Normalized defining polynomial
\( x^{18} + 12 x^{16} + 33 x^{14} - 63 x^{12} - 351 x^{10} - 360 x^{8} + 39 x^{6} + 189 x^{4} + 54 x^{2} - 3 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[6, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(5852021141154776996728512=2^{6}\cdot 3^{31}\cdot 23^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $23.77$ | 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: | $2, 3, 23$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{38} a^{12} - \frac{1}{2} a^{11} - \frac{7}{38} a^{10} - \frac{1}{2} a^{9} - \frac{7}{19} a^{8} + \frac{7}{19} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{19} a^{2} + \frac{7}{38}$, $\frac{1}{38} a^{13} + \frac{6}{19} a^{11} + \frac{5}{38} a^{9} + \frac{7}{19} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} + \frac{1}{19} a^{3} + \frac{7}{38} a - \frac{1}{2}$, $\frac{1}{38} a^{14} + \frac{13}{38} a^{10} - \frac{4}{19} a^{8} - \frac{1}{2} a^{7} - \frac{8}{19} a^{6} - \frac{1}{2} a^{5} + \frac{1}{19} a^{4} - \frac{17}{38} a^{2} - \frac{1}{2} a - \frac{4}{19}$, $\frac{1}{38} a^{15} + \frac{13}{38} a^{11} - \frac{4}{19} a^{9} - \frac{1}{2} a^{8} - \frac{8}{19} a^{7} - \frac{1}{2} a^{6} + \frac{1}{19} a^{5} - \frac{17}{38} a^{3} - \frac{1}{2} a^{2} - \frac{4}{19} a$, $\frac{1}{286634} a^{16} - \frac{1706}{143317} a^{14} - \frac{693}{143317} a^{12} - \frac{1}{2} a^{11} + \frac{99113}{286634} a^{10} - \frac{58419}{143317} a^{8} - \frac{1}{2} a^{7} - \frac{36513}{143317} a^{6} - \frac{1}{2} a^{5} + \frac{14164}{143317} a^{4} - \frac{1}{2} a^{3} - \frac{60067}{143317} a^{2} - \frac{109151}{286634}$, $\frac{1}{286634} a^{17} - \frac{1706}{143317} a^{15} - \frac{693}{143317} a^{13} - \frac{22102}{143317} a^{11} - \frac{1}{2} a^{10} + \frac{26479}{286634} a^{9} - \frac{1}{2} a^{8} - \frac{36513}{143317} a^{7} - \frac{1}{2} a^{6} - \frac{114989}{286634} a^{5} - \frac{60067}{143317} a^{3} - \frac{109151}{286634} a - \frac{1}{2}$
Class group and class number
Trivial group, which has order $1$
Unit group
| Rank: | $11$ | 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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 251719.809953 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_6\times S_4$ (as 18T61):
| A solvable group of order 144 |
| The 30 conjugacy class representatives for $C_6\times S_4$ |
| Character table for $C_6\times S_4$ is not computed |
Intermediate fields
| \(\Q(\zeta_{9})^+\), 3.3.621.1, 6.2.4627692.1, 9.9.174583151469.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| 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 | R | ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ | R | ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.6.6.3 | $x^{6} + 2 x^{4} + x^{2} - 7$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ |
| 2.12.0.1 | $x^{12} - 26 x^{10} + 275 x^{8} - 1500 x^{6} + 4375 x^{4} - 6250 x^{2} + 7221$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
| 3 | Data not computed | ||||||
| $23$ | 23.3.0.1 | $x^{3} - x + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 23.3.0.1 | $x^{3} - x + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 23.6.3.1 | $x^{6} - 46 x^{4} + 529 x^{2} - 194672$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 23.6.3.1 | $x^{6} - 46 x^{4} + 529 x^{2} - 194672$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |