Normalized defining polynomial
\( x^{18} - 3 x^{17} - x^{16} + 16 x^{15} - 41 x^{14} + 36 x^{13} + 28 x^{12} - 134 x^{11} + 182 x^{10} - 46 x^{9} - 96 x^{8} + 84 x^{7} + 148 x^{6} - 420 x^{5} + 341 x^{4} - 140 x^{3} - 116 x^{2} + 53 x + 19 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[6, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(73108484269567621953125=5^{7}\cdot 23^{6}\cdot 43^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $18.63$ | 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: | $5, 23, 43$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5} a^{14} - \frac{2}{5} a^{11} + \frac{2}{5} a^{10} + \frac{1}{5} a^{8} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{15} - \frac{2}{5} a^{12} + \frac{2}{5} a^{11} + \frac{1}{5} a^{9} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{25} a^{16} + \frac{1}{25} a^{15} - \frac{1}{25} a^{14} + \frac{8}{25} a^{13} - \frac{1}{5} a^{12} + \frac{9}{25} a^{11} + \frac{9}{25} a^{10} - \frac{9}{25} a^{9} + \frac{2}{5} a^{8} + \frac{1}{5} a^{7} + \frac{9}{25} a^{6} + \frac{12}{25} a^{4} + \frac{3}{25} a^{3} + \frac{1}{5} a^{2} - \frac{7}{25} a + \frac{11}{25}$, $\frac{1}{32447151532069003525} a^{17} + \frac{519919639683165654}{32447151532069003525} a^{16} + \frac{3149419895917782852}{32447151532069003525} a^{15} + \frac{515416382036463459}{6489430306413800705} a^{14} - \frac{5248821894675130756}{32447151532069003525} a^{13} + \frac{536316819043340969}{32447151532069003525} a^{12} + \frac{2303956856503262706}{32447151532069003525} a^{11} - \frac{5885785364650292902}{32447151532069003525} a^{10} + \frac{12905345788796712108}{32447151532069003525} a^{9} - \frac{484249841728715569}{1297886061282760141} a^{8} + \frac{8105286556882517124}{32447151532069003525} a^{7} + \frac{3181030895267940367}{32447151532069003525} a^{6} - \frac{10776343982188256553}{32447151532069003525} a^{5} - \frac{1249039845371592421}{32447151532069003525} a^{4} - \frac{2813415808724874291}{32447151532069003525} a^{3} - \frac{7605465012371850292}{32447151532069003525} a^{2} + \frac{386034398091858206}{6489430306413800705} a + \frac{766327835231008297}{1707744817477315975}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $11$ | 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: | \( 30090.9556958 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 82944 |
| The 80 conjugacy class representatives for t18n782 are not computed |
| Character table for t18n782 is not computed |
Intermediate fields
| 3.1.23.1, 9.3.562419575.1 |
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 | $18$ | $18$ | R | ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{5}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | R | ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ | R | $18$ | ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.4.2.2 | $x^{4} - 5 x^{2} + 50$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 5.6.5.1 | $x^{6} - 5$ | $6$ | $1$ | $5$ | $D_{6}$ | $[\ ]_{6}^{2}$ | |
| 5.8.0.1 | $x^{8} + x^{2} - 2 x + 3$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| $23$ | 23.6.0.1 | $x^{6} - x + 15$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
| 23.12.6.1 | $x^{12} + 365010 x^{6} - 6436343 x^{2} + 33308075025$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
| 43 | Data not computed | ||||||