Normalized defining polynomial
\( x^{18} - 2 x^{17} + 30 x^{16} - 50 x^{15} + 552 x^{14} - 814 x^{13} + 6983 x^{12} - 8936 x^{11} + 65641 x^{10} - 72082 x^{9} + 465597 x^{8} - 424984 x^{7} + 2471666 x^{6} - 1778572 x^{5} + 9429355 x^{4} - 4814010 x^{3} + 23461605 x^{2} - 6474360 x + 29134601 \)
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: | \(-147682003746622037852672000000000=-\,2^{18}\cdot 5^{9}\cdot 19^{16}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $61.26$ | 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: | \(380=2^{2}\cdot 5\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{380}(1,·)$, $\chi_{380}(321,·)$, $\chi_{380}(201,·)$, $\chi_{380}(139,·)$, $\chi_{380}(81,·)$, $\chi_{380}(339,·)$, $\chi_{380}(199,·)$, $\chi_{380}(159,·)$, $\chi_{380}(161,·)$, $\chi_{380}(99,·)$, $\chi_{380}(101,·)$, $\chi_{380}(39,·)$, $\chi_{380}(359,·)$, $\chi_{380}(301,·)$, $\chi_{380}(239,·)$, $\chi_{380}(119,·)$, $\chi_{380}(121,·)$, $\chi_{380}(61,·)$$\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}{40767810384675598780004268728504409289601071327961} a^{17} - \frac{3848451596101061148825111565909473197318963253616}{40767810384675598780004268728504409289601071327961} a^{16} + \frac{13557930904639962507174207932418999481868451507479}{40767810384675598780004268728504409289601071327961} a^{15} + \frac{16436809795134872185187780238215625594310895499149}{40767810384675598780004268728504409289601071327961} a^{14} - \frac{7869639614170206274322865412354223077780125795126}{40767810384675598780004268728504409289601071327961} a^{13} + \frac{6272362574401288614352247041677687909516486438823}{40767810384675598780004268728504409289601071327961} a^{12} - \frac{17619189252078691904786937208638041913587448565324}{40767810384675598780004268728504409289601071327961} a^{11} - \frac{14632612976790913766928580500511381196484569182550}{40767810384675598780004268728504409289601071327961} a^{10} + \frac{14694412390809046053097235648806267474041114920454}{40767810384675598780004268728504409289601071327961} a^{9} + \frac{3948708257129156478363869554893220567852186058474}{40767810384675598780004268728504409289601071327961} a^{8} - \frac{10485905972673083477753027598181755191146145803399}{40767810384675598780004268728504409289601071327961} a^{7} - \frac{2355443290433853919772017086773062515044336727254}{40767810384675598780004268728504409289601071327961} a^{6} - \frac{5543430930740439444296114881392291404028537943345}{40767810384675598780004268728504409289601071327961} a^{5} - \frac{15820538568165453349202757714394993662034555086327}{40767810384675598780004268728504409289601071327961} a^{4} + \frac{7207022808036547924267533370883873901416185738344}{40767810384675598780004268728504409289601071327961} a^{3} + \frac{6828617425866368328240288057414970644691301583106}{40767810384675598780004268728504409289601071327961} a^{2} + \frac{12294413269756412643332254030930538325542136246322}{40767810384675598780004268728504409289601071327961} a + \frac{16376598755394734163575235713842817685460411566929}{40767810384675598780004268728504409289601071327961}$
Class group and class number
$C_{8582}$, which has order $8582$ (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
| \(\Q(\sqrt{-5}) \), 3.3.361.1, 6.0.1042568000.1, \(\Q(\zeta_{19})^+\) |
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 | R | ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ | R | ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ | $18$ | $18$ | R | ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ | $18$ | $18$ |
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 | ||||||
| 5 | Data not computed | ||||||
| 19 | Data not computed | ||||||