Normalized defining polynomial
\( x^{18} - 9 x^{17} + 225 x^{16} - 1596 x^{15} + 22104 x^{14} - 127512 x^{13} + 1262352 x^{12} - 5986422 x^{11} + 46502352 x^{10} - 180492362 x^{9} + 1149875379 x^{8} - 3577518954 x^{7} + 19116861729 x^{6} - 45526465413 x^{5} + 206139713310 x^{4} - 340247545008 x^{3} + 1307472810363 x^{2} - 1144383031881 x + 3709960454951 \)
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: | \(-421412815345064168757967085957375942091=-\,3^{44}\cdot 7^{9}\cdot 13^{9}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $139.90$ | 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: | $3, 7, 13$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(2457=3^{3}\cdot 7\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{2457}(1,·)$, $\chi_{2457}(1093,·)$, $\chi_{2457}(454,·)$, $\chi_{2457}(2185,·)$, $\chi_{2457}(1546,·)$, $\chi_{2457}(274,·)$, $\chi_{2457}(1366,·)$, $\chi_{2457}(727,·)$, $\chi_{2457}(1819,·)$, $\chi_{2457}(547,·)$, $\chi_{2457}(1639,·)$, $\chi_{2457}(1000,·)$, $\chi_{2457}(2092,·)$, $\chi_{2457}(820,·)$, $\chi_{2457}(181,·)$, $\chi_{2457}(1912,·)$, $\chi_{2457}(1273,·)$, $\chi_{2457}(2365,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{53} a^{15} + \frac{9}{53} a^{14} + \frac{17}{53} a^{13} - \frac{9}{53} a^{12} + \frac{17}{53} a^{11} - \frac{15}{53} a^{10} + \frac{10}{53} a^{9} - \frac{13}{53} a^{8} - \frac{19}{53} a^{7} + \frac{2}{53} a^{6} + \frac{10}{53} a^{5} - \frac{10}{53} a^{4} + \frac{6}{53} a^{3} + \frac{21}{53} a^{2} + \frac{14}{53} a$, $\frac{1}{53} a^{16} - \frac{11}{53} a^{14} - \frac{3}{53} a^{13} - \frac{8}{53} a^{12} - \frac{9}{53} a^{11} - \frac{14}{53} a^{10} + \frac{3}{53} a^{9} - \frac{8}{53} a^{8} + \frac{14}{53} a^{7} - \frac{8}{53} a^{6} + \frac{6}{53} a^{5} - \frac{10}{53} a^{4} + \frac{20}{53} a^{3} - \frac{16}{53} a^{2} - \frac{20}{53} a$, $\frac{1}{65264291491968108645277757355164329728455618909785841099330648975401} a^{17} - \frac{419623535538861732246548222195272242255840949839187274451300667033}{65264291491968108645277757355164329728455618909785841099330648975401} a^{16} + \frac{4028505179086813568585976826110530489641060289249417143028526444}{1231401726263549219722221836889893013744445639807280020742087716517} a^{15} - \frac{14403963765290780067932573645889612807963187386916771508901942908602}{65264291491968108645277757355164329728455618909785841099330648975401} a^{14} - \frac{32400893708323788526896623025505258406537000291069933565516784915908}{65264291491968108645277757355164329728455618909785841099330648975401} a^{13} + \frac{25946471759088718018209378644508471380513917886228903393245920868685}{65264291491968108645277757355164329728455618909785841099330648975401} a^{12} - \frac{7487675093699335393721760445806361856853206782843939696688384498696}{65264291491968108645277757355164329728455618909785841099330648975401} a^{11} + \frac{15919959403252661114707370498986221696024860688379537045274586536778}{65264291491968108645277757355164329728455618909785841099330648975401} a^{10} - \frac{30880940654011952540185933734458100345481605209412510041524166179624}{65264291491968108645277757355164329728455618909785841099330648975401} a^{9} - \frac{15959606035802918184267015840591375415556461013846750754486473944858}{65264291491968108645277757355164329728455618909785841099330648975401} a^{8} + \frac{24877119942680745572684564167134432764374870840865049766414468775268}{65264291491968108645277757355164329728455618909785841099330648975401} a^{7} + \frac{25209536989681017857907480049931965809477272475690843494438452534627}{65264291491968108645277757355164329728455618909785841099330648975401} a^{6} - \frac{16384973766814543759928967651851836537934863723103860266744105905491}{65264291491968108645277757355164329728455618909785841099330648975401} a^{5} - \frac{9571400697604378849192949439950903387412777333586436745178664901821}{65264291491968108645277757355164329728455618909785841099330648975401} a^{4} + \frac{4662525862330453223802354224652240260714101625363114319878039913022}{65264291491968108645277757355164329728455618909785841099330648975401} a^{3} - \frac{8396117893757619265454653017574410220694435072441824461756629127058}{65264291491968108645277757355164329728455618909785841099330648975401} a^{2} + \frac{10039164469000046746707903037953608037677911469295893717141339514707}{65264291491968108645277757355164329728455618909785841099330648975401} a - \frac{369724107224103914635052735884425925933666528430653147255824780059}{1231401726263549219722221836889893013744445639807280020742087716517}$
Class group and class number
$C_{7}\times C_{7}\times C_{249242}$, which has order $12212858$ (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
| \(\Q(\sqrt{-91}) \), \(\Q(\zeta_{9})^+\), 6.0.4944179331.10, \(\Q(\zeta_{27})^+\) |
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 | ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ | R | $18$ | R | ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/53.1.0.1}{1} }^{18}$ | ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$ |
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 | ||||||
| 7 | Data not computed | ||||||
| 13 | Data not computed | ||||||