Normalized defining polynomial
\( x^{20} - 4 x^{19} + 4 x^{18} + 8 x^{17} - 25 x^{16} + 22 x^{15} + 12 x^{14} - 51 x^{13} + 50 x^{12} + 2 x^{11} - 47 x^{10} + 27 x^{9} + 27 x^{8} - 46 x^{7} + 16 x^{6} + 20 x^{5} - 23 x^{4} + 6 x^{3} + 5 x^{2} - 4 x + 1 \)
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: | \(36252013791534423828125=5^{15}\cdot 1039^{2}\cdot 1049^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $13.43$ | 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: | $5, 1039, 1049$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{11768599} a^{19} + \frac{1136101}{11768599} a^{18} - \frac{2837315}{11768599} a^{17} + \frac{2119627}{11768599} a^{16} + \frac{4568232}{11768599} a^{15} + \frac{3751585}{11768599} a^{14} - \frac{3717596}{11768599} a^{13} - \frac{5751516}{11768599} a^{12} + \frac{212036}{11768599} a^{11} + \frac{90411}{287039} a^{10} + \frac{2340356}{11768599} a^{9} - \frac{1187262}{11768599} a^{8} + \frac{3679902}{11768599} a^{7} - \frac{4427289}{11768599} a^{6} - \frac{1262526}{11768599} a^{5} - \frac{5255090}{11768599} a^{4} + \frac{5702816}{11768599} a^{3} - \frac{4341581}{11768599} a^{2} - \frac{1363323}{11768599} a - \frac{993930}{11768599}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $9$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{27541871}{11768599} a^{19} - \frac{109753620}{11768599} a^{18} + \frac{98867302}{11768599} a^{17} + \frac{252145226}{11768599} a^{16} - \frac{694376106}{11768599} a^{15} + \frac{501342869}{11768599} a^{14} + \frac{498202609}{11768599} a^{13} - \frac{1439533298}{11768599} a^{12} + \frac{1160211882}{11768599} a^{11} + \frac{9560914}{287039} a^{10} - \frac{1422768704}{11768599} a^{9} + \frac{494582025}{11768599} a^{8} + \frac{1023621973}{11768599} a^{7} - \frac{1179981135}{11768599} a^{6} + \frac{139404372}{11768599} a^{5} + \frac{715564343}{11768599} a^{4} - \frac{559481004}{11768599} a^{3} - \frac{1797586}{11768599} a^{2} + \frac{178631488}{11768599} a - \frac{71461303}{11768599} \) (order $10$) | 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: | \( 3694.42225582 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 57600 |
| The 70 conjugacy class representatives for t20n654 are not computed |
| Character table for t20n654 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 10.4.3405971875.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 20 siblings: | data not computed |
| Degree 24 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 | $20$ | $20$ | R | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ | ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ | $20$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 5.12.9.2 | $x^{12} - 10 x^{8} + 25 x^{4} - 500$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ | |
| 1039 | Data not computed | ||||||
| 1049 | Data not computed | ||||||