Normalized defining polynomial
\( x^{20} - x^{19} - 12 x^{18} + 13 x^{17} + 35 x^{16} - 48 x^{15} + 2 x^{14} + 36 x^{13} + 247 x^{12} - 302 x^{11} - 1711 x^{10} + 2899 x^{9} + 366 x^{8} - 4888 x^{7} + 6101 x^{6} - 3067 x^{5} - 159 x^{4} + 739 x^{3} - 249 x^{2} + 28 x - 1 \)
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: | \(158013751236884807357861328125=5^{11}\cdot 61^{4}\cdot 97^{2}\cdot 397^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $28.84$ | 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: | $5, 61, 97, 397$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{8897645625342804618653985629} a^{19} + \frac{522145876064400994637940813}{8897645625342804618653985629} a^{18} + \frac{71427191562110483975730409}{145863043038406633092688289} a^{17} - \frac{1785053832721230850172423520}{8897645625342804618653985629} a^{16} - \frac{991577635907407180379134021}{8897645625342804618653985629} a^{15} - \frac{4338428483343702580877328221}{8897645625342804618653985629} a^{14} - \frac{1810473135389146844311771789}{8897645625342804618653985629} a^{13} + \frac{3626638229781543098658818672}{8897645625342804618653985629} a^{12} - \frac{3860135478908136397394870880}{8897645625342804618653985629} a^{11} + \frac{2926968098154752303874359787}{8897645625342804618653985629} a^{10} - \frac{1468678285262139804759454440}{8897645625342804618653985629} a^{9} + \frac{987317151113290063023961477}{8897645625342804618653985629} a^{8} + \frac{3961222910681012761249458523}{8897645625342804618653985629} a^{7} + \frac{3329229253392866799395806168}{8897645625342804618653985629} a^{6} - \frac{3019549824928453966313875122}{8897645625342804618653985629} a^{5} - \frac{2652162405454683739450727780}{8897645625342804618653985629} a^{4} - \frac{4182621559058081770611473533}{8897645625342804618653985629} a^{3} - \frac{3342613483963573526894647586}{8897645625342804618653985629} a^{2} - \frac{3959418586541409627656728190}{8897645625342804618653985629} a - \frac{2385119097458216592944699583}{8897645625342804618653985629}$
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: | \( 26473017.0469 \) (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 t20n802 are not computed |
| Character table for t20n802 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 5.5.24217.1, 10.10.1832697153125.1 |
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 | $20$ | $20$ | R | $20$ | ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 | Data not computed | ||||||
| 61 | Data not computed | ||||||
| $97$ | 97.4.2.2 | $x^{4} - 97 x^{2} + 47045$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 97.4.0.1 | $x^{4} - x + 23$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 97.12.0.1 | $x^{12} - x + 68$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
| 397 | Data not computed | ||||||