Normalized defining polynomial
\( x^{20} - 304 x^{18} + 38949 x^{16} - 2808026 x^{14} + 127489037 x^{12} - 3839138862 x^{10} + 78102567868 x^{8} - 1064005067775 x^{6} + 9311802693856 x^{4} - 47343128591330 x^{2} + 106306843291441 \)
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: | \(4005869174343660274328698855843840000000000=2^{20}\cdot 5^{10}\cdot 13^{8}\cdot 19^{4}\cdot 60662149^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $134.94$ | 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: | $2, 5, 13, 19, 60662149$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{13} a^{6} - \frac{5}{13} a^{4} + \frac{1}{13} a^{2}$, $\frac{1}{13} a^{7} - \frac{5}{13} a^{5} + \frac{1}{13} a^{3}$, $\frac{1}{169} a^{8} - \frac{5}{169} a^{6} - \frac{64}{169} a^{4} + \frac{3}{13} a^{2}$, $\frac{1}{169} a^{9} - \frac{5}{169} a^{7} - \frac{64}{169} a^{5} + \frac{3}{13} a^{3}$, $\frac{1}{2197} a^{10} - \frac{5}{2197} a^{8} - \frac{64}{2197} a^{6} - \frac{75}{169} a^{4} - \frac{1}{13} a^{2}$, $\frac{1}{2197} a^{11} - \frac{5}{2197} a^{9} - \frac{64}{2197} a^{7} - \frac{75}{169} a^{5} - \frac{1}{13} a^{3}$, $\frac{1}{28561} a^{12} - \frac{5}{28561} a^{10} - \frac{64}{28561} a^{8} - \frac{75}{2197} a^{6} - \frac{53}{169} a^{4} + \frac{6}{13} a^{2}$, $\frac{1}{28561} a^{13} - \frac{5}{28561} a^{11} - \frac{64}{28561} a^{9} - \frac{75}{2197} a^{7} - \frac{53}{169} a^{5} + \frac{6}{13} a^{3}$, $\frac{1}{7054567} a^{14} - \frac{3}{371293} a^{12} - \frac{818}{7054567} a^{10} - \frac{105}{542659} a^{8} - \frac{630}{41743} a^{6} - \frac{1156}{3211} a^{4} - \frac{101}{247} a^{2}$, $\frac{1}{7054567} a^{15} - \frac{3}{371293} a^{13} - \frac{818}{7054567} a^{11} - \frac{105}{542659} a^{9} - \frac{630}{41743} a^{7} - \frac{1156}{3211} a^{5} - \frac{101}{247} a^{3}$, $\frac{1}{1742478049} a^{16} - \frac{3}{91709371} a^{14} + \frac{24870}{1742478049} a^{12} + \frac{12492}{134036773} a^{10} - \frac{12581}{10310521} a^{8} - \frac{7483}{793117} a^{6} - \frac{8955}{61009} a^{4} + \frac{29}{247} a^{2}$, $\frac{1}{1742478049} a^{17} - \frac{3}{91709371} a^{15} + \frac{24870}{1742478049} a^{13} + \frac{12492}{134036773} a^{11} - \frac{12581}{10310521} a^{9} - \frac{7483}{793117} a^{7} - \frac{8955}{61009} a^{5} + \frac{29}{247} a^{3}$, $\frac{1}{617061705157393628296619} a^{18} + \frac{248732079837}{4639561692912734047343} a^{16} + \frac{763017256740649}{617061705157393628296619} a^{14} - \frac{712496565091485166}{47466285012107202176663} a^{12} + \frac{76295063469977283}{3651252693239015552051} a^{10} + \frac{721245939758641419}{280865591787616580927} a^{8} + \frac{224530791880522163}{21605045522124352379} a^{6} - \frac{35438879005402416}{87469819927628957} a^{4} - \frac{26867178012837}{354128825617931} a^{2} + \frac{576288742341}{1433719941773}$, $\frac{1}{617061705157393628296619} a^{19} + \frac{248732079837}{4639561692912734047343} a^{17} + \frac{763017256740649}{617061705157393628296619} a^{15} - \frac{712496565091485166}{47466285012107202176663} a^{13} + \frac{76295063469977283}{3651252693239015552051} a^{11} + \frac{721245939758641419}{280865591787616580927} a^{9} + \frac{224530791880522163}{21605045522124352379} a^{7} - \frac{35438879005402416}{87469819927628957} a^{5} - \frac{26867178012837}{354128825617931} a^{3} + \frac{576288742341}{1433719941773} a$
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: | \( 199790513425000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 7372800 |
| The 189 conjugacy class representatives for t20n1030 are not computed |
| Character table for t20n1030 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 10.6.189569215625.1 |
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 32 sibling: | 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 | R | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ | R | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ | R | $16{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ | $16{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ | $16{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ | ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| $5$ | 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $13$ | 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.8.4.1 | $x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $19$ | 19.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 19.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 19.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 19.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 19.4.0.1 | $x^{4} - 2 x + 10$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 19.8.4.2 | $x^{8} - 13718 x^{2} + 1303210$ | $2$ | $4$ | $4$ | $C_8$ | $[\ ]_{2}^{4}$ | |
| 60662149 | Data not computed | ||||||