Normalized defining polynomial
\( x^{18} + 468 x^{16} + 91260 x^{14} + 9596496 x^{12} + 588128112 x^{10} + 21172612032 x^{8} + 428157265536 x^{6} + 4337177495040 x^{4} + 16914992230656 x^{2} + 13993003321856 \)
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: | \(-6765429537581484920649996381709729642447896576=-\,2^{27}\cdot 3^{44}\cdot 13^{15}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $351.66$ | 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: | $2, 3, 13$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(2808=2^{3}\cdot 3^{3}\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{2808}(1,·)$, $\chi_{2808}(907,·)$, $\chi_{2808}(1537,·)$, $\chi_{2808}(883,·)$, $\chi_{2808}(841,·)$, $\chi_{2808}(1291,·)$, $\chi_{2808}(1873,·)$, $\chi_{2808}(2755,·)$, $\chi_{2808}(601,·)$, $\chi_{2808}(2713,·)$, $\chi_{2808}(2779,·)$, $\chi_{2808}(355,·)$, $\chi_{2808}(1819,·)$, $\chi_{2808}(2473,·)$, $\chi_{2808}(1777,·)$, $\chi_{2808}(2227,·)$, $\chi_{2808}(937,·)$, $\chi_{2808}(1843,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{104} a^{6}$, $\frac{1}{104} a^{7}$, $\frac{1}{208} a^{8}$, $\frac{1}{733616} a^{9} + \frac{9}{28216} a^{7} + \frac{351}{14108} a^{5} + \frac{1543}{7054} a^{3} - \frac{1389}{3527} a$, $\frac{1}{2219922016} a^{10} - \frac{1080967}{554980504} a^{8} - \frac{721999}{554980504} a^{6} - \frac{1811335}{21345404} a^{4} + \frac{2474565}{10672702} a^{2} - \frac{382}{1513}$, $\frac{1}{2219922016} a^{11} + \frac{11}{85381616} a^{9} + \frac{1432513}{554980504} a^{7} - \frac{342075}{10672702} a^{5} - \frac{998175}{5336351} a^{3} - \frac{106654}{5336351} a$, $\frac{1}{57717972416} a^{12} - \frac{737845}{1109961008} a^{8} + \frac{240186}{69372563} a^{6} + \frac{2530601}{21345404} a^{4} + \frac{547645}{5336351} a^{2} - \frac{337}{1513}$, $\frac{1}{57717972416} a^{13} + \frac{499}{1109961008} a^{9} - \frac{2410231}{554980504} a^{7} + \frac{104073}{10672702} a^{5} - \frac{809516}{5336351} a^{3} - \frac{2169023}{5336351} a$, $\frac{1}{115435944832} a^{14} - \frac{1548067}{1109961008} a^{8} + \frac{111531}{554980504} a^{6} + \frac{196907}{10672702} a^{4} + \frac{528237}{5336351} a^{2} - \frac{20}{1513}$, $\frac{1}{115435944832} a^{15} - \frac{67}{277490252} a^{9} - \frac{2230}{5336351} a^{7} - \frac{636539}{21345404} a^{5} - \frac{1374917}{10672702} a^{3} + \frac{586102}{5336351} a$, $\frac{1}{230871889664} a^{16} + \frac{119531}{65291824} a^{8} + \frac{59447}{32645912} a^{6} - \frac{17717}{313903} a^{4} + \frac{60647}{313903} a^{2} + \frac{508}{1513}$, $\frac{1}{230871889664} a^{17} + \frac{1}{16322956} a^{9} - \frac{14185}{4080739} a^{7} + \frac{38157}{1255612} a^{5} - \frac{23903}{313903} a^{3} + \frac{1253088}{5336351} a$
Class group and class number
$C_{3}\times C_{59698674}$, which has order $179096022$ (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: | \( 59652214.53290313 \) (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
| \(\Q(\sqrt{-26}) \), \(\Q(\zeta_{9})^+\), 6.0.7380232704.12, 9.9.151470380950257681.1 |
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 | ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ | $18$ | R | ${\href{/LocalNumberField/17.1.0.1}{1} }^{18}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ | $18$ | $18$ | ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ | $18$ | ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ | $18$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| $3$ | 3.9.22.2 | $x^{9} + 9 x^{7} + 3 x^{6} + 18 x^{5} + 51$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ |
| 3.9.22.2 | $x^{9} + 9 x^{7} + 3 x^{6} + 18 x^{5} + 51$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ | |
| 13 | Data not computed | ||||||