ $1777^{3}$ 6.2.5611284433.1 $\PGL(2,5)$ (as 6T14) $1$ $0$
4.1777.5t5.b.a$4$ $1777$ 6.2.5611284433.1 $\PGL(2,5)$ (as 6T14) $1$ $0$
5.3157729.10t13.b.a$5$ $1777^{2}$ 6.2.5611284433.1 $\PGL(2,5)$ (as 6T14) $1$ $1$
* 5.5611284433.6t14.b.a$5$ $1777^{3}$ 6.2.5611284433.1 $\PGL(2,5)$ (as 6T14) $1$ $1$
6.5611284433.20t30.b.a$6$ $1777^{3}$ 6.2.5611284433.1 $\PGL(2,5)$ (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 *.