# Properties

 Label 6.6.158470336.1 Degree $6$ Signature $[6, 0]$ Discriminant $158470336$ Root discriminant $$23.26$$ Ramified primes $2,19$ Class number $1$ Class group trivial Galois group $C_6$ (as 6T1)

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 19*x^4 + 38*x^2 - 19)

gp: K = bnfinit(x^6 - 19*x^4 + 38*x^2 - 19, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^6 - 19*x^4 + 38*x^2 - 19);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^6 - 19*x^4 + 38*x^2 - 19)

$$x^{6} - 19x^{4} + 38x^{2} - 19$$

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: $[6, 0]$ sage: K.signature()  gp: K.sign  magma: Signature(K);  oscar: signature(K) Discriminant: $$158470336$$ 158470336 $$\medspace = 2^{6}\cdot 19^{5}$$ sage: K.disc()  gp: K.disc  magma: OK := Integers(K); Discriminant(OK);  oscar: OK = ring_of_integers(K); discriminant(OK) Root discriminant: $$23.26$$ 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: $$2$$, $$19$$ 2, 19 sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(OK));  oscar: prime_divisors(discriminant((OK))) Discriminant root field: $$\Q(\sqrt{19})$$ $\card{ \Gal(K/\Q) }$: $6$ sage: K.automorphisms()  magma: Automorphisms(K);  oscar: automorphisms(K) This field is Galois and abelian over $\Q$. Conductor: $$76=2^{2}\cdot 19$$ Dirichlet character group: $\lbrace$$\chi_{76}(1,·), \chi_{76}(27,·), \chi_{76}(49,·), \chi_{76}(75,·), \chi_{76}(45,·), \chi_{76}(31,·)$$\rbrace$ This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{7}a^{4}-\frac{2}{7}a^{2}-\frac{3}{7}$, $\frac{1}{7}a^{5}-\frac{2}{7}a^{3}-\frac{3}{7}a$

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: $5$ 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: $\frac{1}{7}a^{4}-\frac{16}{7}a^{2}+\frac{11}{7}$, $\frac{3}{7}a^{4}-\frac{55}{7}a^{2}+\frac{68}{7}$, $\frac{3}{7}a^{5}-\frac{3}{7}a^{4}-\frac{55}{7}a^{3}+\frac{55}{7}a^{2}+\frac{75}{7}a-\frac{68}{7}$, $\frac{1}{7}a^{4}-\frac{16}{7}a^{2}-a+\frac{11}{7}$, $\frac{5}{7}a^{5}+\frac{9}{7}a^{4}-\frac{87}{7}a^{3}-\frac{158}{7}a^{2}+\frac{55}{7}a+\frac{134}{7}$ 1/7*a^4 - 16/7*a^2 + 11/7, 3/7*a^4 - 55/7*a^2 + 68/7, 3/7*a^5 - 3/7*a^4 - 55/7*a^3 + 55/7*a^2 + 75/7*a - 68/7, 1/7*a^4 - 16/7*a^2 - a + 11/7, 5/7*a^5 + 9/7*a^4 - 87/7*a^3 - 158/7*a^2 + 55/7*a + 134/7 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: $$91.4119887805$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);  oscar: regulator(K)

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) = \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}} \approx\frac{2^{6}\cdot(2\pi)^{0}\cdot 91.4119887805 \cdot 1}{2\cdot\sqrt{158470336}}\approx 0.232369512362$

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

x = polygen(QQ); K.<a> = NumberField(x^6 - 19*x^4 + 38*x^2 - 19)

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 - 19*x^4 + 38*x^2 - 19, 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 - 19*x^4 + 38*x^2 - 19);

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 - 19*x^4 + 38*x^2 - 19);

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

$C_6$ (as 6T1):

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 cyclic group of order 6 The 6 conjugacy class representatives for $C_6$ Character table for $C_6$

## 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

 Twin sextic algebra: 3.3.361.1 $\times$ $$\Q(\sqrt{19})$$ $\times$ $$\Q$$

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type R ${\href{/padicField/3.3.0.1}{3} }^{2}$ ${\href{/padicField/5.3.0.1}{3} }^{2}$ ${\href{/padicField/7.2.0.1}{2} }^{3}$ ${\href{/padicField/11.2.0.1}{2} }^{3}$ ${\href{/padicField/13.6.0.1}{6} }$ ${\href{/padicField/17.3.0.1}{3} }^{2}$ R ${\href{/padicField/23.6.0.1}{6} }$ ${\href{/padicField/29.6.0.1}{6} }$ ${\href{/padicField/31.1.0.1}{1} }^{6}$ ${\href{/padicField/37.2.0.1}{2} }^{3}$ ${\href{/padicField/41.6.0.1}{6} }$ ${\href{/padicField/43.6.0.1}{6} }$ ${\href{/padicField/47.6.0.1}{6} }$ ${\href{/padicField/53.6.0.1}{6} }$ ${\href{/padicField/59.3.0.1}{3} }^{2}$

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 $$2$$ 2.6.6.5$x^{6} + 6 x^{5} + 32 x^{4} + 90 x^{3} + 199 x^{2} + 204 x + 149$$2$$3$$6$$C_6$$[2]^{3} $$19$$ 19.6.5.2x^{6} + 152$$6$$1$$5$$C_6$$[\ ]_{6}$## Artin representations Label Dimension Conductor Artin stem field$G$Ind$\chi(c)$* 1.1.1t1.a.a$11$$$\Q$$$C_111$* 1.76.2t1.a.a$1 2^{2} \cdot 19 $$$\Q(\sqrt{19})$$$C_2$(as 2T1)$11$* 1.19.3t1.a.a$1 19 $3.3.361.1$C_3$(as 3T1)$01$* 1.76.6t1.b.a$1 2^{2} \cdot 19 $6.6.158470336.1$C_6$(as 6T1)$01$* 1.19.3t1.a.b$1 19 $3.3.361.1$C_3$(as 3T1)$01$* 1.76.6t1.b.b$1 2^{2} \cdot 19 $6.6.158470336.1$C_6$(as 6T1)$01\$

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 *.