Normalized defining polynomial
\( x^{18} - 9 x^{17} + 72 x^{16} - 372 x^{15} + 1908 x^{14} - 7560 x^{13} + 30192 x^{12} - 97146 x^{11} + 320526 x^{10} - 855266 x^{9} + 2391597 x^{8} - 5261490 x^{7} + 12570585 x^{6} - 22062519 x^{5} + 44899785 x^{4} - 57927960 x^{3} + 99243441 x^{2} - 73205505 x + 103239849 \)
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: | \(-1773722731399331010404115847164903=-\,3^{44}\cdot 23^{9}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $70.33$ | 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: | $3, 23$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(621=3^{3}\cdot 23\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{621}(1,·)$, $\chi_{621}(70,·)$, $\chi_{621}(139,·)$, $\chi_{621}(208,·)$, $\chi_{621}(277,·)$, $\chi_{621}(22,·)$, $\chi_{621}(346,·)$, $\chi_{621}(91,·)$, $\chi_{621}(415,·)$, $\chi_{621}(160,·)$, $\chi_{621}(484,·)$, $\chi_{621}(229,·)$, $\chi_{621}(553,·)$, $\chi_{621}(298,·)$, $\chi_{621}(367,·)$, $\chi_{621}(436,·)$, $\chi_{621}(505,·)$, $\chi_{621}(574,·)$$\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}{4186954850560758341917949366983624269977411396707873} a^{17} + \frac{920239497120926580382772495444568532832559456636671}{4186954850560758341917949366983624269977411396707873} a^{16} + \frac{1643593766375373671084522883402682059142593931000386}{4186954850560758341917949366983624269977411396707873} a^{15} + \frac{818372402193863030519241447704851249909841859341071}{4186954850560758341917949366983624269977411396707873} a^{14} - \frac{1731719798117223921514414807801604375456297185038876}{4186954850560758341917949366983624269977411396707873} a^{13} + \frac{1008289114691467711106289615720671346295512796136978}{4186954850560758341917949366983624269977411396707873} a^{12} + \frac{1559758400059192633044580272686540954908513870412122}{4186954850560758341917949366983624269977411396707873} a^{11} + \frac{1093921227758268021447590045022184738434631231768323}{4186954850560758341917949366983624269977411396707873} a^{10} + \frac{461946685027071124645910763479022305052387778989916}{4186954850560758341917949366983624269977411396707873} a^{9} - \frac{1795020480028047730472672337261371909850057797172211}{4186954850560758341917949366983624269977411396707873} a^{8} + \frac{772060117527225858405310033341219825070396450333289}{4186954850560758341917949366983624269977411396707873} a^{7} - \frac{1008195750418419287731825897270933450368322300458976}{4186954850560758341917949366983624269977411396707873} a^{6} - \frac{1929347758078935650275058957094396550963262632050482}{4186954850560758341917949366983624269977411396707873} a^{5} + \frac{831019264250846636676838734725554442692428874368718}{4186954850560758341917949366983624269977411396707873} a^{4} - \frac{1512026114532912749334463813674809103797656041793034}{4186954850560758341917949366983624269977411396707873} a^{3} - \frac{2994910502113457073683760898595671474632555105591}{6471336708749240095700076301365725301356122715159} a^{2} - \frac{2063818235929425444911401563259193194739080805938255}{4186954850560758341917949366983624269977411396707873} a - \frac{2093363793812175239923786610978062917907035130997}{6471336708749240095700076301365725301356122715159}$
Class group and class number
$C_{37449}$, which has order $37449$ (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
| \(\Q(\sqrt{-23}) \), \(\Q(\zeta_{9})^+\), 6.0.79827687.1, \(\Q(\zeta_{27})^+\) |
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 | ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ | R | $18$ | $18$ | $18$ | ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ | R | ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ | $18$ | ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 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 | $x^{9} + 18 x^{7} + 24 x^{6} + 18 x^{5} + 18 x^{3} + 3$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ | |
| 23 | Data not computed | ||||||