Properties

 Label 6.2.284765625.3 Degree $6$ Signature $[2, 2]$ Discriminant $3^{6}\cdot 5^{8}$ Root discriminant $25.65$ Ramified primes $3, 5$ Class number $1$ Class group Trivial Galois group $A_6$ (as 6T15)

Related objects

Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 5*x^3 + 45*x^2 - 99*x - 15)

gp: K = bnfinit(x^6 - 5*x^3 + 45*x^2 - 99*x - 15, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-15, -99, 45, -5, 0, 0, 1]);

Normalizeddefining polynomial

$$x^{6} - 5 x^{3} + 45 x^{2} - 99 x - 15$$

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: $$284765625=3^{6}\cdot 5^{8}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $25.65$ 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: $3, 5$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{4029} a^{5} + \frac{809}{4029} a^{4} + \frac{1783}{4029} a^{3} + \frac{20}{1343} a^{2} + \frac{1}{17} a - \frac{586}{1343}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

Class group and class number

Trivial group, which has order $1$

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: $$\frac{10}{1343} a^{5} + \frac{32}{1343} a^{4} + \frac{371}{1343} a^{3} + \frac{600}{1343} a^{2} - \frac{4}{17} a - \frac{6836}{1343}$$,  $$\frac{475}{4029} a^{5} + \frac{1520}{4029} a^{4} + \frac{835}{4029} a^{3} - \frac{2587}{1343} a^{2} - \frac{35}{17} a - \frac{349}{1343}$$,  $$\frac{45}{1343} a^{5} + \frac{144}{1343} a^{4} - \frac{345}{1343} a^{3} + \frac{14}{1343} a^{2} + \frac{50}{17} a - \frac{9274}{1343}$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$307.511755298$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

Galois group

$A_6$ (as 6T15):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A non-solvable group of order 360 The 7 conjugacy class representatives for $A_6$ Character table for $A_6$

Intermediate fields

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

Sibling algebras

 Twin sextic algebra: 6.2.23066015625.4 Degree 6 sibling: 6.2.23066015625.4 Degree 10 sibling: 10.2.6568408355712890625.2 Degree 15 siblings: Deg 15, Deg 15 Degree 20 sibling: Deg 20 Degree 30 siblings: data not computed Degree 36 sibling: data not computed Degree 40 sibling: data not computed Degree 45 sibling: data not computed

Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 ${\href{/LocalNumberField/2.4.0.1}{4} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$3$$\Q_{3}$$x + 1$$1$$1$$0Trivial[\ ] 3.2.1.2x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.3.5.2$x^{3} + 21$$3$$1$$5$$S_3$$[5/2]_{2} 5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.5.8.1$x^{5} - 5 x^{4} + 105$$5$$1$$8$$C_5$$[2]$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
5.23066015625.6t15.c.a$5$ $3^{10} \cdot 5^{8}$ $x^{6} - 5 x^{3} + 45 x^{2} - 99 x - 15$ $A_6$ (as 6T15) $1$ $1$
* 5.284765625.6t15.b.a$5$ $3^{6} \cdot 5^{8}$ $x^{6} - 5 x^{3} + 45 x^{2} - 99 x - 15$ $A_6$ (as 6T15) $1$ $1$
8.10509453369140625.36t555.b.a$8$ $3^{16} \cdot 5^{12}$ $x^{6} - 5 x^{3} + 45 x^{2} - 99 x - 15$ $A_6$ (as 6T15) $1$ $0$
8.10509453369140625.36t555.b.b$8$ $3^{16} \cdot 5^{12}$ $x^{6} - 5 x^{3} + 45 x^{2} - 99 x - 15$ $A_6$ (as 6T15) $1$ $0$
9.6568408355712890625.10t26.b.a$9$ $3^{16} \cdot 5^{16}$ $x^{6} - 5 x^{3} + 45 x^{2} - 99 x - 15$ $A_6$ (as 6T15) $1$ $1$
10.59115675201416015625.30t88.b.a$10$ $3^{18} \cdot 5^{16}$ $x^{6} - 5 x^{3} + 45 x^{2} - 99 x - 15$ $A_6$ (as 6T15) $1$ $-2$

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 *.