Normalized defining polynomial
\( x^{18} - 4 x^{17} + 3 x^{16} - 8 x^{15} + 45 x^{14} - 94 x^{13} + 62 x^{12} + 68 x^{11} - 156 x^{10} + 70 x^{9} + 101 x^{8} - 164 x^{7} + 123 x^{6} - 106 x^{5} + 115 x^{4} - 72 x^{3} + 18 x^{2} - 2 x + 1 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[2, 8]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(335405228452106761142272=2^{18}\cdot 13^{3}\cdot 37^{6}\cdot 61^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $20.28$ | 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: | $2, 13, 37, 61$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{23} a^{14} + \frac{6}{23} a^{13} + \frac{9}{23} a^{12} - \frac{9}{23} a^{11} - \frac{1}{23} a^{10} + \frac{8}{23} a^{9} - \frac{7}{23} a^{8} - \frac{8}{23} a^{7} - \frac{1}{23} a^{6} - \frac{10}{23} a^{5} - \frac{11}{23} a^{4} + \frac{8}{23} a^{3} + \frac{2}{23} a^{2} + \frac{10}{23} a - \frac{4}{23}$, $\frac{1}{23} a^{15} - \frac{4}{23} a^{13} + \frac{6}{23} a^{12} + \frac{7}{23} a^{11} - \frac{9}{23} a^{10} - \frac{9}{23} a^{9} + \frac{11}{23} a^{8} + \frac{1}{23} a^{7} - \frac{4}{23} a^{6} + \frac{3}{23} a^{5} + \frac{5}{23} a^{4} - \frac{2}{23} a^{2} + \frac{5}{23} a + \frac{1}{23}$, $\frac{1}{23} a^{16} + \frac{7}{23} a^{13} - \frac{3}{23} a^{12} + \frac{1}{23} a^{11} + \frac{10}{23} a^{10} - \frac{3}{23} a^{9} - \frac{4}{23} a^{8} + \frac{10}{23} a^{7} - \frac{1}{23} a^{6} + \frac{11}{23} a^{5} + \frac{2}{23} a^{4} + \frac{7}{23} a^{3} - \frac{10}{23} a^{2} - \frac{5}{23} a + \frac{7}{23}$, $\frac{1}{368671035925} a^{17} - \frac{1105203476}{73734207185} a^{16} + \frac{4316453933}{368671035925} a^{15} + \frac{1666941984}{368671035925} a^{14} + \frac{162814815336}{368671035925} a^{13} - \frac{201327453}{4337306305} a^{12} - \frac{1939558583}{368671035925} a^{11} + \frac{84429024551}{368671035925} a^{10} - \frac{85616375407}{368671035925} a^{9} - \frac{166672143923}{368671035925} a^{8} - \frac{55375633776}{368671035925} a^{7} - \frac{127676100813}{368671035925} a^{6} + \frac{34734021361}{368671035925} a^{5} - \frac{50213475867}{368671035925} a^{4} - \frac{37292530393}{368671035925} a^{3} + \frac{73367470321}{368671035925} a^{2} - \frac{108864643778}{368671035925} a + \frac{3832858876}{368671035925}$
Class group and class number
$C_{2}$, which has order $2$
Unit group
| Rank: | $9$ | 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: | \( 12438.1504042 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 5184 |
| The 49 conjugacy class representatives for t18n486 |
| Character table for t18n486 is not computed |
Intermediate fields
| 3.3.148.1, 6.2.69479488.1, 9.5.2570741056.1 |
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 | ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ | R | ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ | R | ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$ |
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 | Data not computed | ||||||
| $13$ | 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.12.0.1 | $x^{12} + x^{2} - x + 2$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
| 37 | Data not computed | ||||||
| 61 | Data not computed | ||||||