Normalized defining polynomial
\( x^{18} + 76 x^{16} + 2166 x^{14} + 29716 x^{12} + 209741 x^{10} + 768208 x^{8} + 1451581 x^{6} + 1344364 x^{4} + 521284 x^{2} + 54872 \)
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: | \(-231020214178968501941151058507071488=-\,2^{33}\cdot 11^{6}\cdot 19^{15}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $92.18$ | 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: | $2, 11, 19$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{19} a^{6}$, $\frac{1}{19} a^{7}$, $\frac{1}{19} a^{8}$, $\frac{1}{19} a^{9}$, $\frac{1}{361} a^{10} + \frac{2}{19} a^{4}$, $\frac{1}{361} a^{11} + \frac{2}{19} a^{5}$, $\frac{1}{2527} a^{12} - \frac{2}{2527} a^{10} + \frac{2}{133} a^{8} + \frac{2}{133} a^{6} - \frac{61}{133} a^{4} + \frac{2}{7} a^{2} + \frac{2}{7}$, $\frac{1}{2527} a^{13} - \frac{2}{2527} a^{11} + \frac{2}{133} a^{9} + \frac{2}{133} a^{7} - \frac{61}{133} a^{5} + \frac{2}{7} a^{3} + \frac{2}{7} a$, $\frac{1}{96026} a^{14} + \frac{2}{2527} a^{10} + \frac{59}{2527} a^{8} - \frac{3}{266} a^{6} + \frac{6}{19} a^{4} - \frac{15}{266} a^{2} - \frac{1}{7}$, $\frac{1}{96026} a^{15} + \frac{2}{2527} a^{11} + \frac{59}{2527} a^{9} - \frac{3}{266} a^{7} + \frac{6}{19} a^{5} - \frac{15}{266} a^{3} - \frac{1}{7} a$, $\frac{1}{8912941268} a^{16} + \frac{2881}{4456470634} a^{14} + \frac{3141}{33507298} a^{12} + \frac{42918}{117275543} a^{10} + \frac{143071}{6092236} a^{8} - \frac{54805}{12344794} a^{6} + \frac{8480869}{24689588} a^{4} - \frac{5788511}{12344794} a^{2} - \frac{115443}{324863}$, $\frac{1}{8912941268} a^{17} + \frac{2881}{4456470634} a^{15} + \frac{3141}{33507298} a^{13} + \frac{42918}{117275543} a^{11} + \frac{143071}{6092236} a^{9} - \frac{54805}{12344794} a^{7} + \frac{8480869}{24689588} a^{5} - \frac{5788511}{12344794} a^{3} - \frac{115443}{324863} a$
Class group and class number
$C_{90}\times C_{180}$, which has order $16200$ (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: | \( 654963.5034721284 \) (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
| \(\Q(\sqrt{-38}) \), 3.3.15884.1, 3.3.361.1, 6.0.613597140992.3, 6.0.1267762688.1, 9.9.4007556327104.1 |
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 | R | ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ | R | ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ | R | ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.6.9.3 | $x^{6} - 4 x^{4} + 4 x^{2} + 24$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ |
| 2.12.24.307 | $x^{12} + 28 x^{11} - 2 x^{10} + 16 x^{9} + 26 x^{8} + 8 x^{7} + 20 x^{6} - 24 x^{5} - 8 x^{4} + 32 x^{3} + 32 x^{2} + 32 x + 24$ | $4$ | $3$ | $24$ | $C_6\times C_2$ | $[2, 3]^{3}$ | |
| $11$ | 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 19 | Data not computed | ||||||