Normalized defining polynomial
\( x^{20} - 2 x^{19} + 2 x^{18} - x^{17} - 2 x^{15} - 9 x^{14} + 3 x^{12} + x^{11} + 22 x^{9} + 45 x^{8} + 41 x^{7} + 29 x^{6} + 18 x^{5} + 7 x^{4} - 2 x^{3} - 3 x^{2} + 1 \)
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: | \(522050086690675147528677=3^{10}\cdot 13^{5}\cdot 47^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $15.34$ | 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: | $3, 13, 47$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2327} a^{18} + \frac{606}{2327} a^{17} - \frac{392}{2327} a^{16} + \frac{750}{2327} a^{15} + \frac{154}{2327} a^{14} + \frac{483}{2327} a^{13} + \frac{855}{2327} a^{12} + \frac{699}{2327} a^{11} - \frac{1062}{2327} a^{10} + \frac{618}{2327} a^{9} + \frac{410}{2327} a^{8} - \frac{431}{2327} a^{7} + \frac{477}{2327} a^{6} + \frac{83}{179} a^{5} - \frac{298}{2327} a^{4} - \frac{349}{2327} a^{3} + \frac{970}{2327} a^{2} - \frac{428}{2327} a - \frac{93}{2327}$, $\frac{1}{44624879} a^{19} - \frac{5828}{44624879} a^{18} - \frac{7966992}{44624879} a^{17} - \frac{6701350}{44624879} a^{16} + \frac{6341927}{44624879} a^{15} + \frac{1629331}{3432683} a^{14} + \frac{304615}{44624879} a^{13} + \frac{22197910}{44624879} a^{12} - \frac{10734788}{44624879} a^{11} - \frac{7384444}{44624879} a^{10} - \frac{12222690}{44624879} a^{9} + \frac{8009981}{44624879} a^{8} + \frac{4474568}{44624879} a^{7} - \frac{13062404}{44624879} a^{6} + \frac{11813036}{44624879} a^{5} + \frac{5768168}{44624879} a^{4} - \frac{5344238}{44624879} a^{3} - \frac{19193490}{44624879} a^{2} - \frac{18703608}{44624879} a - \frac{3781052}{44624879}$
Class group and class number
Trivial group, which has order $1$
Unit group
| Rank: | $9$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{101}{179} a^{19} + \frac{3033}{2327} a^{18} - \frac{3686}{2327} a^{17} + \frac{2709}{2327} a^{16} - \frac{1459}{2327} a^{15} + \frac{4347}{2327} a^{14} + \frac{8679}{2327} a^{13} - \frac{753}{2327} a^{12} - \frac{3626}{2327} a^{11} + \frac{666}{2327} a^{10} + \frac{1692}{2327} a^{9} - \frac{32355}{2327} a^{8} - \frac{48901}{2327} a^{7} - \frac{39744}{2327} a^{6} - \frac{2475}{179} a^{5} - \frac{24462}{2327} a^{4} - \frac{11185}{2327} a^{3} + \frac{1774}{2327} a^{2} + \frac{732}{2327} a + \frac{44}{2327} \) (order $6$) | 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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 8452.6678373 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$D_4\times D_5$ (as 20T21):
| A solvable group of order 80 |
| The 20 conjugacy class representatives for $D_4\times D_5$ |
| Character table for $D_4\times D_5$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 4.0.117.1, 5.1.2209.1, 10.0.1185762483.1 |
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 | $20$ | R | ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ | R | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ | R | ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ | $20$ |
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 |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||
| $13$ | 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $47$ | 47.4.0.1 | $x^{4} - x + 39$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 47.8.4.1 | $x^{8} + 172302 x^{4} - 103823 x^{2} + 7421994801$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 47.8.4.1 | $x^{8} + 172302 x^{4} - 103823 x^{2} + 7421994801$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |