Normalized defining polynomial
\( x^{18} - 9 x^{17} + 144 x^{16} - 948 x^{15} + 8820 x^{14} - 45864 x^{13} + 313680 x^{12} - 1326330 x^{11} + 7239870 x^{10} - 25028738 x^{9} + 113150061 x^{8} - 315334674 x^{7} + 1200620121 x^{6} - 2590431399 x^{5} + 8346151809 x^{4} - 12699762504 x^{3} + 34466899041 x^{2} - 28501566129 x + 64264721489 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 9]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-4535231005796548759010580522990234375=-\,3^{44}\cdot 5^{9}\cdot 11^{9}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $108.76$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $3, 5, 11$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1485=3^{3}\cdot 5\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1485}(1,·)$, $\chi_{1485}(331,·)$, $\chi_{1485}(1156,·)$, $\chi_{1485}(769,·)$, $\chi_{1485}(1099,·)$, $\chi_{1485}(274,·)$, $\chi_{1485}(1429,·)$, $\chi_{1485}(604,·)$, $\chi_{1485}(991,·)$, $\chi_{1485}(1264,·)$, $\chi_{1485}(166,·)$, $\chi_{1485}(934,·)$, $\chi_{1485}(1321,·)$, $\chi_{1485}(109,·)$, $\chi_{1485}(496,·)$, $\chi_{1485}(439,·)$, $\chi_{1485}(826,·)$, $\chi_{1485}(661,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{6913542239825672006322818799857609643123406225230784919274088609} a^{17} - \frac{3323506656001913304934137190033527540974210372061046434947345219}{6913542239825672006322818799857609643123406225230784919274088609} a^{16} - \frac{2157246127873425654089145165326306872658413146483346013137774755}{6913542239825672006322818799857609643123406225230784919274088609} a^{15} + \frac{937766368819381388666736240683307455197930611611060886998582383}{6913542239825672006322818799857609643123406225230784919274088609} a^{14} + \frac{3221480813825263436954495348956781622657007816795058602204039846}{6913542239825672006322818799857609643123406225230784919274088609} a^{13} - \frac{1634064467814841595272316729344191503369481533362606140690215070}{6913542239825672006322818799857609643123406225230784919274088609} a^{12} - \frac{2793065088764989240031117943257040970382593379670456206685872877}{6913542239825672006322818799857609643123406225230784919274088609} a^{11} + \frac{1806914943402079350137844325334078651017505043210433230648596856}{6913542239825672006322818799857609643123406225230784919274088609} a^{10} - \frac{1863097914165197760191808828556739573939271035092768257916245878}{6913542239825672006322818799857609643123406225230784919274088609} a^{9} + \frac{1386187872071866514009762128480414728094047257447196565805128}{11658587250970779100038480269574383883850600717083954332671313} a^{8} + \frac{649328433656848330595106335777725353124038796144769718163555445}{6913542239825672006322818799857609643123406225230784919274088609} a^{7} + \frac{135316070567805002735032643760150865915961696205151843974160423}{6913542239825672006322818799857609643123406225230784919274088609} a^{6} + \frac{2003852840374359520680328944445268848100091473836937146195861033}{6913542239825672006322818799857609643123406225230784919274088609} a^{5} + \frac{730977589985291015734116809062528882312790134869520619538921513}{6913542239825672006322818799857609643123406225230784919274088609} a^{4} - \frac{598478080049175665335839794382264879298583359259005807426126689}{6913542239825672006322818799857609643123406225230784919274088609} a^{3} - \frac{3222657681520424336565828770015046911081955173701710098388016943}{6913542239825672006322818799857609643123406225230784919274088609} a^{2} + \frac{3204771673787168940022309440790755443527062420275344812422120600}{6913542239825672006322818799857609643123406225230784919274088609} a - \frac{2426693865015788388754013848405925456188899493165355692506907396}{6913542239825672006322818799857609643123406225230784919274088609}$
Class group and class number
$C_{846748}$, which has order $846748$ (assuming GRH)
Unit group
| Rank: | $8$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 40934.03294431194 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 18 |
| The 18 conjugacy class representatives for $C_{18}$ |
| Character table for $C_{18}$ |
Intermediate fields
| \(\Q(\sqrt{-55}) \), \(\Q(\zeta_{9})^+\), 6.0.1091586375.3, \(\Q(\zeta_{27})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ | R | R | ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ | R | ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ | $18$ | $18$ | ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ | $18$ | ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ | $18$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||
| 5 | Data not computed | ||||||
| 11 | Data not computed | ||||||