Normalized defining polynomial
\( x^{18} - 7 x^{17} + 40 x^{16} - 154 x^{15} + 627 x^{14} - 1979 x^{13} + 6508 x^{12} - 16865 x^{11} + 47380 x^{10} - 103535 x^{9} + 254065 x^{8} - 459765 x^{7} + 999805 x^{6} - 1439914 x^{5} + 2800752 x^{4} - 2907231 x^{3} + 5123532 x^{2} - 2913750 x + 4775041 \)
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: | \(-11088656920413061413017818359375=-\,3^{9}\cdot 5^{9}\cdot 19^{16}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $53.05$ | 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: | $3, 5, 19$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(285=3\cdot 5\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{285}(256,·)$, $\chi_{285}(1,·)$, $\chi_{285}(194,·)$, $\chi_{285}(196,·)$, $\chi_{285}(134,·)$, $\chi_{285}(74,·)$, $\chi_{285}(271,·)$, $\chi_{285}(16,·)$, $\chi_{285}(149,·)$, $\chi_{285}(226,·)$, $\chi_{285}(104,·)$, $\chi_{285}(106,·)$, $\chi_{285}(44,·)$, $\chi_{285}(239,·)$, $\chi_{285}(119,·)$, $\chi_{285}(121,·)$, $\chi_{285}(61,·)$, $\chi_{285}(254,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{7411501356656592181466203321313077133973321331} a^{17} + \frac{1682738557907462545045773902922109084413913964}{7411501356656592181466203321313077133973321331} a^{16} + \frac{807522237229070266785860172624574478394746143}{7411501356656592181466203321313077133973321331} a^{15} + \frac{3398915715717610630137767161978683054574047668}{7411501356656592181466203321313077133973321331} a^{14} + \frac{2803081509103715947408754993061402201436382873}{7411501356656592181466203321313077133973321331} a^{13} - \frac{2298919975841044920010086129573228354087374348}{7411501356656592181466203321313077133973321331} a^{12} + \frac{424812794607696096264682100288037127173099089}{7411501356656592181466203321313077133973321331} a^{11} - \frac{2550839963118580214330867815738611093421574762}{7411501356656592181466203321313077133973321331} a^{10} - \frac{2285797400678778670964227612058408818528450861}{7411501356656592181466203321313077133973321331} a^{9} + \frac{2623412566824055181176797016160328164037023527}{7411501356656592181466203321313077133973321331} a^{8} + \frac{367009032232930439810849131902243081927375967}{7411501356656592181466203321313077133973321331} a^{7} + \frac{139159286028700338682990362052094353250244097}{7411501356656592181466203321313077133973321331} a^{6} + \frac{2816671644350102722163992888380535265358707326}{7411501356656592181466203321313077133973321331} a^{5} + \frac{149896263902877470385714319453385511949720650}{7411501356656592181466203321313077133973321331} a^{4} + \frac{2633693025092469994085482100065244444696004381}{7411501356656592181466203321313077133973321331} a^{3} - \frac{211392910844076151177367916561474670881743050}{7411501356656592181466203321313077133973321331} a^{2} + \frac{3310924754130500478352795312724776283878867870}{7411501356656592181466203321313077133973321331} a - \frac{121516822687688678596241730506059675462611316}{7411501356656592181466203321313077133973321331}$
Class group and class number
$C_{3258}$, which has order $3258$ (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: | \( 22305.8950792 \) (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
| \(\Q(\sqrt{-15}) \), 3.3.361.1, 6.0.439833375.1, \(\Q(\zeta_{19})^+\) |
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 | R | ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ | $18$ | ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ | R | ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ | $18$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ | $18$ | $18$ | ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ | $18$ |
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 | ||||||
| 5 | Data not computed | ||||||
| $19$ | 19.9.8.8 | $x^{9} - 19$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ |
| 19.9.8.8 | $x^{9} - 19$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ | |