# Properties

 Label 6.0.2811072.1 Degree $6$ Signature $[0, 3]$ Discriminant $-2811072$ Root discriminant $$11.88$$ Ramified primes $2,3,11$ Class number $1$ Class group trivial Galois group $S_4\times C_2$ (as 6T11)

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^5 - x^4 + 2*x^3 + 2*x^2 - 8*x + 14)

gp: K = bnfinit(y^6 - 2*y^5 - y^4 + 2*y^3 + 2*y^2 - 8*y + 14, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^6 - 2*x^5 - x^4 + 2*x^3 + 2*x^2 - 8*x + 14);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^6 - 2*x^5 - x^4 + 2*x^3 + 2*x^2 - 8*x + 14)

$$x^{6} - 2x^{5} - x^{4} + 2x^{3} + 2x^{2} - 8x + 14$$

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: $$-2811072$$ -2811072 $$\medspace = -\,2^{6}\cdot 3\cdot 11^{4}$$ sage: K.disc()  gp: K.disc  magma: OK := Integers(K); Discriminant(OK);  oscar: OK = ring_of_integers(K); discriminant(OK) Root discriminant: $$11.88$$ 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)) Galois root discriminant: $2^{3/2}3^{1/2}11^{2/3}\approx 24.230780918292734$ Ramified primes: $$2$$, $$3$$, $$11$$ 2, 3, 11 sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(OK));  oscar: prime_divisors(discriminant((OK))) Discriminant root field: $$\Q(\sqrt{-3})$$ $\card{ \Aut(K/\Q) }$: $2$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{46}a^{5}-\frac{17}{46}a^{4}-\frac{11}{23}a^{3}+\frac{5}{23}a^{2}-\frac{5}{23}a+\frac{2}{23}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

 Monogenic: No Index: $2$ Inessential primes: $2$

## 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: $$-1$$ -1  (order $2$) 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: $\frac{11}{23}a^{5}-\frac{3}{23}a^{4}-\frac{12}{23}a^{3}-\frac{5}{23}a^{2}+\frac{5}{23}a-\frac{71}{23}$, $\frac{11}{23}a^{5}-\frac{3}{23}a^{4}+\frac{11}{23}a^{3}-\frac{5}{23}a^{2}-\frac{110}{23}a-\frac{255}{23}$ 11/23*a^5 - 3/23*a^4 - 12/23*a^3 - 5/23*a^2 + 5/23*a - 71/23, 11/23*a^5 - 3/23*a^4 + 11/23*a^3 - 5/23*a^2 - 110/23*a - 255/23 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: $$34.9279772815$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);  oscar: regulator(K)

## Class number formula

\begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 34.9279772815 \cdot 1}{2\cdot\sqrt{2811072}}\cr\approx \mathstrut & 2.58372959986 \end{aligned}

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

x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^5 - x^4 + 2*x^3 + 2*x^2 - 8*x + 14)

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 - 2*x^5 - x^4 + 2*x^3 + 2*x^2 - 8*x + 14, 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 - 2*x^5 - x^4 + 2*x^3 + 2*x^2 - 8*x + 14);

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 - 2*x^5 - x^4 + 2*x^3 + 2*x^2 - 8*x + 14);

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

$C_2\times S_4$ (as 6T11):

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 48 The 10 conjugacy class representatives for $S_4\times C_2$ Character table for $S_4\times C_2$

## 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

 Twin sextic algebra: 4.2.17424.1 $\times$ $$\Q(\sqrt{3})$$ Degree 6 sibling: 6.2.11244288.1 Degree 8 siblings: 8.4.4857532416.5, 8.0.303595776.4 Degree 12 siblings: deg 12, deg 12, deg 12, deg 12, deg 12, deg 12 Degree 16 sibling: deg 16 Degree 24 siblings: deg 24, deg 24, deg 24, deg 24 Minimal sibling: This field is its own minimal sibling

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type R R ${\href{/padicField/5.6.0.1}{6} }$ ${\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ R ${\href{/padicField/13.3.0.1}{3} }^{2}$ ${\href{/padicField/17.6.0.1}{6} }$ ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.2.0.1}{2} }^{3}$ ${\href{/padicField/29.6.0.1}{6} }$ ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.3.0.1}{3} }^{2}$ ${\href{/padicField/41.6.0.1}{6} }$ ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }$ ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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

# 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 $$2$$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$2.4.6.8$x^{4} + 2 x^{3} + 2$$4$$1$$6$$D_{4}$$[2, 2]^{2} $$3$$ 3.2.1.1x^{2} + 6$$2$$1$$1$$C_2$$[\ ]_{2}$3.4.0.1$x^{4} + 2 x^{3} + 2$$1$$4$$0$$C_4$$[\ ]^{4} $$11$$ 11.6.4.1x^{6} + 21 x^{5} + 153 x^{4} + 449 x^{3} + 537 x^{2} + 1569 x + 3440$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$## Artin representations Label Dimension Conductor Artin stem field$G$Ind$\chi(c)$* 1.1.1t1.a.a$11$$$\Q$$$C_111$1.12.2t1.a.a$1 2^{2} \cdot 3 $$$\Q(\sqrt{3})$$$C_2$(as 2T1)$11$1.4.2t1.a.a$1 2^{2}$$$\Q(\sqrt{-1})$$$C_2$(as 2T1)$1-1$1.3.2t1.a.a$1 3 $$$\Q(\sqrt{-3})$$$C_2$(as 2T1)$1-1$2.4356.6t3.d.a$2 2^{2} \cdot 3^{2} \cdot 11^{2}$6.2.25299648.1$D_{6}$(as 6T3)$10$* 2.484.3t2.b.a$2 2^{2} \cdot 11^{2}$3.1.484.1$S_3$(as 3T2)$10$3.17424.4t5.a.a$3 2^{4} \cdot 3^{2} \cdot 11^{2}$4.2.17424.1$S_4$(as 4T5)$11$* 3.5808.6t11.a.a$3 2^{4} \cdot 3 \cdot 11^{2}$6.0.2811072.1$S_4\times C_2$(as 6T11)$1-1$3.23232.6t11.a.a$3 2^{6} \cdot 3 \cdot 11^{2}$6.0.2811072.1$S_4\times C_2$(as 6T11)$11$3.69696.6t8.a.a$3 2^{6} \cdot 3^{2} \cdot 11^{2}$4.2.17424.1$S_4$(as 4T5)$1-1\$

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 *.