Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 9*x^12 + 4887*x^6 + 27)
gp: K = bnfinit(y^18 + 9*y^12 + 4887*y^6 + 27, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + 9*x^12 + 4887*x^6 + 27);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 + 9*x^12 + 4887*x^6 + 27)
\( x^{18} + 9x^{12} + 4887x^{6} + 27 \)
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: |
\(-5559060566555523000000000000\)
\(\medspace = -\,2^{12}\cdot 3^{33}\cdot 5^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(34.78\) | 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: | $2^{2/3}3^{11/6}5^{2/3}\approx 34.78475645409113$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $18$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{27}a^{6}+\frac{4}{9}$, $\frac{1}{27}a^{7}+\frac{4}{9}a$, $\frac{1}{27}a^{8}+\frac{4}{9}a^{2}$, $\frac{1}{54}a^{9}-\frac{1}{54}a^{6}-\frac{5}{18}a^{3}+\frac{5}{18}$, $\frac{1}{54}a^{10}-\frac{1}{54}a^{7}-\frac{5}{18}a^{4}+\frac{5}{18}a$, $\frac{1}{54}a^{11}-\frac{1}{54}a^{8}-\frac{5}{18}a^{5}+\frac{5}{18}a^{2}$, $\frac{1}{2916}a^{12}-\frac{5}{486}a^{6}-\frac{1}{2}a^{3}-\frac{137}{324}$, $\frac{1}{2916}a^{13}-\frac{5}{486}a^{7}-\frac{1}{2}a^{4}-\frac{137}{324}a$, $\frac{1}{2916}a^{14}-\frac{5}{486}a^{8}-\frac{1}{2}a^{5}-\frac{137}{324}a^{2}$, $\frac{1}{8748}a^{15}-\frac{5}{1458}a^{9}-\frac{1}{54}a^{6}+\frac{187}{972}a^{3}-\frac{2}{9}$, $\frac{1}{8748}a^{16}-\frac{5}{1458}a^{10}-\frac{1}{54}a^{7}+\frac{187}{972}a^{4}-\frac{2}{9}a$, $\frac{1}{8748}a^{17}-\frac{5}{1458}a^{11}-\frac{1}{54}a^{8}+\frac{187}{972}a^{5}-\frac{2}{9}a^{2}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
$C_{2}\times C_{6}$, which has order $12$ (assuming GRH)
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: |
\( \frac{7}{2916} a^{15} + \frac{5}{243} a^{9} + \frac{3775}{324} a^{3} + \frac{1}{2} \)
(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: |
$\frac{1}{729}a^{15}+\frac{1}{1458}a^{12}+\frac{7}{486}a^{9}-\frac{1}{486}a^{6}+\frac{1049}{162}a^{3}+\frac{71}{81}$, $\frac{1}{8748}a^{15}-\frac{1}{2916}a^{12}-\frac{5}{1458}a^{9}-\frac{2}{243}a^{6}+\frac{673}{972}a^{3}+\frac{65}{324}$, $\frac{29}{8748}a^{17}-\frac{17}{8748}a^{15}-\frac{1}{972}a^{14}-\frac{1}{2916}a^{12}+\frac{22}{729}a^{11}-\frac{23}{1458}a^{9}-\frac{1}{162}a^{8}-\frac{2}{243}a^{6}+\frac{15683}{972}a^{5}-\frac{9389}{972}a^{3}-\frac{505}{108}a^{2}+\frac{65}{324}$, $\frac{29}{8748}a^{17}+\frac{13}{8748}a^{15}-\frac{1}{972}a^{14}+\frac{1}{2916}a^{12}+\frac{22}{729}a^{11}+\frac{8}{729}a^{9}-\frac{1}{162}a^{8}+\frac{13}{486}a^{6}+\frac{15683}{972}a^{5}+\frac{6967}{972}a^{3}-\frac{505}{108}a^{2}+\frac{169}{324}$, $\frac{26}{2187}a^{17}+\frac{17}{8748}a^{15}-\frac{5}{1458}a^{14}-\frac{1}{2916}a^{12}+\frac{155}{1458}a^{11}+\frac{23}{1458}a^{9}-\frac{13}{486}a^{8}-\frac{2}{243}a^{6}+\frac{28219}{486}a^{5}+\frac{9389}{972}a^{3}-\frac{1363}{81}a^{2}-\frac{907}{324}$, $\frac{179}{8748}a^{17}+\frac{5}{1458}a^{16}-\frac{11}{8748}a^{15}-\frac{5}{2916}a^{14}-\frac{1}{729}a^{13}-\frac{1}{972}a^{12}+\frac{133}{729}a^{11}+\frac{13}{486}a^{10}-\frac{13}{729}a^{9}-\frac{11}{486}a^{8}-\frac{7}{486}a^{7}-\frac{1}{162}a^{6}+\frac{97193}{972}a^{5}+\frac{1363}{81}a^{4}-\frac{5621}{972}a^{3}-\frac{2357}{324}a^{2}-\frac{725}{162}a-\frac{181}{108}$, $\frac{187}{4374}a^{17}-\frac{11}{1458}a^{15}+\frac{7}{1458}a^{14}-\frac{2}{729}a^{13}+\frac{1}{972}a^{12}+\frac{280}{729}a^{11}-\frac{17}{243}a^{9}+\frac{10}{243}a^{8}-\frac{7}{243}a^{7}+\frac{1}{162}a^{6}+\frac{101551}{486}a^{5}-\frac{2977}{81}a^{3}+\frac{3775}{162}a^{2}-\frac{1049}{81}a+\frac{451}{108}$, $\frac{25}{2916}a^{17}+\frac{4}{2187}a^{16}-\frac{49}{8748}a^{15}-\frac{7}{2916}a^{14}+\frac{1}{1458}a^{13}+\frac{1}{2916}a^{12}+\frac{37}{486}a^{11}+\frac{14}{729}a^{10}-\frac{79}{1458}a^{9}-\frac{5}{243}a^{8}+\frac{4}{243}a^{7}-\frac{7}{243}a^{6}+\frac{13585}{324}a^{5}+\frac{2179}{243}a^{4}-\frac{26821}{972}a^{3}-\frac{3613}{324}a^{2}+\frac{421}{162}a+\frac{439}{324}$
| 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: | \( 3270633.26683 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 3270633.26683 \cdot 12}{6\cdot\sqrt{5559060566555523000000000000}}\cr\approx \mathstrut & 1.33899809092 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 + 9*x^12 + 4887*x^6 + 27)
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 + 9*x^12 + 4887*x^6 + 27, 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 + 9*x^12 + 4887*x^6 + 27);
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 + 9*x^12 + 4887*x^6 + 27);
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
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 18 |
| The 6 conjugacy class representatives for $C_3^2 : C_2$ |
| Character table for $C_3^2 : C_2$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.1.300.1 x3, 3.1.972.1 x3, 3.1.6075.2 x3, 3.1.24300.3 x3, 6.0.270000.1, 6.0.2834352.1, 6.0.110716875.2, 6.0.1771470000.1, 9.1.43046721000000.2 x9 |
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
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 | R | ${\href{/padicField/7.3.0.1}{3} }^{6}$ | ${\href{/padicField/11.2.0.1}{2} }^{9}$ | ${\href{/padicField/13.3.0.1}{3} }^{6}$ | ${\href{/padicField/17.2.0.1}{2} }^{9}$ | ${\href{/padicField/19.3.0.1}{3} }^{6}$ | ${\href{/padicField/23.2.0.1}{2} }^{9}$ | ${\href{/padicField/29.2.0.1}{2} }^{9}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.2.0.1}{2} }^{9}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}$ | ${\href{/padicField/47.2.0.1}{2} }^{9}$ | ${\href{/padicField/53.2.0.1}{2} }^{9}$ | ${\href{/padicField/59.2.0.1}{2} }^{9}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
|
\(3\)
| 3.6.11.8 | $x^{6} + 18 x^{2} + 21$ | $6$ | $1$ | $11$ | $S_3$ | $[5/2]_{2}$ |
| 3.6.11.8 | $x^{6} + 18 x^{2} + 21$ | $6$ | $1$ | $11$ | $S_3$ | $[5/2]_{2}$ | |
| 3.6.11.8 | $x^{6} + 18 x^{2} + 21$ | $6$ | $1$ | $11$ | $S_3$ | $[5/2]_{2}$ | |
|
\(5\)
| 5.6.4.1 | $x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 5.6.4.1 | $x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 5.6.4.1 | $x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |