# Properties

 Label 6.0.3385600.1 Degree $6$ Signature $[0, 3]$ Discriminant $-3385600$ Root discriminant $$12.25$$ Ramified primes $2,5,23$ Class number $1$ Class group trivial Galois group $D_{6}$ (as 6T3)

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 6*x^3 + 16*x^2 - 24*x + 18)

gp: K = bnfinit(x^6 - 6*x^3 + 16*x^2 - 24*x + 18, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^6 - 6*x^3 + 16*x^2 - 24*x + 18);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^6 - 6*x^3 + 16*x^2 - 24*x + 18)

$$x^{6} - 6x^{3} + 16x^{2} - 24x + 18$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

## Invariants

 Degree: $6$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K);  oscar: degree(K) Signature: $[0, 3]$ sage: K.signature()  gp: K.sign  magma: Signature(K);  oscar: signature(K) Discriminant: $$-3385600$$ -3385600 $$\medspace = -\,2^{8}\cdot 5^{2}\cdot 23^{2}$$ sage: K.disc()  gp: K.disc  magma: OK := Integers(K); Discriminant(OK);  oscar: OK = ring_of_integers(K); discriminant(OK) Root discriminant: $$12.25$$ sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol))  magma: Abs(Discriminant(OK))^(1/Degree(K));  oscar: (1.0 * dK)^(1/degree(K)) Ramified primes: $$2$$, $$5$$, $$23$$ 2, 5, 23 sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(OK));  oscar: prime_divisors(discriminant((OK))) Discriminant root field: $$\Q(\sqrt{-1})$$ $\card{ \Aut(K/\Q) }$: $2$ sage: K.automorphisms()  magma: Automorphisms(K);  oscar: automorphisms(K) This field is not Galois over $\Q$. This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{165}a^{5}+\frac{14}{55}a^{4}-\frac{17}{55}a^{3}-\frac{1}{55}a^{2}+\frac{1}{3}a-\frac{8}{55}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

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

## Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

oscar: class_group(K)

## Unit group

sage: UK = K.unit_group()

magma: UK, fUK := UnitGroup(K);

oscar: UK, fUK = unit_group(OK)

 Rank: $2$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K);  oscar: rank(UK) Torsion generator: $$\frac{16}{165} a^{5} + \frac{4}{55} a^{4} + \frac{3}{55} a^{3} - \frac{16}{55} a^{2} + \frac{4}{3} a - \frac{73}{55}$$ (16)/(165)*a^(5) + (4)/(55)*a^(4) + (3)/(55)*a^(3) - (16)/(55)*a^(2) + (4)/(3)*a - (73)/(55)  (order $4$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);  oscar: torsion_units_generator(OK) Fundamental units: $\frac{23}{165}a^{5}-\frac{8}{55}a^{4}-\frac{6}{55}a^{3}-\frac{78}{55}a^{2}+\frac{11}{3}a-\frac{129}{55}$, $\frac{53}{165}a^{5}+\frac{27}{55}a^{4}+\frac{34}{55}a^{3}-\frac{108}{55}a^{2}+\frac{11}{3}a-\frac{259}{55}$ 23/165*a^5 - 8/55*a^4 - 6/55*a^3 - 78/55*a^2 + 11/3*a - 129/55, 53/165*a^5 + 27/55*a^4 + 34/55*a^3 - 108/55*a^2 + 11/3*a - 259/55 sage: UK.fundamental_units()  gp: K.fu  magma: [K|fUK(g): g in Generators(UK)];  oscar: [K(fUK(a)) for a in gens(UK)] Regulator: $$33.8354673478$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);  oscar: regulator(K)

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) = \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}} \approx\frac{2^{0}\cdot(2\pi)^{3}\cdot 33.8354673478 \cdot 1}{4\cdot\sqrt{3385600}}\approx 1.14033898064$

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^6 - 6*x^3 + 16*x^2 - 24*x + 18)

DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()

hK = K.class_number(); wK = K.unit_group().torsion_generator().order();

2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))

# self-contained Pari/GP code snippet to compute the analytic class number formula

K = bnfinit(x^6 - 6*x^3 + 16*x^2 - 24*x + 18, 1);

[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]

/* self-contained Magma code snippet to compute the analytic class number formula */

Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^6 - 6*x^3 + 16*x^2 - 24*x + 18);

OK := Integers(K); DK := Discriminant(OK);

UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);

r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);

hK := #clK; wK := #TorsionSubgroup(UK);

2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));

# self-contained Oscar code snippet to compute the analytic class number formula

Qx, x = PolynomialRing(QQ); K, a = NumberField(x^6 - 6*x^3 + 16*x^2 - 24*x + 18);

OK = ring_of_integers(K); DK = discriminant(OK);

UK, fUK = unit_group(OK); clK, fclK = class_group(OK);

r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);

hK = order(clK); wK = torsion_units_order(K);

2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))

## Galois group

$D_6$ (as 6T3):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: G = GaloisGroup(K);

oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)

 A solvable group of order 12 The 6 conjugacy class representatives for $D_{6}$ Character table for $D_{6}$

## Intermediate fields

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

sage: K.subfields()[1:-1]

gp: L = nfsubfields(K); L[2..length(b)]

magma: L := Subfields(K); L[2..#L];

oscar: subfields(K)[2:end-1]

## Sibling algebras

 Galois closure: deg 12 Twin sextic algebra: 3.1.460.1 $\times$ $$\Q(\sqrt{115})$$ $\times$ $$\Q$$ Degree 6 sibling: 6.2.389344000.2

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type R ${\href{/padicField/3.2.0.1}{2} }^{3}$ R ${\href{/padicField/7.6.0.1}{6} }$ ${\href{/padicField/11.2.0.1}{2} }^{3}$ ${\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.3.0.1}{3} }^{2}$ ${\href{/padicField/19.2.0.1}{2} }^{3}$ R ${\href{/padicField/29.3.0.1}{3} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }$ ${\href{/padicField/37.3.0.1}{3} }^{2}$ ${\href{/padicField/41.3.0.1}{3} }^{2}$ ${\href{/padicField/43.2.0.1}{2} }^{3}$ ${\href{/padicField/47.2.0.1}{2} }^{3}$ ${\href{/padicField/53.3.0.1}{3} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:

p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:

p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))]; # to obtain a list of$[e_i,f_i]$for the factorization of the ideal$p\mathcal{O}_K$for$p=7$in Oscar: p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac] ## Local algebras for ramified primes$p$LabelPolynomial$efc$Galois group Slope content $$2$$ 2.6.8.1$x^{6} + 2 x^{3} + 2$$6$$1$$8$$D_{6}$$[2]_{3}^{2} $$5$$ \Q_{5}$$x + 3$$1$$1$$0Trivial[\ ] \Q_{5}$$x + 3$$1$$1$$0Trivial[\ ] 5.2.1.2x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2} $$23$$ 23.2.0.1x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$23.4.2.1$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$$2$$2$$2$$C_2^2[\ ]_{2}^{2}$## Artin representations Label Dimension Conductor Artin stem field$G$Ind$\chi(c)$* 1.1.1t1.a.a$11$$$\Q$$$C_111$1.115.2t1.a.a$1 5 \cdot 23 $$$\Q(\sqrt{-115})$$$C_2$(as 2T1)$1-1$1.460.2t1.a.a$1 2^{2} \cdot 5 \cdot 23 $$$\Q(\sqrt{115})$$$C_2$(as 2T1)$11$* 1.4.2t1.a.a$1 2^{2}$$$\Q(\sqrt{-1})$$$C_2$(as 2T1)$1-1$* 2.460.3t2.a.a$2 2^{2} \cdot 5 \cdot 23 $3.1.460.1$S_3$(as 3T2)$10$* 2.1840.6t3.a.a$2 2^{4} \cdot 5 \cdot 23 $6.0.3385600.1$D_{6}$(as 6T3)$10\$

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 *.