Normalized defining polynomial
\( x^{18} - 72 x^{16} + 2160 x^{14} - 34944 x^{12} + 329472 x^{10} - 1728 x^{9} - 1824768 x^{8} + 62208 x^{7} + 5677056 x^{6} - 746496 x^{5} - 8847360 x^{4} + 3317760 x^{3} + 5308416 x^{2} - 3981312 x + 733184 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[18, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(42774537890835454566245668796878848=2^{12}\cdot 3^{44}\cdot 13^{9}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $83.94$ | 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: | $2, 3, 13$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{4} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{8} a^{5}$, $\frac{1}{16} a^{6}$, $\frac{1}{16} a^{7}$, $\frac{1}{32} a^{8}$, $\frac{1}{64} a^{9}$, $\frac{1}{64} a^{10}$, $\frac{1}{128} a^{11}$, $\frac{1}{256} a^{12}$, $\frac{1}{256} a^{13}$, $\frac{1}{8704} a^{14} + \frac{1}{4352} a^{13} + \frac{3}{2176} a^{12} + \frac{1}{272} a^{11} + \frac{1}{1088} a^{10} + \frac{5}{1088} a^{9} - \frac{7}{544} a^{8} + \frac{5}{272} a^{7} + \frac{3}{136} a^{6} - \frac{7}{136} a^{5} - \frac{5}{68} a^{4} - \frac{1}{68} a^{3} - \frac{7}{34} a^{2} - \frac{6}{17} a - \frac{4}{17}$, $\frac{1}{17408} a^{15} + \frac{1}{2176} a^{13} + \frac{1}{2176} a^{12} - \frac{7}{2176} a^{11} - \frac{7}{1088} a^{10} + \frac{5}{1088} a^{9} - \frac{5}{544} a^{8} - \frac{1}{136} a^{7} + \frac{1}{68} a^{6} + \frac{1}{68} a^{5} - \frac{1}{17} a^{4} - \frac{3}{34} a^{3} + \frac{1}{34} a^{2} + \frac{4}{17} a + \frac{4}{17}$, $\frac{1}{17408} a^{16} - \frac{1}{2176} a^{13} - \frac{1}{1088} a^{12} + \frac{5}{2176} a^{11} + \frac{1}{1088} a^{10} + \frac{1}{272} a^{9} + \frac{7}{544} a^{8} + \frac{1}{272} a^{7} - \frac{3}{272} a^{6} + \frac{3}{136} a^{5} - \frac{3}{68} a^{4} + \frac{3}{34} a^{3} + \frac{1}{17} a^{2} - \frac{6}{17} a - \frac{1}{17}$, $\frac{1}{34816} a^{17} + \frac{1}{272} a^{10} + \frac{1}{136} a^{8} - \frac{1}{136} a^{6} - \frac{7}{68} a^{4} - \frac{3}{34} a^{2} - \frac{4}{17} a - \frac{8}{17}$
Class group and class number
$C_{3}$, which has order $3$ (assuming GRH)
Unit group
| Rank: | $17$ | 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: | \( 241280047092 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_9\times S_3$ (as 18T16):
| A solvable group of order 54 |
| The 27 conjugacy class representatives for $C_9\times S_3$ |
| Character table for $C_9\times S_3$ is not computed |
Intermediate fields
| \(\Q(\sqrt{13}) \), \(\Q(\zeta_{9})^+\), 6.6.14414517.1 |
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 | R | R | $18$ | $18$ | $18$ | R | ${\href{/LocalNumberField/17.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{9}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ | $18$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ | $18$ | ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ | $18$ | ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ | $18$ |
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 | ||||||
| $3$ | 3.9.22.2 | $x^{9} + 9 x^{7} + 3 x^{6} + 18 x^{5} + 51$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ |
| 3.9.22.6 | $x^{9} + 9 x^{8} + 21 x^{6} + 18 x^{5} + 9 x^{3} + 24$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ | |
| 13 | Data not computed | ||||||