Normalized defining polynomial
\( x^{18} - 2 x^{17} + 8 x^{16} - 20 x^{15} + 36 x^{14} - 84 x^{13} + 124 x^{12} - 242 x^{11} + 322 x^{10} - 466 x^{9} + 684 x^{8} - 648 x^{7} + 1050 x^{6} - 548 x^{5} + 1232 x^{4} - 228 x^{3} + 864 x^{2} + 112 x + 394 \)
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: | \(-54205231577484093435150336=-\,2^{26}\cdot 3^{13}\cdot 47^{7}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $26.89$ | 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: | $2, 3, 47$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{154682298119997224183} a^{17} + \frac{64183778915114408969}{154682298119997224183} a^{16} - \frac{26892702120267330020}{154682298119997224183} a^{15} - \frac{6298290130866051745}{154682298119997224183} a^{14} - \frac{53230504770217703413}{154682298119997224183} a^{13} - \frac{1988753509226388949}{154682298119997224183} a^{12} - \frac{4791406302957947907}{154682298119997224183} a^{11} - \frac{47648676262649110020}{154682298119997224183} a^{10} - \frac{75762872740844618367}{154682298119997224183} a^{9} + \frac{15104939785206457683}{154682298119997224183} a^{8} - \frac{72274238435104609213}{154682298119997224183} a^{7} - \frac{69090279105329132146}{154682298119997224183} a^{6} + \frac{40111400883797098283}{154682298119997224183} a^{5} - \frac{60068955334290800081}{154682298119997224183} a^{4} - \frac{54078190001960331910}{154682298119997224183} a^{3} + \frac{71002267394141391018}{154682298119997224183} a^{2} - \frac{61925201898203890901}{154682298119997224183} a - \frac{21148868019935446805}{154682298119997224183}$
Class group and class number
$C_{2}$, which has order $2$ (assuming GRH)
Unit group
| Rank: | $8$ | 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: | \( 332091.822082 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 165888 |
| The 168 conjugacy class representatives for t18n835 are not computed |
| Character table for t18n835 is not computed |
Intermediate fields
| 3.3.564.1, 9.3.19375863552.1 |
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 | R | ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ | $18$ | ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }$ | R | ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$ |
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 | ||||||
| $3$ | 3.6.7.2 | $x^{6} + 3 x^{2} + 6$ | $6$ | $1$ | $7$ | $D_{6}$ | $[3/2]_{2}^{2}$ |
| 3.6.3.2 | $x^{6} - 9 x^{2} + 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 3.6.3.2 | $x^{6} - 9 x^{2} + 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 47 | Data not computed | ||||||