Normalized defining polynomial
\( x^{18} - 3 x^{15} + 30 x^{12} - 250 x^{9} + 465 x^{6} + 237 x^{3} + 64 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 9]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-13741574276458659759521484375=-\,3^{37}\cdot 5^{15}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $36.58$ | 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: | $3, 5$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{4} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{12} a^{9} + \frac{1}{4} a^{3} + \frac{1}{3}$, $\frac{1}{12} a^{10} - \frac{1}{4} a^{4} - \frac{1}{6} a$, $\frac{1}{48} a^{11} - \frac{1}{16} a^{8} + \frac{1}{16} a^{5} + \frac{7}{48} a^{2}$, $\frac{1}{48} a^{12} + \frac{1}{48} a^{9} + \frac{1}{16} a^{6} + \frac{19}{48} a^{3} + \frac{1}{3}$, $\frac{1}{48} a^{13} + \frac{1}{48} a^{10} + \frac{1}{16} a^{7} - \frac{5}{48} a^{4} - \frac{1}{6} a$, $\frac{1}{144} a^{14} - \frac{1}{144} a^{13} + \frac{1}{144} a^{12} + \frac{1}{144} a^{11} - \frac{1}{144} a^{10} + \frac{1}{144} a^{9} - \frac{1}{16} a^{8} + \frac{1}{16} a^{7} + \frac{3}{16} a^{6} - \frac{29}{144} a^{5} + \frac{29}{144} a^{4} + \frac{43}{144} a^{3} - \frac{5}{36} a^{2} + \frac{5}{36} a + \frac{1}{9}$, $\frac{1}{6624} a^{15} - \frac{2}{207} a^{12} - \frac{103}{3312} a^{9} + \frac{14}{207} a^{6} + \frac{461}{6624} a^{3} + \frac{86}{207}$, $\frac{1}{6624} a^{16} - \frac{2}{207} a^{13} - \frac{103}{3312} a^{10} + \frac{14}{207} a^{7} + \frac{461}{6624} a^{4} + \frac{86}{207} a$, $\frac{1}{6624} a^{17} - \frac{1}{368} a^{14} - \frac{1}{144} a^{13} + \frac{1}{144} a^{12} - \frac{11}{3312} a^{11} - \frac{1}{144} a^{10} + \frac{1}{144} a^{9} - \frac{95}{1656} a^{8} + \frac{1}{16} a^{7} + \frac{3}{16} a^{6} - \frac{51}{736} a^{5} + \frac{29}{144} a^{4} + \frac{43}{144} a^{3} + \frac{1399}{3312} a^{2} + \frac{5}{36} a + \frac{1}{9}$
Class group and class number
$C_{2}\times C_{2}$, which has order $4$ (assuming GRH)
Unit group
| Rank: | $8$ | 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: | \( 47479719.21338162 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_3:S_3$ (as 18T12):
| A solvable group of order 36 |
| The 12 conjugacy class representatives for $C_2\times C_3:S_3$ |
| Character table for $C_2\times C_3:S_3$ |
Intermediate fields
| \(\Q(\sqrt{-15}) \), 3.1.6075.2, 3.1.243.1, 3.1.675.1, 3.1.6075.1, 6.0.553584375.2, 6.0.6834375.1, 6.0.22143375.1, 6.0.553584375.1, 9.1.6053445140625.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
| Degree 18 sibling: | 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 | ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ | R | R | ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$ |
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 | Data not computed | ||||||
| $5$ | 5.6.5.2 | $x^{6} + 10$ | $6$ | $1$ | $5$ | $D_{6}$ | $[\ ]_{6}^{2}$ |
| 5.12.10.1 | $x^{12} + 6 x^{11} + 27 x^{10} + 80 x^{9} + 195 x^{8} + 366 x^{7} + 571 x^{6} + 702 x^{5} + 1005 x^{4} + 1140 x^{3} + 357 x^{2} - 138 x + 44$ | $6$ | $2$ | $10$ | $D_6$ | $[\ ]_{6}^{2}$ | |