# Properties

 Label 6.0.25969216.3 Degree $6$ Signature $[0, 3]$ Discriminant $-25969216$ Root discriminant $$17.21$$ Ramified primes see page Class number $3$ Class group $[3]$ Galois group $S_3\times C_3$ (as 6T5)

# Related objects

Show commands: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^5 - 12*x^4 + 46*x^3 + 23*x^2 - 240*x + 225)

gp: K = bnfinit(x^6 - 2*x^5 - 12*x^4 + 46*x^3 + 23*x^2 - 240*x + 225, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![225, -240, 23, 46, -12, -2, 1]);

$$x^{6} - 2x^{5} - 12x^{4} + 46x^{3} + 23x^{2} - 240x + 225$$

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: $$-25969216$$ -25969216 $$\medspace = -\,2^{6}\cdot 7^{4}\cdot 13^{2}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $$17.21$$ 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: $$2$$, $$7$$, $$13$$ 2, 7, 13 sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $\card{ \Aut(K/\Q) }$: $3$ This field is not Galois over $\Q$. This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{11}a^{4}+\frac{5}{11}a^{3}-\frac{1}{11}a^{2}-\frac{4}{11}a-\frac{3}{11}$, $\frac{1}{165}a^{5}-\frac{2}{165}a^{4}+\frac{21}{55}a^{3}-\frac{74}{165}a^{2}-\frac{52}{165}a-\frac{3}{11}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

 Monogenic: Not computed Index: $1$ Inessential primes: None

## Class group and class number

$C_{3}$, which has order $3$

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: $$-\frac{2}{33} a^{5} + \frac{1}{33} a^{4} + \frac{8}{11} a^{3} - \frac{47}{33} a^{2} - \frac{115}{33} a + 7$$ -(2)/(33)*a^(5) + (1)/(33)*a^(4) + (8)/(11)*a^(3) - (47)/(33)*a^(2) - (115)/(33)*a + 7  (order $4$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: $\frac{2}{165}a^{5}-\frac{4}{165}a^{4}-\frac{13}{55}a^{3}+\frac{17}{165}a^{2}+\frac{226}{165}a-\frac{17}{11}$, $\frac{29}{165}a^{5}+\frac{2}{165}a^{4}-\frac{116}{55}a^{3}+\frac{599}{165}a^{2}+\frac{1882}{165}a-19$ 2/165*a^5 - 4/165*a^4 - 13/55*a^3 + 17/165*a^2 + 226/165*a - 17/11, 29/165*a^5 + 2/165*a^4 - 116/55*a^3 + 599/165*a^2 + 1882/165*a - 19 sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$19.0957855121$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{3}\cdot 19.0957855121 \cdot 3}{4\sqrt{25969216}}\approx 0.697122302609$

## Galois group

$C_3\times S_3$ (as 6T5):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A solvable group of order 18 The 9 conjugacy class representatives for $S_3\times C_3$ Character table for $S_3\times C_3$

## Intermediate fields

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

## Sibling algebras

 Galois closure: 18.0.84535014172552012147112280064.2 Twin sextic algebra: 3.3.8281.2 $\times$ 3.1.676.1 Degree 9 sibling: 9.3.36343632130624.2

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type R ${\href{/padicField/3.2.0.1}{2} }^{3}$ ${\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{3}$ R ${\href{/padicField/11.2.0.1}{2} }^{3}$ R ${\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ ${\href{/padicField/19.2.0.1}{2} }^{3}$ ${\href{/padicField/23.6.0.1}{6} }$ ${\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ ${\href{/padicField/31.6.0.1}{6} }$ ${\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ ${\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ ${\href{/padicField/43.6.0.1}{6} }$ ${\href{/padicField/47.6.0.1}{6} }$ ${\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ ${\href{/padicField/59.6.0.1}{6} }$

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
$$2$$ 2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3} $$7$$ 7.6.4.1x^{6} + 35 x^{3} + 441$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$$13$$ 13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3} 13.3.2.1x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$

## Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $$\Q$$ $C_1$ $1$ $1$
* 1.4.2t1.a.a$1$ $2^{2}$ $$\Q(\sqrt{-1})$$ $C_2$ (as 2T1) $1$ $-1$
1.364.6t1.g.a$1$ $2^{2} \cdot 7 \cdot 13$ 6.0.4388797504.1 $C_6$ (as 6T1) $0$ $-1$
1.364.6t1.g.b$1$ $2^{2} \cdot 7 \cdot 13$ 6.0.4388797504.1 $C_6$ (as 6T1) $0$ $-1$
1.91.3t1.a.a$1$ $7 \cdot 13$ 3.3.8281.2 $C_3$ (as 3T1) $0$ $1$
1.91.3t1.a.b$1$ $7 \cdot 13$ 3.3.8281.2 $C_3$ (as 3T1) $0$ $1$
2.676.3t2.b.a$2$ $2^{2} \cdot 13^{2}$ 3.1.676.1 $S_3$ (as 3T2) $1$ $0$
* 2.2548.6t5.d.a$2$ $2^{2} \cdot 7^{2} \cdot 13$ 6.0.25969216.3 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.2548.6t5.d.b$2$ $2^{2} \cdot 7^{2} \cdot 13$ 6.0.25969216.3 $S_3\times C_3$ (as 6T5) $0$ $0$

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 *.