Normalized defining polynomial
\( x^{18} - 7 x^{17} + 121 x^{16} - 658 x^{15} + 6531 x^{14} - 28817 x^{13} + 209683 x^{12} - 761507 x^{11} + 4439956 x^{10} - 13225274 x^{9} + 64493752 x^{8} - 154238751 x^{7} + 643876978 x^{6} - 1178657248 x^{5} + 4267526217 x^{4} - 5397548424 x^{3} + 17074629129 x^{2} - 11355872994 x + 31522065121 \)
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: | \(-673269902026520824357188276050914131=-\,3^{9}\cdot 17^{9}\cdot 19^{16}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $97.83$ | 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: | $3, 17, 19$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(969=3\cdot 17\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{969}(256,·)$, $\chi_{969}(1,·)$, $\chi_{969}(460,·)$, $\chi_{969}(917,·)$, $\chi_{969}(662,·)$, $\chi_{969}(919,·)$, $\chi_{969}(613,·)$, $\chi_{969}(866,·)$, $\chi_{969}(101,·)$, $\chi_{969}(358,·)$, $\chi_{969}(815,·)$, $\chi_{969}(560,·)$, $\chi_{969}(305,·)$, $\chi_{969}(562,·)$, $\chi_{969}(254,·)$, $\chi_{969}(764,·)$, $\chi_{969}(766,·)$, $\chi_{969}(511,·)$$\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}{469661808795580518838880204416757309958104658061305175082007631} a^{17} + \frac{50973052494159566352769077022799758791907385364247599739092069}{469661808795580518838880204416757309958104658061305175082007631} a^{16} - \frac{194635494032454225451139208488746736743470166154310611111721541}{469661808795580518838880204416757309958104658061305175082007631} a^{15} - \frac{36624120839577090038444838981707236087024020296418889380984797}{469661808795580518838880204416757309958104658061305175082007631} a^{14} + \frac{204935382792749673348353751052120814532688552140290296538083303}{469661808795580518838880204416757309958104658061305175082007631} a^{13} + \frac{218532290051546872150536473622401214731916446499948706338649558}{469661808795580518838880204416757309958104658061305175082007631} a^{12} + \frac{160236147263078038866495221973488614125249280173799030877394610}{469661808795580518838880204416757309958104658061305175082007631} a^{11} + \frac{55902918650615940771277196808917586567086474204235154924148253}{469661808795580518838880204416757309958104658061305175082007631} a^{10} - \frac{136233988877788379833625114603610725963431892397545841436234956}{469661808795580518838880204416757309958104658061305175082007631} a^{9} + \frac{16676135819862676286450239904424846223912989448705411946621423}{469661808795580518838880204416757309958104658061305175082007631} a^{8} + \frac{180258706341204930881746817015199731069779964152573875354330600}{469661808795580518838880204416757309958104658061305175082007631} a^{7} + \frac{133290286424395341829749588656456917900163099280918740853015651}{469661808795580518838880204416757309958104658061305175082007631} a^{6} - \frac{21598228358795931266162145540085990111943636161751383774525905}{469661808795580518838880204416757309958104658061305175082007631} a^{5} - \frac{214648186746361152664341500031972245923375961042822926028998499}{469661808795580518838880204416757309958104658061305175082007631} a^{4} - \frac{204046244430091717851476918999539411686283959076165989159596832}{469661808795580518838880204416757309958104658061305175082007631} a^{3} + \frac{116852127294300389613704702092871036765044264056632967719196321}{469661808795580518838880204416757309958104658061305175082007631} a^{2} - \frac{144105615933554336452129715634346869851096009126228307459665010}{469661808795580518838880204416757309958104658061305175082007631} a - \frac{42122397764379567688238525112004106476682825670323321471690616}{469661808795580518838880204416757309958104658061305175082007631}$
Class group and class number
$C_{784854}$, which has order $784854$ (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
| \(\Q(\sqrt{-51}) \), 3.3.361.1, 6.0.17287210971.4, \(\Q(\zeta_{19})^+\) |
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 | $18$ | R | ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ | R | R | ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ | $18$ | $18$ | $18$ |
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 |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||
| 17 | Data not computed | ||||||
| $19$ | 19.9.8.8 | $x^{9} - 19$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ |
| 19.9.8.8 | $x^{9} - 19$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ | |