Normalized defining polynomial
\( x^{20} - 5 x^{19} + 17 x^{18} - 39 x^{17} + 71 x^{16} - 97 x^{15} + 105 x^{14} + 10 x^{13} - 112 x^{12} + 230 x^{11} - 346 x^{10} + 391 x^{9} - 337 x^{8} + 129 x^{7} + 152 x^{6} - 330 x^{5} + 437 x^{4} - 408 x^{3} + 369 x^{2} - 162 x + 81 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 10]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(56366140167260768414380128125=5^{5}\cdot 31^{2}\cdot 41^{2}\cdot 139^{2}\cdot 4903^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $27.39$ | 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: | $5, 31, 41, 139, 4903$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{16} - \frac{1}{3} a^{15} - \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{18} + \frac{1}{9} a^{17} - \frac{4}{9} a^{16} - \frac{1}{9} a^{14} - \frac{4}{9} a^{13} + \frac{1}{9} a^{11} + \frac{2}{9} a^{10} - \frac{1}{9} a^{9} - \frac{1}{9} a^{8} - \frac{2}{9} a^{7} + \frac{2}{9} a^{6} - \frac{1}{3} a^{5} - \frac{1}{9} a^{4} - \frac{1}{3} a^{3} - \frac{4}{9} a^{2}$, $\frac{1}{22046518830799766280509199} a^{19} - \frac{110705755425349357800890}{22046518830799766280509199} a^{18} - \frac{212644148105215606000927}{22046518830799766280509199} a^{17} + \frac{538767053877188353730428}{2449613203422196253389911} a^{16} - \frac{10349122442411380356413173}{22046518830799766280509199} a^{15} + \frac{9829890694346329761023678}{22046518830799766280509199} a^{14} - \frac{1194705370848965401488697}{2449613203422196253389911} a^{13} + \frac{5347404962022651788665837}{22046518830799766280509199} a^{12} + \frac{2400840373769278175071964}{22046518830799766280509199} a^{11} + \frac{10223749830879658557886274}{22046518830799766280509199} a^{10} - \frac{4523580043478854432352146}{22046518830799766280509199} a^{9} + \frac{3402441543341135929140142}{22046518830799766280509199} a^{8} - \frac{8329072594628707049814352}{22046518830799766280509199} a^{7} + \frac{885119924535965362639952}{7348839610266588760169733} a^{6} - \frac{6364114076538826147298710}{22046518830799766280509199} a^{5} - \frac{2631418440325261535081509}{7348839610266588760169733} a^{4} - \frac{2946281473380173496015811}{22046518830799766280509199} a^{3} - \frac{982636685112132011033944}{2449613203422196253389911} a^{2} + \frac{112839230900638059597899}{2449613203422196253389911} a - \frac{98883852713259780671672}{816537734474065417796637}$
Class group and class number
$C_{2}$, which has order $2$ (assuming GRH)
Unit group
| Rank: | $9$ | 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: | \( 925084.609293 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 245760 |
| The 201 conjugacy class representatives for t20n886 are not computed |
| Character table for t20n886 is not computed |
Intermediate fields
| 5.3.4903.1, 10.2.4247018348621.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $20$ | ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ | R | $20$ | ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ | R | ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ | R | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | $20$ | ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 5.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.6.0.1 | $x^{6} - x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 5.6.3.2 | $x^{6} - 25 x^{2} + 250$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| $31$ | 31.4.2.1 | $x^{4} + 713 x^{2} + 138384$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 31.8.0.1 | $x^{8} - x + 22$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 31.8.0.1 | $x^{8} - x + 22$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| $41$ | 41.4.2.1 | $x^{4} + 943 x^{2} + 242064$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 41.8.0.1 | $x^{8} - x + 12$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 41.8.0.1 | $x^{8} - x + 12$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| $139$ | 139.2.1.2 | $x^{2} + 556$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 139.2.1.2 | $x^{2} + 556$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 139.8.0.1 | $x^{8} - x + 19$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 139.8.0.1 | $x^{8} - x + 19$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 4903 | Data not computed | ||||||