Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 3*x^16 + 9*x^15 + 24*x^14 - 30*x^13 - 78*x^12 + 60*x^11 + 141*x^10 - 56*x^9 - 150*x^8 + 12*x^7 + 81*x^6 + 12*x^5 - 15*x^4 - 3*x^3 + 3*x^2 + 3*x + 1)
gp: K = bnfinit(y^18 - 3*y^17 - 3*y^16 + 9*y^15 + 24*y^14 - 30*y^13 - 78*y^12 + 60*y^11 + 141*y^10 - 56*y^9 - 150*y^8 + 12*y^7 + 81*y^6 + 12*y^5 - 15*y^4 - 3*y^3 + 3*y^2 + 3*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 3*x^17 - 3*x^16 + 9*x^15 + 24*x^14 - 30*x^13 - 78*x^12 + 60*x^11 + 141*x^10 - 56*x^9 - 150*x^8 + 12*x^7 + 81*x^6 + 12*x^5 - 15*x^4 - 3*x^3 + 3*x^2 + 3*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 3*x^17 - 3*x^16 + 9*x^15 + 24*x^14 - 30*x^13 - 78*x^12 + 60*x^11 + 141*x^10 - 56*x^9 - 150*x^8 + 12*x^7 + 81*x^6 + 12*x^5 - 15*x^4 - 3*x^3 + 3*x^2 + 3*x + 1)
\( x^{18} - 3 x^{17} - 3 x^{16} + 9 x^{15} + 24 x^{14} - 30 x^{13} - 78 x^{12} + 60 x^{11} + 141 x^{10} + \cdots + 1 \)
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: |
\(-12213256064722484031\)
\(\medspace = -\,3^{33}\cdot 13^{3}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.49\) | 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: | $3^{103/54}13^{1/2}\approx 29.311381743553262$ | ||
| Ramified primes: |
\(3\), \(13\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-39}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}-\frac{1}{3}$, $\frac{1}{3}a^{10}-\frac{1}{3}a$, $\frac{1}{9}a^{11}+\frac{1}{9}a^{10}+\frac{1}{9}a^{9}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{4}{9}a^{2}-\frac{4}{9}a-\frac{4}{9}$, $\frac{1}{9}a^{12}-\frac{1}{9}a^{9}-\frac{1}{3}a^{6}+\frac{2}{9}a^{3}+\frac{1}{9}$, $\frac{1}{9}a^{13}-\frac{1}{9}a^{10}-\frac{1}{3}a^{7}+\frac{2}{9}a^{4}+\frac{1}{9}a$, $\frac{1}{9}a^{14}+\frac{1}{9}a^{10}+\frac{1}{9}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{4}{9}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{4}{9}a-\frac{4}{9}$, $\frac{1}{9}a^{15}-\frac{1}{9}a^{9}-\frac{1}{9}a^{6}+\frac{1}{3}a^{3}-\frac{2}{9}$, $\frac{1}{9}a^{16}-\frac{1}{9}a^{10}-\frac{1}{9}a^{7}+\frac{1}{3}a^{4}-\frac{2}{9}a$, $\frac{1}{9}a^{17}+\frac{1}{9}a^{10}+\frac{1}{9}a^{9}-\frac{4}{9}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{4}{9}a-\frac{4}{9}$
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: |
\( -\frac{11}{9} a^{17} + \frac{41}{9} a^{16} + \frac{4}{9} a^{15} - \frac{104}{9} a^{14} - 21 a^{13} + \frac{467}{9} a^{12} + \frac{532}{9} a^{11} - \frac{1052}{9} a^{10} - \frac{266}{3} a^{9} + \frac{1187}{9} a^{8} + \frac{769}{9} a^{7} - \frac{649}{9} a^{6} - \frac{358}{9} a^{5} + \frac{26}{3} a^{4} + \frac{46}{9} a^{3} + \frac{23}{9} a^{2} - \frac{25}{9} a - \frac{5}{3} \)
(order $18$)
| 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: |
$\frac{5}{9}a^{17}-\frac{19}{9}a^{16}+5a^{14}+\frac{86}{9}a^{13}-\frac{73}{3}a^{12}-\frac{221}{9}a^{11}+\frac{470}{9}a^{10}+39a^{9}-\frac{545}{9}a^{8}-\frac{347}{9}a^{7}+34a^{6}+\frac{62}{3}a^{5}-\frac{47}{9}a^{4}-\frac{11}{3}a^{3}-\frac{19}{9}a^{2}+\frac{10}{9}a+1$, $\frac{4}{9}a^{17}-\frac{11}{9}a^{16}-\frac{16}{9}a^{15}+\frac{40}{9}a^{14}+\frac{94}{9}a^{13}-\frac{106}{9}a^{12}-\frac{341}{9}a^{11}+\frac{265}{9}a^{10}+\frac{569}{9}a^{9}-\frac{295}{9}a^{8}-\frac{559}{9}a^{7}+\frac{154}{9}a^{6}+\frac{263}{9}a^{5}-\frac{25}{9}a^{4}-\frac{26}{9}a^{3}-\frac{4}{9}a^{2}+\frac{11}{9}a+\frac{10}{9}$, $\frac{23}{9}a^{17}-\frac{85}{9}a^{16}-a^{15}+\frac{211}{9}a^{14}+\frac{403}{9}a^{13}-\frac{968}{9}a^{12}-\frac{1099}{9}a^{11}+\frac{2135}{9}a^{10}+\frac{1711}{9}a^{9}-\frac{2489}{9}a^{8}-\frac{1652}{9}a^{7}+164a^{6}+\frac{803}{9}a^{5}-\frac{343}{9}a^{4}-\frac{130}{9}a^{3}+\frac{43}{9}a^{2}+\frac{58}{9}a+\frac{29}{9}$, $\frac{17}{9}a^{17}-7a^{16}-\frac{2}{3}a^{15}+\frac{52}{3}a^{14}+\frac{296}{9}a^{13}-\frac{718}{9}a^{12}-\frac{806}{9}a^{11}+\frac{1591}{9}a^{10}+\frac{1246}{9}a^{9}-\frac{1853}{9}a^{8}-\frac{410}{3}a^{7}+125a^{6}+\frac{208}{3}a^{5}-\frac{263}{9}a^{4}-\frac{131}{9}a^{3}+\frac{8}{9}a^{2}+\frac{47}{9}a+\frac{32}{9}$, $\frac{1}{9}a^{16}-\frac{2}{3}a^{15}+a^{14}+\frac{2}{3}a^{13}+\frac{1}{9}a^{12}-\frac{79}{9}a^{11}+\frac{55}{9}a^{10}+\frac{154}{9}a^{9}-\frac{38}{3}a^{8}-\frac{187}{9}a^{7}+\frac{41}{3}a^{6}+\frac{35}{3}a^{5}-\frac{17}{3}a^{4}-\frac{10}{9}a^{3}+\frac{10}{9}a^{2}-\frac{10}{9}a+\frac{2}{9}$, $\frac{20}{9}a^{17}-\frac{76}{9}a^{16}-\frac{2}{9}a^{15}+\frac{188}{9}a^{14}+38a^{13}-\frac{890}{9}a^{12}-\frac{922}{9}a^{11}+\frac{1972}{9}a^{10}+\frac{1468}{9}a^{9}-\frac{2330}{9}a^{8}-\frac{1517}{9}a^{7}+\frac{1421}{9}a^{6}+\frac{850}{9}a^{5}-\frac{112}{3}a^{4}-\frac{211}{9}a^{3}+\frac{25}{9}a^{2}+\frac{74}{9}a+\frac{32}{9}$, $\frac{7}{9}a^{17}-\frac{25}{9}a^{16}-a^{15}+\frac{76}{9}a^{14}+14a^{13}-\frac{299}{9}a^{12}-\frac{415}{9}a^{11}+\frac{737}{9}a^{10}+\frac{221}{3}a^{9}-\frac{940}{9}a^{8}-\frac{644}{9}a^{7}+\frac{214}{3}a^{6}+\frac{302}{9}a^{5}-23a^{4}-\frac{10}{9}a^{3}+\frac{22}{9}a^{2}-\frac{8}{9}a+1$, $\frac{10}{9}a^{17}-\frac{14}{3}a^{16}+\frac{17}{9}a^{15}+\frac{28}{3}a^{14}+\frac{131}{9}a^{13}-\frac{482}{9}a^{12}-\frac{236}{9}a^{11}+116a^{10}+\frac{182}{9}a^{9}-\frac{1141}{9}a^{8}-\frac{17}{3}a^{7}+\frac{622}{9}a^{6}-\frac{34}{3}a^{5}-\frac{92}{9}a^{4}+\frac{92}{9}a^{3}+\frac{2}{9}a^{2}+\frac{4}{3}a-\frac{5}{9}$
| 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: | \( 1025.11738531 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 1025.11738531 \cdot 1}{18\cdot\sqrt{12213256064722484031}}\cr\approx \mathstrut & 0.248716259546 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 3*x^16 + 9*x^15 + 24*x^14 - 30*x^13 - 78*x^12 + 60*x^11 + 141*x^10 - 56*x^9 - 150*x^8 + 12*x^7 + 81*x^6 + 12*x^5 - 15*x^4 - 3*x^3 + 3*x^2 + 3*x + 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 - 3*x^17 - 3*x^16 + 9*x^15 + 24*x^14 - 30*x^13 - 78*x^12 + 60*x^11 + 141*x^10 - 56*x^9 - 150*x^8 + 12*x^7 + 81*x^6 + 12*x^5 - 15*x^4 - 3*x^3 + 3*x^2 + 3*x + 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 - 3*x^17 - 3*x^16 + 9*x^15 + 24*x^14 - 30*x^13 - 78*x^12 + 60*x^11 + 141*x^10 - 56*x^9 - 150*x^8 + 12*x^7 + 81*x^6 + 12*x^5 - 15*x^4 - 3*x^3 + 3*x^2 + 3*x + 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 - 3*x^17 - 3*x^16 + 9*x^15 + 24*x^14 - 30*x^13 - 78*x^12 + 60*x^11 + 141*x^10 - 56*x^9 - 150*x^8 + 12*x^7 + 81*x^6 + 12*x^5 - 15*x^4 - 3*x^3 + 3*x^2 + 3*x + 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
$S_3^2:C_6$ (as 18T93):
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 216 |
| The 27 conjugacy class representatives for $S_3^2:C_6$ |
| Character table for $S_3^2:C_6$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{9})\), 6.0.255879.1 |
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 12 siblings: | data not computed |
| Degree 18 sibling: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 27 sibling: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.0.851162814333.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.6.0.1}{6} }$ | R | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.6.0.1}{6} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.6.0.1}{6} }$ | R | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.3.0.1}{3} }^{3}{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{4}$ | ${\href{/padicField/37.3.0.1}{3} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.6.0.1}{6} }$ | ${\href{/padicField/43.3.0.1}{3} }^{6}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.6.0.1}{6} }$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(3\)
| Deg $18$ | $18$ | $1$ | $33$ | |||
|
\(13\)
| 13.3.0.1 | $x^{3} + 2 x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 13.3.0.1 | $x^{3} + 2 x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 13.6.0.1 | $x^{6} + 10 x^{3} + 11 x^{2} + 11 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 13.6.3.2 | $x^{6} + 338 x^{2} - 24167$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |