LMFDB - Number field 18.0.2954312706550833698643.4

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^15 + 24*x^12 + 88*x^9 + 69*x^6 + 12*x^3 + 1)

gp: K = bnfinit(y^18 - 6*y^15 + 24*y^12 + 88*y^9 + 69*y^6 + 12*y^3 + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 6*x^15 + 24*x^12 + 88*x^9 + 69*x^6 + 12*x^3 + 1);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 6*x^15 + 24*x^12 + 88*x^9 + 69*x^6 + 12*x^3 + 1)

$ x^{18} - 6x^{15} + 24x^{12} + 88x^{9} + 69x^{6} + 12x^{3} + 1 $ x^18 - 6*x^15 + 24*x^12 + 88*x^9 + 69*x^6 + 12*x^3 + 1

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$18$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[0, 9]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$-2954312706550833698643$ -2954312706550833698643 $\medspace = -\,3^{45}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$15.59$	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:	$3^{47/18}\approx 17.6123217011059$
Ramified primes:	$3$ 3	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}) $
$\card{ \Aut(K/\Q) }$:	$3$	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{4}+\frac{1}{3}a$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{5}+\frac{1}{3}a^{2}$, $\frac{1}{6}a^{9}-\frac{1}{2}a^{3}-\frac{1}{6}$, $\frac{1}{6}a^{10}-\frac{1}{2}a^{4}-\frac{1}{6}a$, $\frac{1}{18}a^{11}+\frac{1}{18}a^{10}+\frac{1}{18}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{18}a^{2}-\frac{1}{18}a-\frac{1}{18}$, $\frac{1}{18}a^{12}-\frac{1}{18}a^{9}-\frac{1}{6}a^{6}-\frac{2}{9}a^{3}+\frac{7}{18}$, $\frac{1}{18}a^{13}-\frac{1}{18}a^{10}-\frac{1}{6}a^{7}-\frac{2}{9}a^{4}+\frac{7}{18}a$, $\frac{1}{18}a^{14}+\frac{1}{18}a^{10}+\frac{1}{18}a^{9}-\frac{1}{6}a^{8}+\frac{5}{18}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}+\frac{1}{3}a^{2}-\frac{1}{18}a-\frac{1}{18}$, $\frac{1}{18}a^{15}-\frac{1}{18}a^{9}-\frac{1}{18}a^{6}-\frac{4}{9}$, $\frac{1}{54}a^{16}-\frac{1}{54}a^{15}+\frac{1}{54}a^{13}-\frac{1}{54}a^{12}+\frac{1}{54}a^{10}-\frac{1}{54}a^{9}+\frac{4}{27}a^{7}-\frac{4}{27}a^{6}-\frac{19}{54}a^{4}+\frac{19}{54}a^{3}+\frac{4}{27}a-\frac{4}{27}$, $\frac{1}{54}a^{17}-\frac{1}{54}a^{15}+\frac{1}{54}a^{14}-\frac{1}{54}a^{12}+\frac{1}{54}a^{11}-\frac{1}{54}a^{9}+\frac{4}{27}a^{8}-\frac{4}{27}a^{6}-\frac{19}{54}a^{5}+\frac{19}{54}a^{3}+\frac{4}{27}a^{2}-\frac{4}{27}$ 1, a, a^2, a^3, a^4, a^5, 1/3*a^6 + 1/3*a^3 + 1/3, 1/3*a^7 + 1/3*a^4 + 1/3*a, 1/3*a^8 + 1/3*a^5 + 1/3*a^2, 1/6*a^9 - 1/2*a^3 - 1/6, 1/6*a^10 - 1/2*a^4 - 1/6*a, 1/18*a^11 + 1/18*a^10 + 1/18*a^9 - 1/2*a^5 - 1/2*a^4 - 1/2*a^3 - 1/18*a^2 - 1/18*a - 1/18, 1/18*a^12 - 1/18*a^9 - 1/6*a^6 - 2/9*a^3 + 7/18, 1/18*a^13 - 1/18*a^10 - 1/6*a^7 - 2/9*a^4 + 7/18*a, 1/18*a^14 + 1/18*a^10 + 1/18*a^9 - 1/6*a^8 + 5/18*a^5 - 1/2*a^4 - 1/2*a^3 + 1/3*a^2 - 1/18*a - 1/18, 1/18*a^15 - 1/18*a^9 - 1/18*a^6 - 4/9, 1/54*a^16 - 1/54*a^15 + 1/54*a^13 - 1/54*a^12 + 1/54*a^10 - 1/54*a^9 + 4/27*a^7 - 4/27*a^6 - 19/54*a^4 + 19/54*a^3 + 4/27*a - 4/27, 1/54*a^17 - 1/54*a^15 + 1/54*a^14 - 1/54*a^12 + 1/54*a^11 - 1/54*a^9 + 4/27*a^8 - 4/27*a^6 - 19/54*a^5 + 19/54*a^3 + 4/27*a^2 - 4/27

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:	$8$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ \frac{1}{3} a^{15} - \frac{13}{6} a^{12} + \frac{55}{6} a^{9} + \frac{145}{6} a^{6} + \frac{41}{3} a^{3} + \frac{11}{6} $ (1)/(3)a^(15) - (13)/(6)a^(12) + (55)/(6)a^(9) + (145)/(6)a^(6) + (41)/(3)*a^(3) + (11)/(6) (order $6$)	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:	$\frac{1}{18}a^{17}-\frac{1}{2}a^{14}+\frac{22}{9}a^{11}+\frac{1}{9}a^{8}-\frac{22}{3}a^{5}-\frac{77}{18}a^{2}$, $a$, $\frac{2}{27}a^{17}+\frac{11}{54}a^{15}-\frac{13}{27}a^{14}-\frac{67}{54}a^{12}+\frac{53}{27}a^{11}+\frac{269}{54}a^{9}+\frac{160}{27}a^{8}+\frac{476}{27}a^{6}+\frac{13}{27}a^{5}+\frac{625}{54}a^{3}-\frac{80}{27}a^{2}+\frac{23}{27}$, $\frac{2}{27}a^{17}-\frac{1}{54}a^{15}-\frac{13}{27}a^{14}-\frac{1}{54}a^{12}+\frac{53}{27}a^{11}+\frac{13}{27}a^{9}+\frac{160}{27}a^{8}-\frac{157}{27}a^{6}+\frac{13}{27}a^{5}-\frac{211}{27}a^{3}-\frac{80}{27}a^{2}-\frac{71}{54}$, $\frac{16}{27}a^{17}+\frac{1}{54}a^{16}-\frac{193}{54}a^{14}+\frac{1}{54}a^{13}+\frac{385}{27}a^{11}-\frac{13}{27}a^{10}+\frac{2821}{54}a^{8}+\frac{157}{27}a^{7}+\frac{1993}{54}a^{5}+\frac{211}{27}a^{4}+\frac{56}{27}a^{2}+\frac{71}{54}a+1$, $\frac{16}{27}a^{17}+\frac{5}{27}a^{16}-\frac{1}{3}a^{15}-\frac{193}{54}a^{14}-\frac{34}{27}a^{13}+\frac{13}{6}a^{12}+\frac{385}{27}a^{11}+\frac{295}{54}a^{10}-\frac{55}{6}a^{9}+\frac{2821}{54}a^{8}+\frac{319}{27}a^{7}-\frac{145}{6}a^{6}+\frac{1993}{54}a^{5}+\frac{203}{54}a^{4}-\frac{41}{3}a^{3}+\frac{56}{27}a^{2}-\frac{25}{54}a-\frac{11}{6}$, $\frac{7}{9}a^{17}-\frac{11}{54}a^{16}+\frac{11}{54}a^{15}-\frac{29}{6}a^{14}+\frac{67}{54}a^{13}-\frac{67}{54}a^{12}+\frac{355}{18}a^{11}-\frac{269}{54}a^{10}+\frac{269}{54}a^{9}+\frac{1153}{18}a^{8}-\frac{476}{27}a^{7}+\frac{476}{27}a^{6}+\frac{122}{3}a^{5}-\frac{625}{54}a^{4}+\frac{625}{54}a^{3}+\frac{29}{18}a^{2}-\frac{23}{27}a+\frac{23}{27}$, $\frac{13}{18}a^{17}-\frac{5}{27}a^{16}-\frac{11}{54}a^{15}-\frac{13}{3}a^{14}+\frac{34}{27}a^{13}+\frac{67}{54}a^{12}+\frac{311}{18}a^{11}-\frac{295}{54}a^{10}-\frac{269}{54}a^{9}+\frac{1151}{18}a^{8}-\frac{319}{27}a^{7}-\frac{476}{27}a^{6}+48a^{5}-\frac{203}{54}a^{4}-\frac{625}{54}a^{3}+\frac{53}{9}a^{2}+\frac{25}{54}a-\frac{23}{27}$ 1/18a^17 - 1/2a^14 + 22/9a^11 + 1/9a^8 - 22/3a^5 - 77/18a^2, a, 2/27a^17 + 11/54a^15 - 13/27a^14 - 67/54a^12 + 53/27a^11 + 269/54a^9 + 160/27a^8 + 476/27a^6 + 13/27a^5 + 625/54a^3 - 80/27a^2 + 23/27, 2/27a^17 - 1/54a^15 - 13/27a^14 - 1/54a^12 + 53/27a^11 + 13/27a^9 + 160/27a^8 - 157/27a^6 + 13/27a^5 - 211/27a^3 - 80/27a^2 - 71/54, 16/27a^17 + 1/54a^16 - 193/54a^14 + 1/54a^13 + 385/27a^11 - 13/27a^10 + 2821/54a^8 + 157/27a^7 + 1993/54a^5 + 211/27a^4 + 56/27a^2 + 71/54a + 1, 16/27a^17 + 5/27a^16 - 1/3a^15 - 193/54a^14 - 34/27a^13 + 13/6a^12 + 385/27a^11 + 295/54a^10 - 55/6a^9 + 2821/54a^8 + 319/27a^7 - 145/6a^6 + 1993/54a^5 + 203/54a^4 - 41/3a^3 + 56/27a^2 - 25/54a - 11/6, 7/9a^17 - 11/54a^16 + 11/54a^15 - 29/6a^14 + 67/54a^13 - 67/54a^12 + 355/18a^11 - 269/54a^10 + 269/54a^9 + 1153/18a^8 - 476/27a^7 + 476/27a^6 + 122/3a^5 - 625/54a^4 + 625/54a^3 + 29/18a^2 - 23/27a + 23/27, 13/18a^17 - 5/27a^16 - 11/54a^15 - 13/3a^14 + 34/27a^13 + 67/54a^12 + 311/18a^11 - 295/54a^10 - 269/54a^9 + 1151/18a^8 - 319/27a^7 - 476/27a^6 + 48a^5 - 203/54a^4 - 625/54a^3 + 53/9a^2 + 25/54*a - 23/27	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:	$ 9573.44755541 $	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 9573.44755541 \cdot 1}{6\cdot\sqrt{2954312706550833698643}}\cr\approx \mathstrut & 0.448030619915 \end{aligned}\]

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

x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^15 + 24*x^12 + 88*x^9 + 69*x^6 + 12*x^3 + 1)

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^18 - 6*x^15 + 24*x^12 + 88*x^9 + 69*x^6 + 12*x^3 + 1, 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^18 - 6*x^15 + 24*x^12 + 88*x^9 + 69*x^6 + 12*x^3 + 1);

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^18 - 6*x^15 + 24*x^12 + 88*x^9 + 69*x^6 + 12*x^3 + 1);

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

$C_3^2:C_6$ (as 18T22):

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 54

The 10 conjugacy class representatives for $C_3^2:C_6$

Character table for $C_3^2:C_6$

Intermediate fields

$\Q(\sqrt{-3}) $, 6.0.177147.1

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 fields

Degree 9 siblings:	data not computed
Degree 18 siblings:	data not computed
Degree 27 sibling:	data not computed
Minimal sibling:	9.1.31381059609.1

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	${\href{/padicField/2.6.0.1}{6} }^{3}$	R	${\href{/padicField/5.6.0.1}{6} }^{3}$	${\href{/padicField/7.3.0.1}{3} }^{5}{,}\,{\href{/padicField/7.1.0.1}{1} }^{3}$	${\href{/padicField/11.6.0.1}{6} }^{3}$	${\href{/padicField/13.3.0.1}{3} }^{5}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$	${\href{/padicField/17.2.0.1}{2} }^{9}$	${\href{/padicField/19.3.0.1}{3} }^{6}$	${\href{/padicField/23.6.0.1}{6} }^{3}$	${\href{/padicField/29.6.0.1}{6} }^{3}$	${\href{/padicField/31.3.0.1}{3} }^{5}{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$	${\href{/padicField/37.3.0.1}{3} }^{6}$	${\href{/padicField/41.6.0.1}{6} }^{3}$	${\href{/padicField/43.3.0.1}{3} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$	${\href{/padicField/47.6.0.1}{6} }^{3}$	${\href{/padicField/53.2.0.1}{2} }^{9}$	${\href{/padicField/59.6.0.1}{6} }^{3}$

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

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$3$ 3		Deg $18$	$18$	$1$	$45$

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