Normalized defining polynomial
\( x^{18} - 7 x^{17} - 21 x^{16} + 230 x^{15} - 39 x^{14} - 2523 x^{13} + 2827 x^{12} + 12069 x^{11} - 19596 x^{10} - 25476 x^{9} + 52436 x^{8} + 21175 x^{7} - 56437 x^{6} - 10039 x^{5} + 24080 x^{4} + 4861 x^{3} - 1984 x^{2} - 565 x - 31 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[18, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(24096702957455403051316876953125=5^{9}\cdot 37^{16}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $55.39$ | 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: | $5, 37$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(185=5\cdot 37\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{185}(1,·)$, $\chi_{185}(71,·)$, $\chi_{185}(9,·)$, $\chi_{185}(144,·)$, $\chi_{185}(81,·)$, $\chi_{185}(84,·)$, $\chi_{185}(174,·)$, $\chi_{185}(86,·)$, $\chi_{185}(26,·)$, $\chi_{185}(16,·)$, $\chi_{185}(34,·)$, $\chi_{185}(164,·)$, $\chi_{185}(44,·)$, $\chi_{185}(46,·)$, $\chi_{185}(49,·)$, $\chi_{185}(181,·)$, $\chi_{185}(121,·)$, $\chi_{185}(149,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{31} a^{15} + \frac{13}{31} a^{14} + \frac{12}{31} a^{13} - \frac{10}{31} a^{12} - \frac{11}{31} a^{11} + \frac{8}{31} a^{10} - \frac{6}{31} a^{9} + \frac{2}{31} a^{8} - \frac{7}{31} a^{7} - \frac{7}{31} a^{6} - \frac{11}{31} a^{5} + \frac{8}{31} a^{4} + \frac{6}{31} a^{3} - \frac{11}{31} a^{2} + \frac{13}{31} a$, $\frac{1}{1333} a^{16} + \frac{11}{1333} a^{15} - \frac{107}{1333} a^{14} + \frac{555}{1333} a^{13} - \frac{425}{1333} a^{12} - \frac{652}{1333} a^{11} - \frac{611}{1333} a^{10} + \frac{479}{1333} a^{9} + \frac{268}{1333} a^{8} + \frac{348}{1333} a^{7} + \frac{3}{1333} a^{6} - \frac{590}{1333} a^{5} - \frac{227}{1333} a^{4} - \frac{333}{1333} a^{3} + \frac{252}{1333} a^{2} + \frac{253}{1333} a - \frac{5}{43}$, $\frac{1}{172299010580371339307} a^{17} + \frac{6573078984395758}{172299010580371339307} a^{16} - \frac{2723752179336767600}{172299010580371339307} a^{15} + \frac{3133086374563131394}{172299010580371339307} a^{14} + \frac{79348700309983218247}{172299010580371339307} a^{13} + \frac{61955208774448455651}{172299010580371339307} a^{12} + \frac{62137731596639599683}{172299010580371339307} a^{11} - \frac{40919747140837764227}{172299010580371339307} a^{10} + \frac{43364931567961704814}{172299010580371339307} a^{9} + \frac{1023019638921373524}{4006953734427240449} a^{8} - \frac{77953124837792168978}{172299010580371339307} a^{7} + \frac{10134127444845417560}{172299010580371339307} a^{6} - \frac{52770295421201019494}{172299010580371339307} a^{5} - \frac{31772977212411374714}{172299010580371339307} a^{4} - \frac{24631431209204360366}{172299010580371339307} a^{3} - \frac{11004992178911770158}{172299010580371339307} a^{2} + \frac{6937517732456204277}{172299010580371339307} a - \frac{1870339387661925986}{5558032599366817397}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $17$ | 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: | \( 8441875446.42 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 18 |
| The 18 conjugacy class representatives for $C_{18}$ |
| Character table for $C_{18}$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 3.3.1369.1, 6.6.234270125.1, 9.9.3512479453921.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $18$ | $18$ | R | $18$ | ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ | $18$ | $18$ | ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/31.1.0.1}{1} }^{18}$ | R | ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ | $18$ | ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 5 | Data not computed | ||||||
| 37 | Data not computed | ||||||