Normalized defining polynomial
\( x^{20} - 5 x^{19} - 9 x^{18} + 90 x^{17} - 58 x^{16} - 590 x^{15} + 1074 x^{14} + 1444 x^{13} - 5372 x^{12} + 1043 x^{11} + 11414 x^{10} - 11526 x^{9} - 6983 x^{8} + 17325 x^{7} - 6552 x^{6} - 4366 x^{5} + 3643 x^{4} - 406 x^{3} - 185 x^{2} + 18 x + 1 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[12, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(2857413128106559934928689217=3^{4}\cdot 17^{9}\cdot 4153^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $23.59$ | 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: | $3, 17, 4153$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{55742154150747055051} a^{19} - \frac{4864836082062757631}{55742154150747055051} a^{18} - \frac{3004675601626079778}{55742154150747055051} a^{17} - \frac{1791448670530413610}{55742154150747055051} a^{16} - \frac{19375016804949626636}{55742154150747055051} a^{15} + \frac{3846046625109735375}{55742154150747055051} a^{14} + \frac{11556278126568000430}{55742154150747055051} a^{13} - \frac{7963816858307488874}{55742154150747055051} a^{12} + \frac{21071962321551511125}{55742154150747055051} a^{11} - \frac{3213065192953451511}{55742154150747055051} a^{10} - \frac{2570781878219114501}{55742154150747055051} a^{9} - \frac{20984989969804423724}{55742154150747055051} a^{8} + \frac{2933311146720041999}{55742154150747055051} a^{7} - \frac{18894682975058949496}{55742154150747055051} a^{6} + \frac{13377629414717157652}{55742154150747055051} a^{5} - \frac{13711652749415549805}{55742154150747055051} a^{4} - \frac{12413723731604664083}{55742154150747055051} a^{3} - \frac{4554958935476004059}{55742154150747055051} a^{2} - \frac{25764272672565317376}{55742154150747055051} a + \frac{9639891312976292411}{55742154150747055051}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $15$ | 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: | \( 2628564.39825 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 122880 |
| The 138 conjugacy class representatives for t20n804 are not computed |
| Character table for t20n804 is not computed |
Intermediate fields
| 5.5.70601.1, 10.6.44860510809.1 |
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 | ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ | ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ | R | ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ | ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ | ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 3.12.0.1 | $x^{12} - x^{4} - x^{3} - x^{2} + x - 1$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
| $17$ | 17.3.0.1 | $x^{3} - x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 17.3.0.1 | $x^{3} - x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 17.4.3.3 | $x^{4} + 51$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 17.4.3.4 | $x^{4} + 459$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 17.6.3.1 | $x^{6} - 34 x^{4} + 289 x^{2} - 44217$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 4153 | Data not computed | ||||||