Normalized defining polynomial
\( x^{18} - 4 x^{17} - 18 x^{16} + 78 x^{15} + 92 x^{14} - 508 x^{13} - 144 x^{12} + 1432 x^{11} + 134 x^{10} - 2144 x^{9} - 912 x^{8} + 3464 x^{7} + 1916 x^{6} - 6060 x^{5} - 380 x^{4} + 4196 x^{3} - 1172 x^{2} + 44 x + 52 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[10, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(238480221168530754513103814656=2^{16}\cdot 19^{8}\cdot 37^{8}\cdot 61\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $42.86$ | 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: | $2, 19, 37, 61$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{2} a^{13}$, $\frac{1}{2} a^{14}$, $\frac{1}{2} a^{15}$, $\frac{1}{62} a^{16} - \frac{7}{62} a^{15} + \frac{7}{31} a^{14} - \frac{5}{31} a^{13} - \frac{3}{62} a^{12} + \frac{11}{62} a^{11} + \frac{7}{62} a^{10} + \frac{13}{62} a^{9} - \frac{7}{31} a^{8} - \frac{3}{31} a^{7} + \frac{3}{31} a^{6} - \frac{15}{31} a^{5} + \frac{13}{31} a^{4} - \frac{10}{31} a^{3} + \frac{14}{31} a^{2} - \frac{7}{31} a - \frac{8}{31}$, $\frac{1}{1806317453349567395791396} a^{17} + \frac{3327101976815929710563}{903158726674783697895698} a^{16} - \frac{154568140015267939233985}{903158726674783697895698} a^{15} + \frac{38451768896042014500929}{903158726674783697895698} a^{14} + \frac{34725525862740801031136}{451579363337391848947849} a^{13} + \frac{13959564667185503774203}{451579363337391848947849} a^{12} + \frac{125000685954872052254071}{903158726674783697895698} a^{11} - \frac{104597920364080310848941}{451579363337391848947849} a^{10} - \frac{95930069403515297520598}{451579363337391848947849} a^{9} - \frac{418565733350583977338289}{903158726674783697895698} a^{8} + \frac{72343242100907191618083}{451579363337391848947849} a^{7} - \frac{110089907903443368791940}{451579363337391848947849} a^{6} + \frac{147016463899472299301318}{451579363337391848947849} a^{5} + \frac{66453501666391223658316}{451579363337391848947849} a^{4} + \frac{6708386738287061561064}{451579363337391848947849} a^{3} - \frac{212466677467822074576990}{451579363337391848947849} a^{2} + \frac{184106718931528300791709}{451579363337391848947849} a - \frac{94601131272783259616916}{451579363337391848947849}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $13$ | 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: | \( 211462412.834 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 27648 |
| The 88 conjugacy class representatives for t18n656 are not computed |
| Character table for t18n656 is not computed |
Intermediate fields
| 3.3.148.1, 9.9.62526089134336.1 |
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 | ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ | $18$ | $18$ | ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | R | ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ | R | ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ | $18$ | ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$ |
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 | Data not computed | ||||||
| $19$ | $\Q_{19}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{19}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 19.4.0.1 | $x^{4} - 2 x + 10$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 19.12.8.1 | $x^{12} - 114 x^{9} + 4332 x^{6} - 54872 x^{3} + 130321000$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ | |
| 37 | Data not computed | ||||||
| 61 | Data not computed | ||||||