Normalized defining polynomial
\( x^{18} + 6 x^{16} + 48 x^{14} + 166 x^{12} + 374 x^{10} + 660 x^{8} + 469 x^{6} + 76 x^{4} + 16 x^{2} + 1 \)
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: | \(-87445981790876136737050624=-\,2^{12}\cdot 23^{6}\cdot 229^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $27.62$ | 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: | $2, 23, 229$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{7} a^{14} - \frac{3}{7} a^{12} - \frac{1}{7} a^{8} - \frac{2}{7} a^{6} - \frac{3}{7} a^{4} + \frac{2}{7} a^{2} + \frac{3}{7}$, $\frac{1}{14} a^{15} - \frac{1}{14} a^{14} - \frac{3}{14} a^{13} - \frac{2}{7} a^{12} - \frac{1}{2} a^{11} + \frac{3}{7} a^{9} + \frac{1}{14} a^{8} + \frac{5}{14} a^{7} - \frac{5}{14} a^{6} + \frac{2}{7} a^{5} + \frac{3}{14} a^{4} - \frac{5}{14} a^{3} + \frac{5}{14} a^{2} - \frac{2}{7} a - \frac{3}{14}$, $\frac{1}{706944518} a^{16} - \frac{16165176}{353472259} a^{14} - \frac{1}{2} a^{13} + \frac{198746015}{706944518} a^{12} - \frac{1}{2} a^{11} + \frac{151168209}{353472259} a^{10} - \frac{1}{2} a^{9} + \frac{6982691}{50496037} a^{8} + \frac{46158743}{706944518} a^{6} - \frac{1}{2} a^{5} + \frac{118703897}{353472259} a^{4} + \frac{102097505}{706944518} a^{2} - \frac{1}{2} a + \frac{44676801}{706944518}$, $\frac{1}{706944518} a^{17} + \frac{18165685}{706944518} a^{15} + \frac{23628952}{353472259} a^{13} - \frac{1}{2} a^{12} - \frac{51135841}{706944518} a^{11} - \frac{1}{2} a^{10} - \frac{153105311}{353472259} a^{9} - \frac{1}{2} a^{8} + \frac{21331352}{50496037} a^{7} - \frac{133776288}{353472259} a^{5} - \frac{1}{2} a^{4} - \frac{10741620}{50496037} a^{3} - \frac{157307347}{706944518} a - \frac{1}{2}$
Class group and class number
$C_{2}$, which has order $2$ (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: | \( 219096.079363 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times S_3\times S_4$ (as 18T111):
| A solvable group of order 288 |
| The 30 conjugacy class representatives for $C_2\times S_3\times S_4$ |
| Character table for $C_2\times S_3\times S_4$ is not computed |
Intermediate fields
| 3.3.229.1, 3.1.23.1, 6.0.839056.1, 9.3.146113369163.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 18 siblings: | 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 | ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }$ | ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ | R | ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }$ | ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.3.0.1 | $x^{3} - x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 2.3.0.1 | $x^{3} - x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 2.12.12.24 | $x^{12} - 100 x^{10} - 59 x^{8} + 104 x^{6} + 387 x^{4} + 444 x^{2} + 439$ | $2$ | $6$ | $12$ | $D_4 \times C_3$ | $[2, 2]^{6}$ | |
| $23$ | $\Q_{23}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{23}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 23.2.1.2 | $x^{2} + 46$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.2 | $x^{2} + 46$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 23.8.4.1 | $x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 229 | Data not computed | ||||||