Normalized defining polynomial
\( x^{18} - 17 x^{15} + 193 x^{12} - 1057 x^{9} + 3187 x^{6} - 4125 x^{3} + 15625 \)
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: | \(-535422519938359574117644671=-\,3^{21}\cdot 13^{15}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $30.54$ | 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: | $3, 13$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{6} + \frac{1}{3}$, $\frac{1}{15} a^{10} - \frac{4}{15} a^{7} + \frac{2}{5} a^{4} + \frac{1}{15} a$, $\frac{1}{75} a^{11} + \frac{11}{75} a^{8} - \frac{8}{25} a^{5} - \frac{29}{75} a^{2}$, $\frac{1}{375} a^{12} - \frac{14}{375} a^{9} + \frac{151}{375} a^{6} - \frac{104}{375} a^{3} + \frac{1}{3}$, $\frac{1}{375} a^{13} + \frac{11}{375} a^{10} + \frac{17}{125} a^{7} + \frac{46}{375} a^{4} + \frac{2}{5} a$, $\frac{1}{375} a^{14} + \frac{1}{375} a^{11} - \frac{59}{375} a^{8} - \frac{89}{375} a^{5} + \frac{13}{75} a^{2}$, $\frac{1}{18428625} a^{15} + \frac{1}{1125} a^{14} + \frac{1}{1125} a^{13} - \frac{20978}{18428625} a^{12} + \frac{2}{375} a^{11} + \frac{11}{1125} a^{10} - \frac{2304853}{18428625} a^{9} - \frac{379}{1125} a^{8} + \frac{17}{375} a^{7} - \frac{1786181}{6142875} a^{6} + \frac{166}{1125} a^{5} + \frac{46}{1125} a^{4} - \frac{5469994}{18428625} a^{3} - \frac{16}{225} a^{2} + \frac{2}{15} a - \frac{23266}{147429}$, $\frac{1}{92143125} a^{16} - \frac{28834}{30714375} a^{13} + \frac{1}{1125} a^{12} - \frac{1}{225} a^{11} + \frac{220036}{30714375} a^{10} + \frac{37}{375} a^{9} - \frac{86}{225} a^{8} - \frac{13957264}{30714375} a^{7} - \frac{349}{1125} a^{6} + \frac{8}{75} a^{5} - \frac{23226998}{92143125} a^{4} - \frac{104}{1125} a^{3} - \frac{46}{225} a^{2} + \frac{15004}{147429} a + \frac{2}{9}$, $\frac{1}{460715625} a^{17} - \frac{36913}{51190625} a^{14} - \frac{1}{1125} a^{13} - \frac{1}{1125} a^{12} - \frac{226973}{51190625} a^{11} - \frac{4}{125} a^{10} + \frac{139}{1125} a^{9} + \frac{12579956}{153571875} a^{8} + \frac{49}{1125} a^{7} + \frac{158}{375} a^{6} - \frac{126673013}{460715625} a^{5} + \frac{554}{1125} a^{4} - \frac{271}{1125} a^{3} - \frac{956983}{3685725} a^{2} - \frac{7}{45} a$
Class group and class number
$C_{12}$, which has order $12$ (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: | \( 305948.58795593923 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_6\times S_3$ (as 18T6):
| A solvable group of order 36 |
| The 18 conjugacy class representatives for $S_3 \times C_6$ |
| Character table for $S_3 \times C_6$ |
Intermediate fields
| \(\Q(\sqrt{-39}) \), 3.3.169.1, 3.1.4563.1, 6.0.812017791.2, 6.0.10024911.1, 9.3.95006081547.1 |
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 12 sibling: | 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 | ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }^{2}$ | R | ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ | R | ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$ |
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 | Data not computed | ||||||
| $13$ | 13.6.5.2 | $x^{6} - 13$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
| 13.6.5.2 | $x^{6} - 13$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
| 13.6.5.2 | $x^{6} - 13$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |