Normalized defining polynomial
\( x^{18} - 27 x^{16} + 289 x^{14} - 1645 x^{12} + 5523 x^{10} - 11298 x^{8} + 13839 x^{6} - 9334 x^{4} + 2678 x^{2} - 13 \)
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: | \(92384419220890933620945190912=2^{18}\cdot 7^{16}\cdot 13^{9}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $40.66$ | 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: | $2, 7, 13$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{853} a^{16} + \frac{49}{853} a^{14} - \frac{252}{853} a^{12} - \frac{325}{853} a^{10} - \frac{411}{853} a^{8} + \frac{116}{853} a^{6} - \frac{376}{853} a^{4} - \frac{378}{853} a^{2} + \frac{393}{853}$, $\frac{1}{853} a^{17} + \frac{49}{853} a^{15} - \frac{252}{853} a^{13} - \frac{325}{853} a^{11} - \frac{411}{853} a^{9} + \frac{116}{853} a^{7} - \frac{376}{853} a^{5} - \frac{378}{853} a^{3} + \frac{393}{853} a$
Class group and class number
Trivial group, which has order $1$ (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: | \( 494241653.327 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_3^2.A_4$ (as 18T92):
| A solvable group of order 216 |
| The 40 conjugacy class representatives for $C_2\times C_3^2.A_4$ |
| Character table for $C_2\times C_3^2.A_4$ is not computed |
Intermediate fields
| \(\Q(\zeta_{7})^+\), 6.6.1997632.1, 9.9.164648481361.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 18 siblings: | data not computed |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ | $18$ | R | $18$ | R | ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ | $18$ | ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ | $18$ | $18$ | ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ | $18$ | ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ | $18$ |
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 | ||||||
| 7 | Data not computed | ||||||
| $13$ | 13.3.2.3 | $x^{3} - 52$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 13.3.2.3 | $x^{3} - 52$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 13.6.5.4 | $x^{6} + 26$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
| 13.6.0.1 | $x^{6} + x^{2} - 2 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |