Normalized defining polynomial
\( x^{20} - 3 x^{19} - x^{18} + 4 x^{17} + 13 x^{16} - 26 x^{15} + 3 x^{14} + 16 x^{13} + 4 x^{12} - 21 x^{11} + 21 x^{10} - 21 x^{9} + 4 x^{8} + 16 x^{7} + 3 x^{6} - 26 x^{5} + 13 x^{4} + 4 x^{3} - x^{2} - 3 x + 1 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[4, 8]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(10922267596307822265625=5^{10}\cdot 5783^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $12.64$ | 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: | $5, 5783$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{23123} a^{18} - \frac{9}{1217} a^{17} + \frac{5603}{23123} a^{16} + \frac{6914}{23123} a^{15} - \frac{10992}{23123} a^{14} - \frac{10124}{23123} a^{13} + \frac{725}{23123} a^{12} + \frac{3955}{23123} a^{11} + \frac{5406}{23123} a^{10} - \frac{10387}{23123} a^{9} + \frac{5406}{23123} a^{8} + \frac{3955}{23123} a^{7} + \frac{725}{23123} a^{6} - \frac{10124}{23123} a^{5} - \frac{10992}{23123} a^{4} + \frac{6914}{23123} a^{3} + \frac{5603}{23123} a^{2} - \frac{9}{1217} a + \frac{1}{23123}$, $\frac{1}{254353} a^{19} - \frac{4}{254353} a^{18} - \frac{22954}{254353} a^{17} - \frac{121043}{254353} a^{16} - \frac{58750}{254353} a^{15} + \frac{50298}{254353} a^{14} - \frac{25127}{254353} a^{13} - \frac{36831}{254353} a^{12} - \frac{74045}{254353} a^{11} + \frac{83110}{254353} a^{10} - \frac{64367}{254353} a^{9} - \frac{64409}{254353} a^{8} + \frac{83135}{254353} a^{7} - \frac{74033}{254353} a^{6} - \frac{36844}{254353} a^{5} - \frac{1324}{13387} a^{4} + \frac{50337}{254353} a^{3} - \frac{58759}{254353} a^{2} - \frac{121048}{254353} a - \frac{22956}{254353}$
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: | \( 761.500833891 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 3840 |
| The 36 conjugacy class representatives for t20n279 |
| Character table for t20n279 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 5.3.5783.1, 10.6.104509653125.1, 10.2.836077225.1, 10.2.4180386125.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 10 siblings: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 30 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Degree 40 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 | ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 5.12.6.1 | $x^{12} + 500 x^{6} - 3125 x^{2} + 62500$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
| 5783 | Data not computed | ||||||