Normalized defining polynomial
\( x^{18} - 2 x^{17} + 75 x^{16} - 130 x^{15} + 2812 x^{14} - 4244 x^{13} + 67463 x^{12} - 87766 x^{11} + 1128241 x^{10} - 1240582 x^{9} + 13547997 x^{8} - 12178574 x^{7} + 116358536 x^{6} - 80761652 x^{5} + 687949530 x^{4} - 330301660 x^{3} + 2540118205 x^{2} - 638418210 x + 4468927451 \)
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: | \(-75613185918270483380568064000000000=-\,2^{27}\cdot 5^{9}\cdot 19^{16}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $86.64$ | 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: | $2, 5, 19$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(760=2^{3}\cdot 5\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{760}(1,·)$, $\chi_{760}(579,·)$, $\chi_{760}(161,·)$, $\chi_{760}(201,·)$, $\chi_{760}(139,·)$, $\chi_{760}(81,·)$, $\chi_{760}(339,·)$, $\chi_{760}(441,·)$, $\chi_{760}(539,·)$, $\chi_{760}(481,·)$, $\chi_{760}(419,·)$, $\chi_{760}(681,·)$, $\chi_{760}(619,·)$, $\chi_{760}(321,·)$, $\chi_{760}(99,·)$, $\chi_{760}(499,·)$, $\chi_{760}(739,·)$, $\chi_{760}(121,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{37} a^{15} - \frac{1}{37} a^{14} + \frac{14}{37} a^{13} - \frac{3}{37} a^{12} - \frac{8}{37} a^{11} + \frac{18}{37} a^{8} + \frac{18}{37} a^{7} + \frac{2}{37} a^{6} - \frac{10}{37} a^{5} - \frac{14}{37} a^{4} + \frac{1}{37} a^{2} - \frac{18}{37} a$, $\frac{1}{37} a^{16} + \frac{13}{37} a^{14} + \frac{11}{37} a^{13} - \frac{11}{37} a^{12} - \frac{8}{37} a^{11} + \frac{18}{37} a^{9} - \frac{1}{37} a^{8} - \frac{17}{37} a^{7} - \frac{8}{37} a^{6} + \frac{13}{37} a^{5} - \frac{14}{37} a^{4} + \frac{1}{37} a^{3} - \frac{17}{37} a^{2} - \frac{18}{37} a$, $\frac{1}{109188427699025625508936188858907331869590835011767098649} a^{17} + \frac{1380918577771432311547978139531753556960663299332673580}{109188427699025625508936188858907331869590835011767098649} a^{16} + \frac{518958109665705948707097636994235663410602890869468904}{109188427699025625508936188858907331869590835011767098649} a^{15} - \frac{17565800385034824468822904303011526389386207987587162409}{109188427699025625508936188858907331869590835011767098649} a^{14} + \frac{20173890945665179613809613935035308901723102289021097161}{109188427699025625508936188858907331869590835011767098649} a^{13} + \frac{1921357114440923773650586014251805037233631499924328101}{109188427699025625508936188858907331869590835011767098649} a^{12} + \frac{35057290348676054513326182537018047128027442125219297446}{109188427699025625508936188858907331869590835011767098649} a^{11} + \frac{5395968982232994379589247799818445178689548265634531273}{109188427699025625508936188858907331869590835011767098649} a^{10} - \frac{39697772327757090428004968085487437463335833253596009267}{109188427699025625508936188858907331869590835011767098649} a^{9} + \frac{48638515449551262497277840540660761006605155739246289149}{109188427699025625508936188858907331869590835011767098649} a^{8} + \frac{25895633018441478860784304323496268558788833573782023385}{109188427699025625508936188858907331869590835011767098649} a^{7} - \frac{51781488236501277415171203947331348547977432756461734157}{109188427699025625508936188858907331869590835011767098649} a^{6} - \frac{9751230376029849378822449258557350575662586628857343341}{109188427699025625508936188858907331869590835011767098649} a^{5} - \frac{41971222283760108748444443846102134746576910907688199631}{109188427699025625508936188858907331869590835011767098649} a^{4} + \frac{13877836931075514820095536333146756807839921061226109675}{109188427699025625508936188858907331869590835011767098649} a^{3} - \frac{33525122662908955666108546761512562609750564805514335651}{109188427699025625508936188858907331869590835011767098649} a^{2} - \frac{15709157532819411878509631926091211453866662328521494069}{109188427699025625508936188858907331869590835011767098649} a - \frac{34729910209668094300832019124007573868511167862614197}{2951038586460152040782059158348846807286238784101813477}$
Class group and class number
$C_{324558}$, which has order $324558$ (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
| \(\Q(\sqrt{-10}) \), 3.3.361.1, 6.0.8340544000.3, \(\Q(\zeta_{19})^+\) |
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 | R | $18$ | R | ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ | $18$ | R | ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ | $18$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/37.1.0.1}{1} }^{18}$ | ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ | $18$ | ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | 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 | $x^{9} - 19$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ | |