Normalized defining polynomial
\( x^{18} + 684 x^{16} + 145692 x^{14} + 10156944 x^{12} + 313986096 x^{10} + 4682204352 x^{8} + 31999796352 x^{6} + 81048287232 x^{4} + 69983771904 x^{2} + 14550451712 \)
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: | \(-724363080171793764995208566309796540433577438871552=-\,2^{27}\cdot 3^{44}\cdot 19^{17}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $669.20$ | 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: | $2, 3, 19$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(4104=2^{3}\cdot 3^{3}\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{4104}(1,·)$, $\chi_{4104}(3397,·)$, $\chi_{4104}(577,·)$, $\chi_{4104}(3265,·)$, $\chi_{4104}(1933,·)$, $\chi_{4104}(13,·)$, $\chi_{4104}(3157,·)$, $\chi_{4104}(1849,·)$, $\chi_{4104}(2137,·)$, $\chi_{4104}(2461,·)$, $\chi_{4104}(3937,·)$, $\chi_{4104}(169,·)$, $\chi_{4104}(1405,·)$, $\chi_{4104}(3121,·)$, $\chi_{4104}(3637,·)$, $\chi_{4104}(505,·)$, $\chi_{4104}(3517,·)$, $\chi_{4104}(2197,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{128} a^{14}$, $\frac{1}{156544} a^{15} - \frac{529}{78272} a^{13} - \frac{579}{39136} a^{11} + \frac{577}{19568} a^{9} - \frac{41}{9784} a^{7} - \frac{295}{2446} a^{5} + \frac{202}{1223} a^{3} + \frac{179}{1223} a$, $\frac{1}{247235226298755706361612377917717768448} a^{16} - \frac{30205739758914087225065445385537531}{15452201643672231647600773619857360528} a^{14} + \frac{79470886894272670700350289035173107}{61808806574688926590403094479429442112} a^{12} + \frac{149200454790906762864858099124611215}{30904403287344463295201547239714721056} a^{10} - \frac{294387531887214918474643619445427693}{15452201643672231647600773619857360528} a^{8} - \frac{377013121243007214785072914714745985}{7726100821836115823800386809928680264} a^{6} + \frac{55649335023325373249384575584497380}{965762602729514477975048351241085033} a^{4} - \frac{79830856427952728772069462000027890}{965762602729514477975048351241085033} a^{2} - \frac{26776916925545949079546845484586}{789666886941549041680333893083471}$, $\frac{1}{247235226298755706361612377917717768448} a^{17} - \frac{3925333599345523170695900379061}{61808806574688926590403094479429442112} a^{15} - \frac{266403209586125809555635956135387191}{61808806574688926590403094479429442112} a^{13} + \frac{138168411794248079287695927955525023}{15452201643672231647600773619857360528} a^{11} + \frac{60962567236482150281506632442134257}{15452201643672231647600773619857360528} a^{9} + \frac{84803686303244441504747481577990553}{1931525205459028955950096702482170066} a^{7} - \frac{376759827095334229637835122512364969}{3863050410918057911900193404964340132} a^{5} - \frac{79905357274809004334425200798625209}{1931525205459028955950096702482170066} a^{3} - \frac{238851226891686995602852938122434609}{965762602729514477975048351241085033} a$
Class group and class number
$C_{9}\times C_{271391526}$, which has order $2442523734$ (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: | \( 658443956.5015022 \) (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
| \(\Q(\sqrt{-38}) \), 3.3.29241.2, 6.0.8317790995968.4, 9.9.532962204162830310969.8 |
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 | ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ | $18$ | ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ | R | ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ | $18$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ | $18$ | ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/59.1.0.1}{1} }^{18}$ |
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 | ||||||
| $3$ | 3.9.22.5 | $x^{9} + 9 x^{8} + 12 x^{6} + 18 x^{5} + 9 x^{3} + 60$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ |
| 3.9.22.5 | $x^{9} + 9 x^{8} + 12 x^{6} + 18 x^{5} + 9 x^{3} + 60$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ | |
| 19 | Data not computed | ||||||