Normalized defining polynomial
\( x^{18} - 9 x^{16} - 3 x^{14} + 255 x^{12} - 936 x^{10} + 1251 x^{8} - 510 x^{6} - 18 x^{4} + 36 x^{2} - 3 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[14, 2]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(374529353033905727790624768=2^{12}\cdot 3^{31}\cdot 23^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $29.94$ | 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: | $2, 3, 23$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{20} a^{14} + \frac{1}{4} a^{12} - \frac{1}{10} a^{10} + \frac{1}{10} a^{8} - \frac{1}{2} a^{6} - \frac{7}{20} a^{4} + \frac{1}{10} a^{2} - \frac{7}{20}$, $\frac{1}{40} a^{15} - \frac{1}{40} a^{14} + \frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{20} a^{11} + \frac{1}{20} a^{10} + \frac{1}{20} a^{9} - \frac{1}{20} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{13}{40} a^{5} - \frac{13}{40} a^{4} + \frac{1}{20} a^{3} - \frac{1}{20} a^{2} - \frac{7}{40} a + \frac{7}{40}$, $\frac{1}{762800} a^{16} - \frac{3903}{190700} a^{14} + \frac{260833}{762800} a^{12} - \frac{59761}{190700} a^{10} - \frac{72351}{190700} a^{8} - \frac{204137}{762800} a^{6} - \frac{303699}{762800} a^{4} + \frac{101879}{762800} a^{2} + \frac{57199}{762800}$, $\frac{1}{1525600} a^{17} - \frac{1}{1525600} a^{16} - \frac{3903}{381400} a^{15} + \frac{3903}{381400} a^{14} + \frac{260833}{1525600} a^{13} - \frac{260833}{1525600} a^{12} - \frac{59761}{381400} a^{11} + \frac{59761}{381400} a^{10} - \frac{72351}{381400} a^{9} + \frac{72351}{381400} a^{8} - \frac{204137}{1525600} a^{7} + \frac{204137}{1525600} a^{6} - \frac{303699}{1525600} a^{5} + \frac{303699}{1525600} a^{4} - \frac{660921}{1525600} a^{3} + \frac{660921}{1525600} a^{2} - \frac{705601}{1525600} a + \frac{705601}{1525600}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $15$ | 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: | \( 18472088.079 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 9216 |
| The 96 conjugacy class representatives for t18n544 are not computed |
| Character table for t18n544 is not computed |
Intermediate fields
| \(\Q(\zeta_{9})^+\), 3.3.621.1, 9.9.174583151469.1 |
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 | ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ | R | ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }$ |
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 | $x^{3} - x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 2.12.12.16 | $x^{12} - 16 x^{10} - 23 x^{8} + 24 x^{6} - 29 x^{4} - 8 x^{2} - 13$ | $2$ | $6$ | $12$ | 12T134 | $[2, 2, 2, 2, 2, 2]^{6}$ | |
| 3 | Data not computed | ||||||
| $23$ | 23.3.0.1 | $x^{3} - x + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 23.3.0.1 | $x^{3} - x + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 23.12.6.1 | $x^{12} + 365010 x^{6} - 6436343 x^{2} + 33308075025$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |