Normalized defining polynomial
\( x^{18} - 9 x^{17} + 63 x^{16} - 300 x^{15} + 1368 x^{14} - 5040 x^{13} + 18192 x^{12} - 54630 x^{11} + 164196 x^{10} - 411866 x^{9} + 1056267 x^{8} - 2200338 x^{7} + 4844985 x^{6} - 8116569 x^{5} + 15278400 x^{4} - 19032684 x^{3} + 30278943 x^{2} - 21799593 x + 28658393 \)
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: | \(-317773455265378312682733715471299=-\,3^{44}\cdot 19^{9}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $63.93$ | 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: | $3, 19$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(513=3^{3}\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{513}(1,·)$, $\chi_{513}(322,·)$, $\chi_{513}(265,·)$, $\chi_{513}(343,·)$, $\chi_{513}(400,·)$, $\chi_{513}(151,·)$, $\chi_{513}(94,·)$, $\chi_{513}(229,·)$, $\chi_{513}(208,·)$, $\chi_{513}(37,·)$, $\chi_{513}(172,·)$, $\chi_{513}(493,·)$, $\chi_{513}(115,·)$, $\chi_{513}(436,·)$, $\chi_{513}(286,·)$, $\chi_{513}(457,·)$, $\chi_{513}(58,·)$, $\chi_{513}(379,·)$$\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}{12440870317958056479097007757878329477720025547457} a^{17} - \frac{1937003492585187251396144797716488230758304787094}{12440870317958056479097007757878329477720025547457} a^{16} + \frac{3071201333118363804279824776341312267414834301228}{12440870317958056479097007757878329477720025547457} a^{15} - \frac{5436789091039797883005870576779988239423234150999}{12440870317958056479097007757878329477720025547457} a^{14} + \frac{4688620761565292111559679548976014643365832882068}{12440870317958056479097007757878329477720025547457} a^{13} - \frac{4426432798127403288064430287458856922376629965000}{12440870317958056479097007757878329477720025547457} a^{12} + \frac{5605351058966048807269613032255501889754283899099}{12440870317958056479097007757878329477720025547457} a^{11} - \frac{5427800159024562851307030800916274084412569601642}{12440870317958056479097007757878329477720025547457} a^{10} - \frac{4704387881728437074195030618007271765267447957674}{12440870317958056479097007757878329477720025547457} a^{9} - \frac{5580290417238958473076632401562818223747885372145}{12440870317958056479097007757878329477720025547457} a^{8} - \frac{2920241800497606644075829256679935042547648124424}{12440870317958056479097007757878329477720025547457} a^{7} - \frac{1127510100111857455114421359823379807433365637936}{12440870317958056479097007757878329477720025547457} a^{6} - \frac{5808636448350361662156625053648864331180539372835}{12440870317958056479097007757878329477720025547457} a^{5} - \frac{4757632188556443488921656267155023445572648265609}{12440870317958056479097007757878329477720025547457} a^{4} + \frac{3456745603328810046436977984933852058291309332809}{12440870317958056479097007757878329477720025547457} a^{3} + \frac{2326176202463518013802375672494452892413685963293}{12440870317958056479097007757878329477720025547457} a^{2} + \frac{2706091179963956547489512282289278619165442056714}{12440870317958056479097007757878329477720025547457} a - \frac{5736577488520219750784576244421760849024391517121}{12440870317958056479097007757878329477720025547457}$
Class group and class number
$C_{2}\times C_{3746}$, which has order $7492$ (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
| \(\Q(\sqrt{-19}) \), \(\Q(\zeta_{9})^+\), 6.0.45001899.1, \(\Q(\zeta_{27})^+\) |
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 | $18$ | R | ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ | $18$ | ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ | R | ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ | $18$ | $18$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ | $18$ | ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ | $18$ |
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 |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||
| $19$ | 19.6.3.2 | $x^{6} - 361 x^{2} + 27436$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 19.6.3.2 | $x^{6} - 361 x^{2} + 27436$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 19.6.3.2 | $x^{6} - 361 x^{2} + 27436$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |