Normalized defining polynomial
\( x^{18} - 9 x^{17} + 99 x^{16} - 588 x^{15} + 3960 x^{14} - 18144 x^{13} + 92400 x^{12} - 342630 x^{11} + 1416780 x^{10} - 4312778 x^{9} + 14929371 x^{8} - 36981954 x^{7} + 108462921 x^{6} - 210931569 x^{5} + 524296044 x^{4} - 733296564 x^{3} + 1530279351 x^{2} - 1193424489 x + 2049933909 \)
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: | \(-77615347601079862368096390169921875=-\,3^{44}\cdot 5^{9}\cdot 7^{9}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $86.76$ | 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: | $3, 5, 7$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(945=3^{3}\cdot 5\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{945}(1,·)$, $\chi_{945}(454,·)$, $\chi_{945}(769,·)$, $\chi_{945}(841,·)$, $\chi_{945}(139,·)$, $\chi_{945}(526,·)$, $\chi_{945}(211,·)$, $\chi_{945}(664,·)$, $\chi_{945}(349,·)$, $\chi_{945}(736,·)$, $\chi_{945}(34,·)$, $\chi_{945}(421,·)$, $\chi_{945}(874,·)$, $\chi_{945}(559,·)$, $\chi_{945}(244,·)$, $\chi_{945}(631,·)$, $\chi_{945}(316,·)$, $\chi_{945}(106,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{109} a^{16} + \frac{6}{109} a^{15} + \frac{15}{109} a^{14} + \frac{10}{109} a^{13} - \frac{26}{109} a^{12} + \frac{23}{109} a^{10} - \frac{25}{109} a^{9} - \frac{19}{109} a^{8} - \frac{52}{109} a^{7} - \frac{13}{109} a^{6} + \frac{26}{109} a^{5} + \frac{53}{109} a^{3} - \frac{28}{109} a^{2} + \frac{14}{109} a - \frac{30}{109}$, $\frac{1}{22095789080844660822518693516790949765794861210280688101} a^{17} + \frac{64751859086685796180155069850270251582421724953562175}{22095789080844660822518693516790949765794861210280688101} a^{16} - \frac{3417576433363253460081377604352228172424857753458657058}{22095789080844660822518693516790949765794861210280688101} a^{15} - \frac{7258572543546173223769487375793613544814643732543961172}{22095789080844660822518693516790949765794861210280688101} a^{14} + \frac{3898212959671491656104041688335619617407463028778290046}{22095789080844660822518693516790949765794861210280688101} a^{13} - \frac{5253758196281990188296628715387496459787639034529570927}{22095789080844660822518693516790949765794861210280688101} a^{12} + \frac{9920127807612233943117714645115974621761311424011169728}{22095789080844660822518693516790949765794861210280688101} a^{11} - \frac{4200596618411459846849822312667996016010826530764511649}{22095789080844660822518693516790949765794861210280688101} a^{10} - \frac{272586505164772130900321507523244232198986923054597016}{22095789080844660822518693516790949765794861210280688101} a^{9} - \frac{3151279588865256998658793187490243093491698390411965432}{22095789080844660822518693516790949765794861210280688101} a^{8} + \frac{4051504912058856389097276495198265719817039145014555031}{22095789080844660822518693516790949765794861210280688101} a^{7} - \frac{755731808374761284661213683382645641842596359186399332}{22095789080844660822518693516790949765794861210280688101} a^{6} + \frac{10748275921264310567977748677390296509959846820245187860}{22095789080844660822518693516790949765794861210280688101} a^{5} - \frac{6247592087240663043075873532800472306749843788089011770}{22095789080844660822518693516790949765794861210280688101} a^{4} + \frac{2021125168029431200613340242440899554096207340624263933}{22095789080844660822518693516790949765794861210280688101} a^{3} - \frac{10353378888115628488142436817866766740317160228409001856}{22095789080844660822518693516790949765794861210280688101} a^{2} + \frac{8166390487164151480842557481434895228751784631950655751}{22095789080844660822518693516790949765794861210280688101} a + \frac{5656935318739044805516334951692476495531693616668407772}{22095789080844660822518693516790949765794861210280688101}$
Class group and class number
$C_{9}\times C_{27558}$, which has order $248022$ (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: | \( 40934.0329443 \) (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
| \(\Q(\sqrt{-35}) \), \(\Q(\zeta_{9})^+\), 6.0.281302875.3, \(\Q(\zeta_{27})^+\) |
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 | R | R | ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ | $18$ | ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ | $18$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ | $18$ | $18$ | ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ | $18$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.9.22.8 | $x^{9} + 18 x^{7} + 24 x^{6} + 18 x^{5} + 18 x^{3} + 3$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ |
| 3.9.22.8 | $x^{9} + 18 x^{7} + 24 x^{6} + 18 x^{5} + 18 x^{3} + 3$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ | |
| 5 | Data not computed | ||||||
| 7 | Data not computed | ||||||