Normalized defining polynomial
\( x^{18} - 4 x^{17} + 19 x^{16} - 30 x^{15} + 93 x^{14} - 96 x^{13} + 327 x^{12} - 219 x^{11} + 608 x^{10} - 393 x^{9} + 826 x^{8} - 468 x^{7} + 618 x^{6} - 393 x^{5} + 330 x^{4} - 138 x^{3} + 49 x^{2} - 8 x + 1 \)
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: | \(-5319993268633121382752523=-\,3^{9}\cdot 37^{4}\cdot 229^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $23.64$ | 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: | $3, 37, 229$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{237507873853561063} a^{17} + \frac{18778136346416529}{237507873853561063} a^{16} + \frac{102465461214169847}{237507873853561063} a^{15} + \frac{29957995639999119}{237507873853561063} a^{14} + \frac{56079891845069186}{237507873853561063} a^{13} + \frac{97656513073038942}{237507873853561063} a^{12} + \frac{110443594955115955}{237507873853561063} a^{11} + \frac{36137537703247373}{237507873853561063} a^{10} + \frac{57277611991355225}{237507873853561063} a^{9} - \frac{103760166574876010}{237507873853561063} a^{8} - \frac{70458707733252915}{237507873853561063} a^{7} + \frac{43490462656660459}{237507873853561063} a^{6} - \frac{52968339527894156}{237507873853561063} a^{5} + \frac{111193816432338531}{237507873853561063} a^{4} - \frac{72591624962490916}{237507873853561063} a^{3} + \frac{72164213775072092}{237507873853561063} a^{2} - \frac{29780175487836087}{237507873853561063} a - \frac{101835249520522307}{237507873853561063}$
Class group and class number
$C_{6}$, which has order $6$
Unit group
| Rank: | $8$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{38340763460767805}{237507873853561063} a^{17} - \frac{148394097666019470}{237507873853561063} a^{16} + \frac{711351652523116222}{237507873853561063} a^{15} - \frac{1067058198511944776}{237507873853561063} a^{14} + \frac{3469775042345795240}{237507873853561063} a^{13} - \frac{3304362782597081610}{237507873853561063} a^{12} + \frac{12318183431505834455}{237507873853561063} a^{11} - \frac{7038703287014700634}{237507873853561063} a^{10} + \frac{23113980230217725948}{237507873853561063} a^{9} - \frac{12646430371866783876}{237507873853561063} a^{8} + \frac{31348867518603374858}{237507873853561063} a^{7} - \frac{14797073407134640580}{237507873853561063} a^{6} + \frac{23551058927405630616}{237507873853561063} a^{5} - \frac{13055873253786622604}{237507873853561063} a^{4} + \frac{12233313946759345246}{237507873853561063} a^{3} - \frac{4246491809675198221}{237507873853561063} a^{2} + \frac{1942249127161256972}{237507873853561063} a - \frac{82871362187861909}{237507873853561063} \) (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: | \( 22714.2286235 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_3\wr S_3$ (as 18T119):
| A solvable group of order 324 |
| The 44 conjugacy class representatives for $C_2\times C_3\wr S_3$ |
| Character table for $C_2\times C_3\wr S_3$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.3.229.1, 6.0.1415907.1, 9.9.16440305941.1 |
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 | ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ | R | $18$ | ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ | $18$ | ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ | R | ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||
| $37$ | 37.3.0.1 | $x^{3} - x + 2$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 37.3.0.1 | $x^{3} - x + 2$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 37.3.2.2 | $x^{3} + 74$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 37.3.0.1 | $x^{3} - x + 2$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 37.3.0.1 | $x^{3} - x + 2$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 37.3.2.2 | $x^{3} + 74$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 229 | Data not computed | ||||||