Normalized defining polynomial
\( x^{18} - 108 x^{16} + 5247 x^{14} - 143388 x^{12} + 2370627 x^{10} - 11662 x^{9} - 23508324 x^{8} + 1574370 x^{7} + 135435594 x^{6} - 52164126 x^{5} - 366226488 x^{4} + 539274204 x^{3} + 798965721 x^{2} - 1323415422 x + 4642192563 \)
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: | \(-1829454323323333626595491241240421203968=-\,2^{27}\cdot 3^{44}\cdot 7^{12}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $151.79$ | 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: | $2, 3, 7$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1512=2^{3}\cdot 3^{3}\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1512}(1,·)$, $\chi_{1512}(1411,·)$, $\chi_{1512}(1033,·)$, $\chi_{1512}(907,·)$, $\chi_{1512}(499,·)$, $\chi_{1512}(529,·)$, $\chi_{1512}(403,·)$, $\chi_{1512}(121,·)$, $\chi_{1512}(25,·)$, $\chi_{1512}(1507,·)$, $\chi_{1512}(1009,·)$, $\chi_{1512}(1387,·)$, $\chi_{1512}(1003,·)$, $\chi_{1512}(625,·)$, $\chi_{1512}(883,·)$, $\chi_{1512}(1129,·)$, $\chi_{1512}(505,·)$, $\chi_{1512}(379,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} - \frac{1}{7} a^{4} - \frac{2}{7} a^{2} + \frac{1}{7}$, $\frac{1}{7} a^{7} - \frac{1}{7} a^{5} - \frac{2}{7} a^{3} + \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{3}{7} a^{4} - \frac{1}{7} a^{2} + \frac{1}{7}$, $\frac{1}{133} a^{9} + \frac{8}{133} a^{8} + \frac{6}{133} a^{7} - \frac{2}{133} a^{6} - \frac{30}{133} a^{5} - \frac{29}{133} a^{4} + \frac{29}{133} a^{3} - \frac{4}{133} a^{2} + \frac{4}{19} a - \frac{15}{133}$, $\frac{1}{133} a^{10} - \frac{1}{133} a^{8} + \frac{1}{19} a^{7} + \frac{5}{133} a^{6} + \frac{3}{19} a^{5} - \frac{62}{133} a^{4} + \frac{7}{19} a^{3} - \frac{5}{19} a^{2} - \frac{7}{19} a + \frac{9}{19}$, $\frac{1}{133} a^{11} - \frac{4}{133} a^{8} - \frac{8}{133} a^{7} + \frac{60}{133} a^{5} - \frac{37}{133} a^{4} + \frac{32}{133} a^{3} + \frac{4}{133} a^{2} - \frac{61}{133} a - \frac{53}{133}$, $\frac{1}{15827} a^{12} - \frac{3}{931} a^{10} + \frac{18}{15827} a^{8} - \frac{2}{133} a^{7} + \frac{16}{931} a^{6} + \frac{9}{133} a^{5} + \frac{349}{833} a^{4} + \frac{60}{133} a^{3} + \frac{267}{931} a^{2} - \frac{9}{133} a - \frac{594}{15827}$, $\frac{1}{15827} a^{13} - \frac{3}{931} a^{11} + \frac{18}{15827} a^{9} - \frac{2}{133} a^{8} + \frac{16}{931} a^{7} + \frac{9}{133} a^{6} + \frac{349}{833} a^{5} + \frac{60}{133} a^{4} + \frac{267}{931} a^{3} - \frac{9}{133} a^{2} - \frac{594}{15827} a$, $\frac{1}{15827} a^{14} + \frac{5}{2261} a^{10} + \frac{4}{133} a^{8} - \frac{3}{133} a^{7} - \frac{6}{119} a^{6} + \frac{66}{133} a^{5} + \frac{2}{19} a^{4} - \frac{50}{133} a^{3} + \frac{8}{323} a^{2} + \frac{39}{133} a + \frac{129}{931}$, $\frac{1}{15827} a^{15} + \frac{5}{2261} a^{11} + \frac{3}{133} a^{8} + \frac{124}{2261} a^{7} - \frac{2}{133} a^{6} - \frac{37}{133} a^{5} + \frac{4}{19} a^{4} - \frac{947}{2261} a^{3} + \frac{36}{133} a^{2} - \frac{389}{931} a + \frac{22}{133}$, $\frac{1}{300713} a^{16} - \frac{9}{300713} a^{15} + \frac{5}{300713} a^{14} + \frac{9}{300713} a^{13} + \frac{6}{300713} a^{12} - \frac{298}{300713} a^{11} - \frac{845}{300713} a^{10} + \frac{995}{300713} a^{9} + \frac{13079}{300713} a^{8} - \frac{14408}{300713} a^{7} + \frac{18229}{300713} a^{6} + \frac{20647}{300713} a^{5} + \frac{134510}{300713} a^{4} - \frac{123684}{300713} a^{3} - \frac{2845}{15827} a^{2} + \frac{32394}{300713} a + \frac{115537}{300713}$, $\frac{1}{33593203078604072255990625668151377630097576083} a^{17} - \frac{6820565482944358022668920037151180463292}{4799029011229153179427232238307339661442510869} a^{16} + \frac{268738704057336199788061732675798254000185}{33593203078604072255990625668151377630097576083} a^{15} - \frac{188665621809735634783238960543093015688904}{33593203078604072255990625668151377630097576083} a^{14} - \frac{684166985020556115694420023732081007362165}{33593203078604072255990625668151377630097576083} a^{13} + \frac{860157159365464547762872189429861986528833}{33593203078604072255990625668151377630097576083} a^{12} + \frac{1105332733457510633063967477268797623666120}{33593203078604072255990625668151377630097576083} a^{11} + \frac{96325404678497468964909110689550954098400699}{33593203078604072255990625668151377630097576083} a^{10} + \frac{95710162625355526675603964975862686854636358}{33593203078604072255990625668151377630097576083} a^{9} + \frac{1588281796366124971364369128725542470578747860}{33593203078604072255990625668151377630097576083} a^{8} - \frac{124141587870132177373751756474461526039380618}{1768063319926530118736348719376388296320925057} a^{7} - \frac{33848440516735521320983191846425001388882786}{33593203078604072255990625668151377630097576083} a^{6} - \frac{14156538943753006277763005950672926668492267436}{33593203078604072255990625668151377630097576083} a^{5} - \frac{11633153447843430250430097739255043512048553957}{33593203078604072255990625668151377630097576083} a^{4} - \frac{7696832498845957658848009758874938140698814791}{33593203078604072255990625668151377630097576083} a^{3} + \frac{9459516329329225725264144050412897313033867658}{33593203078604072255990625668151377630097576083} a^{2} - \frac{8565291844621088763183670826313963690762897469}{33593203078604072255990625668151377630097576083} a + \frac{7772352261366765673587428022526394574627095430}{33593203078604072255990625668151377630097576083}$
Class group and class number
$C_{3}\times C_{425907}$, which has order $1277721$ (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: | \( 10392888.21418944 \) (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
| \(\Q(\sqrt{-2}) \), \(\Q(\zeta_{9})^+\), 6.0.3359232.1, 9.9.3691950281939241.2 |
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 | R | $18$ | R | ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ | $18$ | ${\href{/LocalNumberField/17.1.0.1}{1} }^{18}$ | ${\href{/LocalNumberField/19.1.0.1}{1} }^{18}$ | $18$ | $18$ | $18$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ | $18$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$ |
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 | ||||||
| $3$ | 3.9.22.6 | $x^{9} + 9 x^{8} + 21 x^{6} + 18 x^{5} + 9 x^{3} + 24$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ |
| 3.9.22.6 | $x^{9} + 9 x^{8} + 21 x^{6} + 18 x^{5} + 9 x^{3} + 24$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ | |
| 7 | Data not computed | ||||||