Normalized defining polynomial
\( x^{20} - 20 x^{18} - 16 x^{17} + 136 x^{16} + 204 x^{15} - 312 x^{14} - 808 x^{13} + 180 x^{12} + 1668 x^{11} - 268 x^{10} - 4624 x^{9} - 3716 x^{8} + 3116 x^{7} + 7564 x^{6} + 5988 x^{5} + 2328 x^{4} + 288 x^{3} - 88 x^{2} - 20 x + 2 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[10, 5]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-375319694708943511165994008576=-\,2^{40}\cdot 13^{10}\cdot 19^{5}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $30.11$ | 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: | $2, 13, 19$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{2552514642519022039029072121} a^{19} - \frac{1240912680988555589934765841}{2552514642519022039029072121} a^{18} - \frac{477319630993645959064754951}{2552514642519022039029072121} a^{17} + \frac{62929745132885027953783797}{2552514642519022039029072121} a^{16} - \frac{88749960740155458006307633}{2552514642519022039029072121} a^{15} - \frac{639976354602206571078653185}{2552514642519022039029072121} a^{14} + \frac{1219660011662125528912712138}{2552514642519022039029072121} a^{13} + \frac{664003642960147265947701521}{2552514642519022039029072121} a^{12} - \frac{206653425198661745493770754}{2552514642519022039029072121} a^{11} - \frac{29222357489940291830059898}{232046785683547458093552011} a^{10} - \frac{208773596323624864528962574}{2552514642519022039029072121} a^{9} - \frac{186559385528989912798740922}{2552514642519022039029072121} a^{8} - \frac{631200340441291347525859871}{2552514642519022039029072121} a^{7} - \frac{628926711949604126365068348}{2552514642519022039029072121} a^{6} + \frac{136792047326530369918763496}{2552514642519022039029072121} a^{5} + \frac{68077141833419368841958211}{2552514642519022039029072121} a^{4} + \frac{549253775850945279920717526}{2552514642519022039029072121} a^{3} - \frac{249603951150689640963130621}{2552514642519022039029072121} a^{2} - \frac{711228064862167039110744858}{2552514642519022039029072121} a - \frac{96359214568888199587262457}{2552514642519022039029072121}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $14$ | 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: | \( 47795305.5366 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 1966080 |
| The 265 conjugacy class representatives for t20n989 are not computed |
| Character table for t20n989 is not computed |
Intermediate fields
| 5.3.51376.1, 10.6.8784233955328.1 |
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 | R | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ | ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ | R | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ | ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ | ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$ |
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 | ||||||
| $13$ | 13.8.0.1 | $x^{8} + 4 x^{2} - x + 6$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |
| 13.12.10.2 | $x^{12} + 39 x^{6} + 676$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ | |
| $19$ | 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 19.4.3.1 | $x^{4} + 76$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 19.12.0.1 | $x^{12} - x + 15$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |