Normalized defining polynomial
\( x^{18} - 3 x^{17} + x^{15} + 3 x^{14} + 3 x^{13} + 3 x^{12} + 6 x^{11} - 15 x^{10} - 3 x^{9} - 15 x^{8} + 6 x^{7} + 3 x^{6} + 3 x^{5} + 3 x^{4} + x^{3} - 3 x + 1 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[8, 5]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-66852546470398177734375=-\,3^{15}\cdot 5^{9}\cdot 13^{4}\cdot 17^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $18.54$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{5} + \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{6} + \frac{1}{3} a$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{7} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{8} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{9} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{15} + \frac{1}{3}$, $\frac{1}{711} a^{16} + \frac{37}{711} a^{15} + \frac{19}{237} a^{14} + \frac{37}{237} a^{13} - \frac{13}{79} a^{12} - \frac{16}{237} a^{11} + \frac{32}{237} a^{10} + \frac{113}{237} a^{9} + \frac{59}{237} a^{8} - \frac{15}{79} a^{7} + \frac{37}{79} a^{6} + \frac{21}{79} a^{5} + \frac{40}{237} a^{4} + \frac{37}{237} a^{3} - \frac{20}{79} a^{2} + \frac{274}{711} a + \frac{238}{711}$, $\frac{1}{711} a^{17} + \frac{110}{711} a^{15} - \frac{34}{237} a^{14} + \frac{14}{237} a^{13} + \frac{5}{237} a^{12} - \frac{8}{237} a^{11} + \frac{35}{237} a^{10} - \frac{14}{237} a^{9} - \frac{95}{237} a^{8} + \frac{39}{79} a^{7} - \frac{94}{237} a^{6} - \frac{1}{3} a^{5} - \frac{100}{237} a^{4} - \frac{7}{237} a^{3} - \frac{176}{711} a^{2} + \frac{97}{237} a + \frac{200}{711}$
Class group and class number
Trivial group, which has order $1$
Unit group
| Rank: | $12$ | 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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 46678.2467071 \) | 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 | $18$ | $18$ | R | R | ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ | $18$ | ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/37.14.0.1}{14} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.6.3.2 | $x^{6} - 9 x^{2} + 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 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 | $[3/2, 3/2]_{2}^{4}$ | |
| $5$ | 5.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 5.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.12.8.2 | $x^{12} + 25 x^{6} - 250 x^{3} + 1250$ | $3$ | $4$ | $8$ | $C_3\times (C_3 : C_4)$ | $[\ ]_{3}^{12}$ | |
| $13$ | $\Q_{13}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{13}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 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 | $x^{6} + 39 x^{3} + 676$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| $17$ | 17.3.2.1 | $x^{3} - 17$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 17.3.2.1 | $x^{3} - 17$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 17.6.0.1 | $x^{6} - x + 12$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 17.6.0.1 | $x^{6} - x + 12$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |