Normalized defining polynomial
\( x^{18} - 7 x^{17} + 58 x^{16} - 266 x^{15} + 1435 x^{14} - 5199 x^{13} + 21866 x^{12} - 64937 x^{11} + 227650 x^{10} - 559943 x^{9} + 1686287 x^{8} - 3386581 x^{7} + 8904619 x^{6} - 13996282 x^{5} + 32437184 x^{4} - 36161351 x^{3} + 74452082 x^{2} - 44882086 x + 83092129 \)
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: | \(-519527019723470744090368949777303=-\,19^{16}\cdot 23^{9}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $65.70$ | 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: | $19, 23$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(437=19\cdot 23\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{437}(1,·)$, $\chi_{437}(68,·)$, $\chi_{437}(321,·)$, $\chi_{437}(137,·)$, $\chi_{437}(139,·)$, $\chi_{437}(206,·)$, $\chi_{437}(275,·)$, $\chi_{437}(277,·)$, $\chi_{437}(24,·)$, $\chi_{437}(346,·)$, $\chi_{437}(47,·)$, $\chi_{437}(93,·)$, $\chi_{437}(415,·)$, $\chi_{437}(229,·)$, $\chi_{437}(45,·)$, $\chi_{437}(367,·)$, $\chi_{437}(252,·)$, $\chi_{437}(254,·)$$\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}{3025743360303391197691220861266214157467082796519307} a^{17} - \frac{937858130649084129601115226447188760413049046379353}{3025743360303391197691220861266214157467082796519307} a^{16} + \frac{1144033288541598726482007602613613078961575227879445}{3025743360303391197691220861266214157467082796519307} a^{15} + \frac{139691210679902972610754876382721390026695310249609}{3025743360303391197691220861266214157467082796519307} a^{14} - \frac{187556376987564632536237889402408304846916809830541}{3025743360303391197691220861266214157467082796519307} a^{13} - \frac{532970375455828326759143811717220658466351968322803}{3025743360303391197691220861266214157467082796519307} a^{12} - \frac{1385272115629972317201980735062799153618245447236812}{3025743360303391197691220861266214157467082796519307} a^{11} + \frac{1389112396085474014255533810533918689564986695862642}{3025743360303391197691220861266214157467082796519307} a^{10} + \frac{1097170949063078611207764493885843265490751859410991}{3025743360303391197691220861266214157467082796519307} a^{9} - \frac{277905326609999143831612630937982729238309628528510}{3025743360303391197691220861266214157467082796519307} a^{8} - \frac{1363662781215295106538982862464016143900586109187588}{3025743360303391197691220861266214157467082796519307} a^{7} - \frac{1353282542506337593588665782486524561157006352432100}{3025743360303391197691220861266214157467082796519307} a^{6} - \frac{787505792596128634336560674855900812673528544006595}{3025743360303391197691220861266214157467082796519307} a^{5} - \frac{727182847883182461234768993517820341298577694354528}{3025743360303391197691220861266214157467082796519307} a^{4} - \frac{256881053783768706843480827206857290114463440470043}{3025743360303391197691220861266214157467082796519307} a^{3} + \frac{763291018007543709837580156025408780401223355743137}{3025743360303391197691220861266214157467082796519307} a^{2} + \frac{978372938840210286380652928490540793174910588282737}{3025743360303391197691220861266214157467082796519307} a - \frac{1369821399556250261421867749182491659366227638789236}{3025743360303391197691220861266214157467082796519307}$
Class group and class number
$C_{13149}$, which has order $13149$ (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
| \(\Q(\sqrt{-23}) \), 3.3.361.1, 6.0.1585615607.1, \(\Q(\zeta_{19})^+\) |
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 | ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ | $18$ | ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ | $18$ | R | R | ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ | $18$ | ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ | $18$ | ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 19 | Data not computed | ||||||
| 23 | Data not computed | ||||||