Normalized defining polynomial
\( x^{18} + 3 x^{16} - 9 x^{15} + 21 x^{14} - 39 x^{13} + 66 x^{12} - 84 x^{11} + 105 x^{10} - 112 x^{9} + 105 x^{8} - 84 x^{7} + 66 x^{6} - 39 x^{5} + 21 x^{4} - 9 x^{3} + 3 x^{2} + 1 \)
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: | \(-125429122538468299827=-\,3^{25}\cdot 23^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $13.08$ | 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: | $3, 23$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{4} a^{15} + \frac{1}{4} a^{14} + \frac{1}{4} a^{13} + \frac{1}{4} a^{11} + \frac{1}{4} a^{10} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{1888} a^{16} - \frac{87}{944} a^{15} + \frac{389}{944} a^{14} + \frac{257}{1888} a^{13} - \frac{635}{1888} a^{12} - \frac{245}{944} a^{11} + \frac{765}{1888} a^{10} - \frac{9}{236} a^{9} + \frac{19}{118} a^{8} - \frac{9}{236} a^{7} - \frac{887}{1888} a^{6} - \frac{245}{944} a^{5} + \frac{73}{1888} a^{4} - \frac{451}{1888} a^{3} - \frac{83}{944} a^{2} + \frac{385}{944} a - \frac{235}{1888}$, $\frac{1}{7552} a^{17} - \frac{1}{7552} a^{16} + \frac{221}{1888} a^{15} - \frac{2973}{7552} a^{14} + \frac{201}{3776} a^{13} - \frac{841}{7552} a^{12} + \frac{955}{7552} a^{11} - \frac{1775}{7552} a^{10} + \frac{369}{944} a^{9} + \frac{193}{944} a^{8} + \frac{1761}{7552} a^{7} + \frac{2763}{7552} a^{6} + \frac{2151}{7552} a^{5} - \frac{1463}{3776} a^{4} - \frac{781}{7552} a^{3} - \frac{851}{1888} a^{2} + \frac{815}{7552} a - \frac{2895}{7552}$
Class group and class number
Trivial group, which has order $1$
Unit group
| Rank: | $8$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{1555}{7552} a^{17} - \frac{907}{7552} a^{16} + \frac{1115}{1888} a^{15} - \frac{14359}{7552} a^{14} + \frac{20111}{3776} a^{13} - \frac{72899}{7552} a^{12} + \frac{114001}{7552} a^{11} - \frac{138517}{7552} a^{10} + \frac{19499}{944} a^{9} - \frac{18877}{944} a^{8} + \frac{146675}{7552} a^{7} - \frac{114727}{7552} a^{6} + \frac{85781}{7552} a^{5} - \frac{29137}{3776} a^{4} + \frac{49009}{7552} a^{3} - \frac{4997}{1888} a^{2} + \frac{10445}{7552} a + \frac{1803}{7552} \) (order $6$) | 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: | \( 2115.40608662 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_3.S_3^2$ (as 18T57):
| A solvable group of order 108 |
| The 11 conjugacy class representatives for $C_3.S_3^2$ |
| Character table for $C_3.S_3^2$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.3.621.1, 6.0.1156923.1, 9.3.6466042647.1 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{3}$ | R | ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ | R | ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$ |
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 | ||||||
| $23$ | 23.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 23.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.4.2.1 | $x^{4} + 299 x^{2} + 25921$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.2.1 | $x^{4} + 299 x^{2} + 25921$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.2.1 | $x^{4} + 299 x^{2} + 25921$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |