Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^15 - 3*x^12 + 20*x^9 + 33*x^6 - 12*x^3 + 1)
gp: K = bnfinit(y^18 - 3*y^15 - 3*y^12 + 20*y^9 + 33*y^6 - 12*y^3 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 3*x^15 - 3*x^12 + 20*x^9 + 33*x^6 - 12*x^3 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 3*x^15 - 3*x^12 + 20*x^9 + 33*x^6 - 12*x^3 + 1)
\( x^{18} - 3x^{15} - 3x^{12} + 20x^{9} + 33x^{6} - 12x^{3} + 1 \)
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: |
\(-16369786239449258043\)
\(\medspace = -\,3^{30}\cdot 43^{3}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.68\) | 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: | $3^{187/108}43^{1/2}\approx 43.94008014870428$ | ||
| Ramified primes: |
\(3\), \(43\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-43}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{4}+\frac{1}{3}a$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{5}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{9}+\frac{1}{3}$, $\frac{1}{3}a^{10}+\frac{1}{3}a$, $\frac{1}{9}a^{11}-\frac{1}{9}a^{10}+\frac{1}{9}a^{9}+\frac{1}{9}a^{2}-\frac{1}{9}a+\frac{1}{9}$, $\frac{1}{9}a^{12}+\frac{1}{9}a^{9}+\frac{1}{9}a^{3}+\frac{1}{9}$, $\frac{1}{9}a^{13}+\frac{1}{9}a^{10}+\frac{1}{9}a^{4}+\frac{1}{9}a$, $\frac{1}{9}a^{14}+\frac{1}{9}a^{10}-\frac{1}{9}a^{9}+\frac{1}{9}a^{5}+\frac{1}{9}a-\frac{1}{9}$, $\frac{1}{9}a^{15}-\frac{1}{9}a^{9}+\frac{1}{9}a^{6}-\frac{1}{9}$, $\frac{1}{27}a^{16}+\frac{1}{27}a^{15}-\frac{1}{27}a^{13}-\frac{1}{27}a^{12}+\frac{1}{27}a^{10}+\frac{1}{27}a^{9}+\frac{1}{27}a^{7}+\frac{1}{27}a^{6}-\frac{1}{27}a^{4}-\frac{1}{27}a^{3}+\frac{1}{27}a+\frac{1}{27}$, $\frac{1}{27}a^{17}-\frac{1}{27}a^{15}-\frac{1}{27}a^{14}+\frac{1}{27}a^{12}+\frac{1}{27}a^{11}-\frac{1}{27}a^{9}+\frac{1}{27}a^{8}-\frac{1}{27}a^{6}-\frac{1}{27}a^{5}+\frac{1}{27}a^{3}+\frac{1}{27}a^{2}-\frac{1}{27}$
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: |
\( \frac{16}{3} a^{17} - \frac{137}{9} a^{14} - \frac{164}{9} a^{11} + 104 a^{8} + \frac{1720}{9} a^{5} - \frac{320}{9} a^{2} \)
(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: |
$\frac{4}{9}a^{15}-\frac{11}{9}a^{12}-\frac{5}{3}a^{9}+\frac{79}{9}a^{6}+\frac{148}{9}a^{3}-\frac{7}{3}$, $4a^{17}+\frac{28}{27}a^{16}-\frac{26}{27}a^{15}-\frac{34}{3}a^{14}-\frac{79}{27}a^{13}+\frac{74}{27}a^{12}-\frac{125}{9}a^{11}-\frac{98}{27}a^{10}+\frac{88}{27}a^{9}+\frac{233}{3}a^{8}+\frac{541}{27}a^{7}-\frac{503}{27}a^{6}+145a^{5}+\frac{1028}{27}a^{4}-\frac{934}{27}a^{3}-\frac{218}{9}a^{2}-\frac{152}{27}a+\frac{151}{27}$, $\frac{11}{3}a^{17}-\frac{28}{27}a^{16}+\frac{8}{27}a^{15}-\frac{94}{9}a^{14}+\frac{79}{27}a^{13}-\frac{23}{27}a^{12}-\frac{113}{9}a^{11}+\frac{98}{27}a^{10}-\frac{28}{27}a^{9}+\frac{214}{3}a^{8}-\frac{541}{27}a^{7}+\frac{161}{27}a^{6}+\frac{1187}{9}a^{5}-\frac{1028}{27}a^{4}+\frac{283}{27}a^{3}-\frac{215}{9}a^{2}+\frac{152}{27}a-\frac{64}{27}$, $\frac{13}{9}a^{15}-\frac{37}{9}a^{12}-\frac{44}{9}a^{9}+\frac{250}{9}a^{6}+\frac{473}{9}a^{3}-\frac{77}{9}$, $\frac{73}{27}a^{17}+\frac{43}{27}a^{16}-\frac{208}{27}a^{14}-\frac{121}{27}a^{13}-\frac{251}{27}a^{11}-\frac{152}{27}a^{10}+\frac{1423}{27}a^{8}+\frac{835}{27}a^{7}+\frac{2627}{27}a^{5}+\frac{1571}{27}a^{4}-\frac{494}{27}a^{2}-\frac{224}{27}a$, $\frac{107}{27}a^{17}-\frac{46}{27}a^{16}-\frac{2}{3}a^{15}-\frac{305}{27}a^{14}+\frac{130}{27}a^{13}+\frac{17}{9}a^{12}-\frac{367}{27}a^{11}+\frac{161}{27}a^{10}+\frac{7}{3}a^{9}+\frac{2087}{27}a^{8}-\frac{892}{27}a^{7}-13a^{6}+\frac{3844}{27}a^{5}-\frac{1670}{27}a^{4}-\frac{214}{9}a^{3}-\frac{709}{27}a^{2}+\frac{260}{27}a+4$, $\frac{100}{27}a^{17}+4a^{16}+\frac{26}{27}a^{15}-\frac{283}{27}a^{14}-\frac{34}{3}a^{13}-\frac{74}{27}a^{12}-\frac{347}{27}a^{11}-\frac{125}{9}a^{10}-\frac{88}{27}a^{9}+\frac{1936}{27}a^{8}+\frac{233}{3}a^{7}+\frac{503}{27}a^{6}+\frac{3632}{27}a^{5}+145a^{4}+\frac{934}{27}a^{3}-\frac{563}{27}a^{2}-\frac{218}{9}a-\frac{151}{27}$, $\frac{71}{27}a^{16}-\frac{43}{27}a^{15}-\frac{203}{27}a^{13}+\frac{121}{27}a^{12}-\frac{241}{27}a^{10}+\frac{152}{27}a^{9}+\frac{1385}{27}a^{7}-\frac{835}{27}a^{6}+\frac{2533}{27}a^{4}-\frac{1571}{27}a^{3}-\frac{493}{27}a+\frac{251}{27}$
| 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: | \( 1206.62977052 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 1206.62977052 \cdot 1}{18\cdot\sqrt{16369786239449258043}}\cr\approx \mathstrut & 0.252870892961 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^15 - 3*x^12 + 20*x^9 + 33*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 - 3*x^15 - 3*x^12 + 20*x^9 + 33*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 - 3*x^15 - 3*x^12 + 20*x^9 + 33*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 - 3*x^15 - 3*x^12 + 20*x^9 + 33*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
$S_3^2:C_6$ (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
| \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{9})\), 6.0.31347.1 |
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: | 12.0.92407922456769.1 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{3}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.6.0.1}{6} }$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.3.0.1}{3} }^{3}{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.6.0.1}{6} }$ | R | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.6.0.1}{6} }$ |
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 $18$ | $18$ | $1$ | $30$ | |||
|
\(43\)
| 43.3.0.1 | $x^{3} + x + 40$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 43.3.0.1 | $x^{3} + x + 40$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 43.6.3.1 | $x^{6} + 1849 x^{2} - 3180280$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 43.6.0.1 | $x^{6} + 19 x^{3} + 28 x^{2} + 21 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |