Properties

 Label 6.2.4950625.1 Degree $6$ Signature $[2, 2]$ Discriminant $4950625$ Root discriminant $13.05$ Ramified primes $5, 89$ Class number $1$ Class group trivial Galois group $C_3^2:C_4$ (as 6T10)

Related objects

Show commands for: SageMath / Pari/GP / Magma

Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 + 4*x^4 - 10*x^3 + 15*x - 5)

gp: K = bnfinit(x^6 - x^5 + 4*x^4 - 10*x^3 + 15*x - 5, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5, 15, 0, -10, 4, -1, 1]);

$$x^{6} - x^{5} + 4 x^{4} - 10 x^{3} + 15 x - 5$$

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: $$4950625$$$$\medspace = 5^{4}\cdot 89^{2}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $13.05$ 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: $5, 89$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{86} a^{5} - \frac{6}{43} a^{4} - \frac{18}{43} a^{3} + \frac{21}{43} a^{2} - \frac{16}{43} a + \frac{23}{86}$

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: $$\frac{15}{86} a^{5} - \frac{4}{43} a^{4} + \frac{31}{43} a^{3} - \frac{72}{43} a^{2} - \frac{25}{43} a + \frac{173}{86}$$,  $$\frac{17}{43} a^{5} + \frac{11}{43} a^{4} + \frac{76}{43} a^{3} - \frac{60}{43} a^{2} - \frac{157}{43} a + \frac{4}{43}$$,  $$\frac{7}{86} a^{5} + \frac{1}{43} a^{4} + \frac{3}{43} a^{3} - \frac{25}{43} a^{2} + \frac{17}{43} a - \frac{11}{86}$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$13.5880018727$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{2}\cdot 13.5880018727 \cdot 1}{2\sqrt{4950625}}\approx 0.482186797608$

Galois group

$C_3:S_3.C_2$ (as 6T10):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A solvable group of order 36 The 6 conjugacy class representatives for $C_3^2:C_4$ Character table for $C_3^2:C_4$

Intermediate fields

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling algebras

 Galois closure: data not computed Twin sextic algebra: 6.2.39213900625.1 Degree 6 sibling: 6.2.39213900625.1 Degree 9 sibling: 9.1.7765332671265625.2 Degree 12 siblings: Deg 12, Deg 12 Degree 18 sibling: Deg 18

Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ R ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2} 5.4.3.2x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$89$$\Q_{89}$$x + 3$$1$$1$$0Trivial[\ ] 89.2.0.1x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.3.2.1$x^{3} - 89$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.5.2t1.a.a$1$ $5$ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
1.5.4t1.a.a$1$ $5$ $x^{4} - x^{3} + x^{2} - x + 1$ $C_4$ (as 4T1) $0$ $-1$
1.5.4t1.a.b$1$ $5$ $x^{4} - x^{3} + x^{2} - x + 1$ $C_4$ (as 4T1) $0$ $-1$
* 4.990125.6t10.a.a$4$ $5^{3} \cdot 89^{2}$ $x^{6} - x^{5} + 4 x^{4} - 10 x^{3} + 15 x - 5$ $C_3^2:C_4$ (as 6T10) $1$ $0$
4.7842780125.6t10.a.a$4$ $5^{3} \cdot 89^{4}$ $x^{6} - x^{5} + 4 x^{4} - 10 x^{3} + 15 x - 5$ $C_3^2:C_4$ (as 6T10) $1$ $0$

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 *.