Normalized defining polynomial
\( x^{18} - 5 x^{17} - x^{16} + 51 x^{15} - 81 x^{14} - 142 x^{13} + 416 x^{12} + 126 x^{11} - 916 x^{10} - 74 x^{9} + 1382 x^{8} + 195 x^{7} - 1530 x^{6} - 329 x^{5} + 941 x^{4} + 311 x^{3} - 220 x^{2} - 117 x - 13 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[10, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(13680011824035182666015625=3^{12}\cdot 5^{11}\cdot 13^{5}\cdot 17^{5}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $24.91$ | 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: | $3, 5, 13, 17$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{11062841278538600261} a^{17} - \frac{18500374140478995}{11062841278538600261} a^{16} - \frac{1315502832899873310}{11062841278538600261} a^{15} + \frac{4644028677286478215}{11062841278538600261} a^{14} + \frac{316239073765375415}{11062841278538600261} a^{13} + \frac{3145191482276385339}{11062841278538600261} a^{12} + \frac{2711658235436249662}{11062841278538600261} a^{11} - \frac{4074560052624556911}{11062841278538600261} a^{10} + \frac{5507605317608112662}{11062841278538600261} a^{9} - \frac{2063477734987174492}{11062841278538600261} a^{8} + \frac{2783019719889904188}{11062841278538600261} a^{7} - \frac{3597871152906711505}{11062841278538600261} a^{6} - \frac{2388461724193190698}{11062841278538600261} a^{5} + \frac{1875550216673467867}{11062841278538600261} a^{4} + \frac{249508261803980511}{11062841278538600261} a^{3} + \frac{2125287802830749564}{11062841278538600261} a^{2} + \frac{1343699791650390757}{11062841278538600261} a + \frac{1774983194433485237}{11062841278538600261}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $13$ | 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: | \( 1093656.11823 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 92897280 |
| The 168 conjugacy class representatives for t18n966 are not computed |
| Character table for t18n966 is not computed |
Intermediate fields
| 9.5.22253180625.1 |
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 | ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ | R | R | $18$ | ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ | R | R | $18$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ | $18$ | ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/37.14.0.1}{14} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ | $18$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.6.0.1 | $x^{6} - x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
| 3.12.12.12 | $x^{12} + 165 x^{10} - 312 x^{9} - 288 x^{8} - 180 x^{7} - 36 x^{6} - 135 x^{5} - 243 x^{4} + 54 x^{3} + 81 x^{2} + 81 x - 162$ | $3$ | $4$ | $12$ | 12T41 | $[3/2, 3/2]_{2}^{4}$ | |
| $5$ | 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.12.8.2 | $x^{12} + 25 x^{6} - 250 x^{3} + 1250$ | $3$ | $4$ | $8$ | $C_3\times (C_3 : C_4)$ | $[\ ]_{3}^{12}$ | |
| $13$ | 13.2.1.1 | $x^{2} - 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 13.5.0.1 | $x^{5} - 2 x + 6$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 13.5.0.1 | $x^{5} - 2 x + 6$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 13.6.4.1 | $x^{6} + 39 x^{3} + 676$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| $17$ | 17.6.5.1 | $x^{6} - 17$ | $6$ | $1$ | $5$ | $D_{6}$ | $[\ ]_{6}^{2}$ |
| 17.12.0.1 | $x^{12} + 3 x^{2} - 2 x + 5$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ |