Normalized defining polynomial
\( x^{18} - 9 x^{14} + 117 x^{12} - 333 x^{10} - 216 x^{8} + 1206 x^{6} - 945 x^{4} + 243 x^{2} - 27 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[6, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(10685367389464913648812032=2^{24}\cdot 3^{27}\cdot 17^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $24.57$ | 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: | $2, 3, 17$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6}$, $\frac{1}{3} a^{7}$, $\frac{1}{3} a^{8}$, $\frac{1}{3} a^{9}$, $\frac{1}{3} a^{10}$, $\frac{1}{3} a^{11}$, $\frac{1}{18} a^{12} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{18} a^{13} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{18} a^{14} - \frac{1}{6} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{36} a^{15} - \frac{1}{36} a^{13} - \frac{1}{36} a^{12} - \frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} + \frac{1}{12} a^{7} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{928914444} a^{16} - \frac{218227}{309638148} a^{14} - \frac{1}{36} a^{13} + \frac{8204281}{464457222} a^{12} - \frac{1}{6} a^{11} + \frac{7484365}{51606358} a^{10} - \frac{1}{6} a^{9} + \frac{35469013}{309638148} a^{8} - \frac{13482629}{103212716} a^{6} + \frac{1}{4} a^{5} + \frac{17957805}{103212716} a^{4} - \frac{8697077}{103212716} a^{2} + \frac{1}{4} a + \frac{14991483}{51606358}$, $\frac{1}{928914444} a^{17} - \frac{218227}{309638148} a^{15} - \frac{1}{36} a^{14} + \frac{8204281}{464457222} a^{13} + \frac{7484365}{51606358} a^{11} - \frac{1}{6} a^{10} + \frac{35469013}{309638148} a^{9} - \frac{13482629}{103212716} a^{7} - \frac{1}{12} a^{6} + \frac{17957805}{103212716} a^{5} - \frac{1}{2} a^{4} - \frac{8697077}{103212716} a^{3} + \frac{1}{4} a^{2} + \frac{14991483}{51606358} a - \frac{1}{2}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $11$ | 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: | \( 527001.336579 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 41472 |
| The 64 conjugacy class representatives for t18n705 are not computed |
| Character table for t18n705 is not computed |
Intermediate fields
| \(\Q(\zeta_{9})^+\), 9.5.9829532736.1 |
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 | R | $18$ | $18$ | ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ | R | ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ | $18$ | $18$ | ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.3.0.1 | $x^{3} - x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 2.3.0.1 | $x^{3} - x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 2.12.24.15 | $x^{12} + 20 x^{11} - 10 x^{10} - 28 x^{9} - 26 x^{8} + 32 x^{7} - 16 x^{6} + 32 x^{5} + 28 x^{4} - 16 x^{3} - 8 x^{2} - 16 x + 8$ | $4$ | $3$ | $24$ | 12T92 | $[2, 2, 2, 3, 3, 3]^{3}$ | |
| 3 | Data not computed | ||||||
| $17$ | 17.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 17.3.2.1 | $x^{3} - 17$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 17.3.2.1 | $x^{3} - 17$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 17.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 17.6.0.1 | $x^{6} - x + 12$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |