Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 24*x^16 - 68*x^15 + 159*x^14 - 300*x^13 + 479*x^12 - 630*x^11 + 702*x^10 - 636*x^9 + 480*x^8 - 270*x^7 + 114*x^6 - 12*x^5 - 12*x^4 + 14*x^3 - 3*x^2 + 1)
gp: K = bnfinit(y^18 - 6*y^17 + 24*y^16 - 68*y^15 + 159*y^14 - 300*y^13 + 479*y^12 - 630*y^11 + 702*y^10 - 636*y^9 + 480*y^8 - 270*y^7 + 114*y^6 - 12*y^5 - 12*y^4 + 14*y^3 - 3*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 6*x^17 + 24*x^16 - 68*x^15 + 159*x^14 - 300*x^13 + 479*x^12 - 630*x^11 + 702*x^10 - 636*x^9 + 480*x^8 - 270*x^7 + 114*x^6 - 12*x^5 - 12*x^4 + 14*x^3 - 3*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 6*x^17 + 24*x^16 - 68*x^15 + 159*x^14 - 300*x^13 + 479*x^12 - 630*x^11 + 702*x^10 - 636*x^9 + 480*x^8 - 270*x^7 + 114*x^6 - 12*x^5 - 12*x^4 + 14*x^3 - 3*x^2 + 1)
\( x^{18} - 6 x^{17} + 24 x^{16} - 68 x^{15} + 159 x^{14} - 300 x^{13} + 479 x^{12} - 630 x^{11} + 702 x^{10} + \cdots + 1 \)
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: |
\(-1999004627104432128\)
\(\medspace = -\,2^{18}\cdot 3^{27}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(10.39\) | 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: | $2^{3/2}3^{3/2}\approx 14.696938456699069$ | ||
| Ramified primes: |
\(2\), \(3\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{10}a^{15}-\frac{3}{10}a^{14}+\frac{1}{10}a^{13}-\frac{2}{5}a^{11}+\frac{1}{10}a^{10}-\frac{1}{5}a^{9}-\frac{1}{5}a^{8}-\frac{3}{10}a^{7}-\frac{3}{10}a^{6}-\frac{3}{10}a^{5}+\frac{2}{5}a^{4}-\frac{1}{10}a^{3}+\frac{2}{5}a+\frac{3}{10}$, $\frac{1}{10}a^{16}+\frac{1}{5}a^{14}+\frac{3}{10}a^{13}-\frac{2}{5}a^{12}-\frac{1}{10}a^{11}+\frac{1}{10}a^{10}+\frac{1}{5}a^{9}+\frac{1}{10}a^{8}-\frac{1}{5}a^{7}-\frac{1}{5}a^{6}-\frac{1}{2}a^{5}+\frac{1}{10}a^{4}-\frac{3}{10}a^{3}+\frac{2}{5}a^{2}-\frac{1}{2}a-\frac{1}{10}$, $\frac{1}{7570}a^{17}-\frac{28}{757}a^{16}+\frac{287}{7570}a^{15}-\frac{746}{3785}a^{14}+\frac{1701}{7570}a^{13}+\frac{2209}{7570}a^{12}-\frac{3729}{7570}a^{11}+\frac{1437}{7570}a^{10}+\frac{1361}{7570}a^{9}+\frac{204}{3785}a^{8}-\frac{1547}{7570}a^{7}-\frac{334}{757}a^{6}-\frac{1862}{3785}a^{5}-\frac{2343}{7570}a^{4}-\frac{3751}{7570}a^{3}-\frac{195}{1514}a^{2}-\frac{831}{7570}a+\frac{573}{1514}$
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: |
\( \frac{11131}{3785} a^{17} - \frac{12437}{757} a^{16} + \frac{465669}{7570} a^{15} - \frac{1220299}{7570} a^{14} + \frac{2648237}{7570} a^{13} - \frac{2269981}{3785} a^{12} + \frac{3238771}{3785} a^{11} - \frac{7271311}{7570} a^{10} + \frac{3324951}{3785} a^{9} - \frac{2210992}{3785} a^{8} + \frac{2203271}{7570} a^{7} - \frac{64853}{1514} a^{6} - \frac{152233}{7570} a^{5} + \frac{142547}{3785} a^{4} + \frac{11263}{7570} a^{3} - \frac{2497}{757} a^{2} + \frac{19604}{3785} a + \frac{1433}{1514} \)
(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: |
$\frac{691}{1514}a^{17}-\frac{601}{757}a^{16}-\frac{3449}{3785}a^{15}+\frac{119159}{7570}a^{14}-\frac{217834}{3785}a^{13}+\frac{239519}{1514}a^{12}-\frac{2409843}{7570}a^{11}+\frac{1986961}{3785}a^{10}-\frac{5079699}{7570}a^{9}+\frac{2641983}{3785}a^{8}-\frac{2037698}{3785}a^{7}+\frac{2394399}{7570}a^{6}-\frac{680991}{7570}a^{5}-\frac{52697}{7570}a^{4}+\frac{147304}{3785}a^{3}-\frac{21191}{1514}a^{2}+\frac{16093}{7570}a+\frac{10998}{3785}$, $\frac{2861}{1514}a^{17}-\frac{37528}{3785}a^{16}+\frac{270573}{7570}a^{15}-\frac{337723}{3785}a^{14}+\frac{282169}{1514}a^{13}-\frac{2282061}{7570}a^{12}+\frac{3054669}{7570}a^{11}-\frac{3080283}{7570}a^{10}+\frac{2448717}{7570}a^{9}-\frac{577606}{3785}a^{8}+\frac{227403}{7570}a^{7}+\frac{218829}{3785}a^{6}-\frac{168162}{3785}a^{5}+\frac{266781}{7570}a^{4}-\frac{9463}{1514}a^{3}+\frac{17761}{7570}a^{2}-\frac{5583}{7570}a-\frac{31}{1514}$, $\frac{3254}{3785}a^{17}-\frac{45561}{7570}a^{16}+\frac{188013}{7570}a^{15}-\frac{549479}{7570}a^{14}+\frac{638021}{3785}a^{13}-\frac{1202502}{3785}a^{12}+\frac{3724771}{7570}a^{11}-\frac{2349719}{3785}a^{10}+\frac{2393121}{3785}a^{9}-\frac{3836011}{7570}a^{8}+\frac{462269}{1514}a^{7}-\frac{924519}{7570}a^{6}+\frac{55421}{3785}a^{5}+\frac{119613}{7570}a^{4}-\frac{49091}{3785}a^{3}+\frac{5993}{3785}a^{2}+\frac{729}{1514}a-\frac{1259}{3785}$, $\frac{1719}{757}a^{17}-\frac{56872}{3785}a^{16}+\frac{458903}{7570}a^{15}-\frac{1318257}{7570}a^{14}+\frac{3042691}{7570}a^{13}-\frac{2850067}{3785}a^{12}+\frac{4412421}{3785}a^{11}-\frac{11182041}{7570}a^{10}+\frac{5784868}{3785}a^{9}-\frac{951936}{757}a^{8}+\frac{6008789}{7570}a^{7}-\frac{2579791}{7570}a^{6}+\frac{422551}{7570}a^{5}+\frac{147244}{3785}a^{4}-\frac{288361}{7570}a^{3}+\frac{30902}{3785}a^{2}+\frac{5906}{3785}a-\frac{9397}{7570}$, $\frac{4461}{3785}a^{17}-\frac{36366}{3785}a^{16}+\frac{317623}{7570}a^{15}-\frac{196473}{1514}a^{14}+\frac{473425}{1514}a^{13}-\frac{2331837}{3785}a^{12}+\frac{3757754}{3785}a^{11}-\frac{9964029}{7570}a^{10}+\frac{1066215}{757}a^{9}-\frac{4557637}{3785}a^{8}+\frac{5850143}{7570}a^{7}-\frac{2584573}{7570}a^{6}+\frac{67819}{1514}a^{5}+\frac{180699}{3785}a^{4}-\frac{67869}{1514}a^{3}+\frac{32041}{3785}a^{2}+\frac{16592}{3785}a-\frac{19771}{7570}$, $\frac{6246}{3785}a^{17}-\frac{15981}{1514}a^{16}+\frac{321777}{7570}a^{15}-\frac{914423}{7570}a^{14}+\frac{209982}{757}a^{13}-\frac{1940616}{3785}a^{12}+\frac{1188961}{1514}a^{11}-\frac{3680024}{3785}a^{10}+\frac{3702183}{3785}a^{9}-\frac{1147641}{1514}a^{8}+\frac{3326587}{7570}a^{7}-\frac{1077679}{7570}a^{6}-\frac{20961}{3785}a^{5}+\frac{360981}{7570}a^{4}-\frac{16119}{757}a^{3}+\frac{3828}{757}a^{2}+\frac{5731}{1514}a+\frac{67}{3785}$, $\frac{9711}{7570}a^{17}-\frac{29491}{3785}a^{16}+\frac{229911}{7570}a^{15}-\frac{317099}{3785}a^{14}+\frac{1426859}{7570}a^{13}-\frac{2580123}{7570}a^{12}+\frac{3876887}{7570}a^{11}-\frac{4715941}{7570}a^{10}+\frac{4711019}{7570}a^{9}-\frac{1845591}{3785}a^{8}+\frac{455815}{1514}a^{7}-\frac{477069}{3785}a^{6}+\frac{125482}{3785}a^{5}+\frac{11631}{7570}a^{4}-\frac{9719}{7570}a^{3}-\frac{16323}{7570}a^{2}+\frac{2681}{1514}a-\frac{3791}{7570}$, $\frac{2337}{1514}a^{17}-\frac{52279}{7570}a^{16}+\frac{85962}{3785}a^{15}-\frac{73491}{1514}a^{14}+\frac{668067}{7570}a^{13}-\frac{797899}{7570}a^{12}+\frac{321124}{3785}a^{11}+\frac{39581}{1514}a^{10}-\frac{1157981}{7570}a^{9}+\frac{2117223}{7570}a^{8}-\frac{1009227}{3785}a^{7}+\frac{1587421}{7570}a^{6}-\frac{264726}{3785}a^{5}+\frac{62306}{3785}a^{4}+\frac{207283}{7570}a^{3}-\frac{74211}{7570}a^{2}+\frac{17313}{3785}a+\frac{16611}{7570}$
| 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: | \( 269.804731118 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 269.804731118 \cdot 1}{18\cdot\sqrt{1999004627104432128}}\cr\approx \mathstrut & 0.161803894792 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 24*x^16 - 68*x^15 + 159*x^14 - 300*x^13 + 479*x^12 - 630*x^11 + 702*x^10 - 636*x^9 + 480*x^8 - 270*x^7 + 114*x^6 - 12*x^5 - 12*x^4 + 14*x^3 - 3*x^2 + 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^17 + 24*x^16 - 68*x^15 + 159*x^14 - 300*x^13 + 479*x^12 - 630*x^11 + 702*x^10 - 636*x^9 + 480*x^8 - 270*x^7 + 114*x^6 - 12*x^5 - 12*x^4 + 14*x^3 - 3*x^2 + 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^17 + 24*x^16 - 68*x^15 + 159*x^14 - 300*x^13 + 479*x^12 - 630*x^11 + 702*x^10 - 636*x^9 + 480*x^8 - 270*x^7 + 114*x^6 - 12*x^5 - 12*x^4 + 14*x^3 - 3*x^2 + 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^17 + 24*x^16 - 68*x^15 + 159*x^14 - 300*x^13 + 479*x^12 - 630*x^11 + 702*x^10 - 636*x^9 + 480*x^8 - 270*x^7 + 114*x^6 - 12*x^5 - 12*x^4 + 14*x^3 - 3*x^2 + 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
$C_6\times S_3$ (as 18T6):
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 36 |
| The 18 conjugacy class representatives for $S_3 \times C_6$ |
| Character table for $S_3 \times C_6$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), 3.1.648.1, \(\Q(\zeta_{9})\), 6.0.1259712.1, 9.3.272097792.1 |
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
| Galois closure: | data not computed |
| Degree 12 sibling: | 12.0.139314069504.1 |
| Degree 18 sibling: | 18.6.1023490369077469249536.1 |
| Minimal sibling: | 12.0.139314069504.1 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{3}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.3.0.1}{3} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}{,}\,{\href{/padicField/37.1.0.1}{1} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.2.0.1}{2} }^{9}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.0.1 | $x^{6} + x^{4} + x^{3} + x + 1$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
| 2.12.18.15 | $x^{12} - 12 x^{11} - 56 x^{9} + 2372 x^{8} + 8992 x^{7} + 115648 x^{6} + 164160 x^{5} + 1305648 x^{4} - 282560 x^{3} + 5591296 x^{2} - 9640320 x + 8497856$ | $2$ | $6$ | $18$ | $C_6\times C_2$ | $[3]^{6}$ | |
|
\(3\)
| Deg $18$ | $6$ | $3$ | $27$ |