Normalized defining polynomial
\( x^{18} - 12691 x^{9} + 40353607 \)
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: | \(-40891490652933723328906846968243=-\,3^{45}\cdot 7^{12}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $57.04$ | 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: | $3, 7$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(189=3^{3}\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{189}(128,·)$, $\chi_{189}(1,·)$, $\chi_{189}(2,·)$, $\chi_{189}(67,·)$, $\chi_{189}(4,·)$, $\chi_{189}(134,·)$, $\chi_{189}(71,·)$, $\chi_{189}(8,·)$, $\chi_{189}(130,·)$, $\chi_{189}(142,·)$, $\chi_{189}(79,·)$, $\chi_{189}(16,·)$, $\chi_{189}(158,·)$, $\chi_{189}(95,·)$, $\chi_{189}(32,·)$, $\chi_{189}(65,·)$, $\chi_{189}(64,·)$, $\chi_{189}(127,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{7} a^{3}$, $\frac{1}{7} a^{4}$, $\frac{1}{7} a^{5}$, $\frac{1}{49} a^{6}$, $\frac{1}{49} a^{7}$, $\frac{1}{49} a^{8}$, $\frac{1}{343} a^{9}$, $\frac{1}{343} a^{10}$, $\frac{1}{2401} a^{11} - \frac{2}{7} a^{2}$, $\frac{1}{16807} a^{12} - \frac{2}{49} a^{3}$, $\frac{1}{16807} a^{13} - \frac{2}{49} a^{4}$, $\frac{1}{117649} a^{14} + \frac{12}{343} a^{5}$, $\frac{1}{823543} a^{15} + \frac{12}{2401} a^{6}$, $\frac{1}{823543} a^{16} + \frac{12}{2401} a^{7}$, $\frac{1}{5764801} a^{17} - \frac{37}{16807} a^{8}$
Class group and class number
$C_{543}$, which has order $543$ (assuming GRH)
Unit group
| Rank: | $8$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{3}{823543} a^{15} + \frac{62}{2401} a^{6} \) (order $18$) | 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: | \( 4392158.29124 \) (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
| \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{9})\), 9.9.3691950281939241.1 |
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 | $18$ | R | $18$ | ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ | $18$ | $18$ | ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/37.1.0.1}{1} }^{18}$ | $18$ | ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ | $18$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ | $18$ |
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 | ||||||
| $7$ | 7.9.6.2 | $x^{9} - 49 x^{3} + 686$ | $3$ | $3$ | $6$ | $C_9$ | $[\ ]_{3}^{3}$ |
| 7.9.6.2 | $x^{9} - 49 x^{3} + 686$ | $3$ | $3$ | $6$ | $C_9$ | $[\ ]_{3}^{3}$ | |