Normalized defining polynomial
\( x^{18} - 3 x^{17} - 9 x^{16} + 41 x^{15} - 5 x^{14} - 132 x^{13} + 128 x^{12} - 17 x^{11} - 15 x^{10} + 232 x^{9} - 405 x^{8} + 141 x^{7} + 326 x^{6} - 240 x^{5} - 135 x^{4} + 67 x^{3} + 29 x^{2} - 2 x - 1 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[12, 3]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-699600837060733684914085599=-\,3^{3}\cdot 13^{9}\cdot 367^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $31.00$ | 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: | $3, 13, 367$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5} a^{14} - \frac{2}{5} a^{13} - \frac{1}{5} a^{12} - \frac{2}{5} a^{11} + \frac{1}{5} a^{10} - \frac{2}{5} a^{9} + \frac{1}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5}$, $\frac{1}{5} a^{15} + \frac{1}{5} a^{12} + \frac{2}{5} a^{11} + \frac{2}{5} a^{9} + \frac{1}{5} a^{8} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{2} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{565} a^{16} - \frac{13}{565} a^{15} - \frac{16}{565} a^{14} - \frac{52}{565} a^{13} - \frac{1}{113} a^{12} + \frac{36}{565} a^{11} + \frac{1}{565} a^{10} - \frac{213}{565} a^{9} - \frac{169}{565} a^{8} + \frac{254}{565} a^{7} + \frac{207}{565} a^{6} - \frac{23}{113} a^{5} + \frac{11}{565} a^{4} + \frac{30}{113} a^{3} - \frac{204}{565} a^{2} - \frac{24}{565} a - \frac{33}{565}$, $\frac{1}{8564058109745} a^{17} - \frac{3829150103}{8564058109745} a^{16} + \frac{8539136622}{1712811621949} a^{15} + \frac{107687912200}{1712811621949} a^{14} + \frac{1072371405046}{8564058109745} a^{13} - \frac{70943660379}{1712811621949} a^{12} + \frac{1268498009034}{8564058109745} a^{11} + \frac{2172868984134}{8564058109745} a^{10} - \frac{793048675001}{8564058109745} a^{9} - \frac{1313390281343}{8564058109745} a^{8} + \frac{210686990463}{1712811621949} a^{7} - \frac{1877976088613}{8564058109745} a^{6} - \frac{667415762898}{8564058109745} a^{5} + \frac{4210877333738}{8564058109745} a^{4} + \frac{2398785838137}{8564058109745} a^{3} - \frac{3023345070523}{8564058109745} a^{2} + \frac{1658473704091}{8564058109745} a - \frac{2224581461544}{8564058109745}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $14$ | 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: | \( 9218935.82046 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 648 |
| The 54 conjugacy class representatives for t18n189 are not computed |
| Character table for t18n189 is not computed |
Intermediate fields
| \(\Q(\sqrt{13}) \), 6.6.295911733.1, 6.4.887735199.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }$ | R | ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ | R | ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.3.0.1 | $x^{3} - x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 3.3.0.1 | $x^{3} - x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 3.6.3.1 | $x^{6} - 6 x^{4} + 9 x^{2} - 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 3.6.0.1 | $x^{6} - x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| $13$ | 13.6.3.1 | $x^{6} - 52 x^{4} + 676 x^{2} - 79092$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 13.12.6.1 | $x^{12} + 338 x^{8} + 8788 x^{6} + 28561 x^{4} + 19307236$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
| 367 | Data not computed | ||||||