Normalized defining polynomial
\( x^{18} + 27 x^{16} + 495 x^{14} + 6219 x^{12} + 59292 x^{10} - 2 x^{9} + 430038 x^{8} + 378 x^{7} + 2361801 x^{6} - 8244 x^{5} + 9427185 x^{4} + 39510 x^{3} + 24883956 x^{2} - 34668 x + 33385281 \)
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: | \(-504202701918008951235072000000000=-\,2^{18}\cdot 3^{44}\cdot 5^{9}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $65.59$ | 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: | $2, 3, 5$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(540=2^{2}\cdot 3^{3}\cdot 5\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{540}(1,·)$, $\chi_{540}(259,·)$, $\chi_{540}(199,·)$, $\chi_{540}(139,·)$, $\chi_{540}(79,·)$, $\chi_{540}(19,·)$, $\chi_{540}(481,·)$, $\chi_{540}(421,·)$, $\chi_{540}(361,·)$, $\chi_{540}(301,·)$, $\chi_{540}(241,·)$, $\chi_{540}(499,·)$, $\chi_{540}(181,·)$, $\chi_{540}(439,·)$, $\chi_{540}(121,·)$, $\chi_{540}(379,·)$, $\chi_{540}(61,·)$, $\chi_{540}(319,·)$$\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}{58470156136560885192272482601249317018645708669009} a^{17} + \frac{19729955646141637448614677758616097926384504947124}{58470156136560885192272482601249317018645708669009} a^{16} - \frac{9466671784283424155896270635979362352871127558046}{58470156136560885192272482601249317018645708669009} a^{15} + \frac{24231436461898208897896352924313680334484059663187}{58470156136560885192272482601249317018645708669009} a^{14} + \frac{28670491505809160019453414393806824992390288777451}{58470156136560885192272482601249317018645708669009} a^{13} + \frac{3554502070053123205155468091413102601392282716921}{58470156136560885192272482601249317018645708669009} a^{12} - \frac{19904339165908078887643910398806863021307563098919}{58470156136560885192272482601249317018645708669009} a^{11} + \frac{1969962903523027387373243812725080630841062124686}{58470156136560885192272482601249317018645708669009} a^{10} + \frac{20026241565168442634733270546886039296608752973786}{58470156136560885192272482601249317018645708669009} a^{9} - \frac{28519401045287036181342735145196152486553855217851}{58470156136560885192272482601249317018645708669009} a^{8} - \frac{9155265956013516176417054612028133011750449126738}{58470156136560885192272482601249317018645708669009} a^{7} - \frac{23861742304153858844035694204193525805353271856153}{58470156136560885192272482601249317018645708669009} a^{6} - \frac{23451317063478653278817311886561563668017197620380}{58470156136560885192272482601249317018645708669009} a^{5} + \frac{27116021886895225035088287151483514248793959509889}{58470156136560885192272482601249317018645708669009} a^{4} + \frac{16731345769519962638809483573653163355924358746335}{58470156136560885192272482601249317018645708669009} a^{3} + \frac{25306229818617207095207117429728287676611193911200}{58470156136560885192272482601249317018645708669009} a^{2} + \frac{27467841857739674785370003611362423350218198963832}{58470156136560885192272482601249317018645708669009} a - \frac{26219106214527490209148304311993035429353375166824}{58470156136560885192272482601249317018645708669009}$
Class group and class number
$C_{24966}$, which has order $24966$ (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: | \( 40934.0329443 \) (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
| \(\Q(\sqrt{-5}) \), \(\Q(\zeta_{9})^+\), 6.0.52488000.1, \(\Q(\zeta_{27})^+\) |
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 | R | R | R | ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ | $18$ | $18$ | ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ | $18$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ | $18$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| $3$ | 3.9.22.8 | $x^{9} + 18 x^{7} + 24 x^{6} + 18 x^{5} + 18 x^{3} + 3$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ |
| 3.9.22.8 | $x^{9} + 18 x^{7} + 24 x^{6} + 18 x^{5} + 18 x^{3} + 3$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ | |
| 5 | Data not computed | ||||||