Normalized defining polynomial
\( x^{20} - x^{19} - x^{18} + 4 x^{17} - 4 x^{16} + 7 x^{15} - 12 x^{14} + 19 x^{13} + 16 x^{12} - 73 x^{11} + 71 x^{10} + 73 x^{9} + 16 x^{8} - 19 x^{7} - 12 x^{6} - 7 x^{5} - 4 x^{4} - 4 x^{3} - x^{2} + x + 1 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[4, 8]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(17327212977905840634765625=5^{10}\cdot 36497^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $18.28$ | 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: | $5, 36497$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{45749077} a^{18} + \frac{9635513}{45749077} a^{17} - \frac{10706041}{45749077} a^{16} - \frac{2072604}{4159007} a^{15} + \frac{18564460}{45749077} a^{14} - \frac{21901888}{45749077} a^{13} - \frac{12592684}{45749077} a^{12} - \frac{21887889}{45749077} a^{11} + \frac{21280767}{45749077} a^{10} - \frac{8067038}{45749077} a^{9} - \frac{21280767}{45749077} a^{8} - \frac{21887889}{45749077} a^{7} + \frac{12592684}{45749077} a^{6} - \frac{21901888}{45749077} a^{5} - \frac{18564460}{45749077} a^{4} - \frac{2072604}{4159007} a^{3} + \frac{10706041}{45749077} a^{2} + \frac{9635513}{45749077} a - \frac{1}{45749077}$, $\frac{1}{8280582937} a^{19} - \frac{20}{8280582937} a^{18} - \frac{3111015675}{8280582937} a^{17} - \frac{2577643401}{8280582937} a^{16} - \frac{1087251350}{8280582937} a^{15} - \frac{650017839}{8280582937} a^{14} - \frac{3911461995}{8280582937} a^{13} + \frac{491853589}{8280582937} a^{12} - \frac{1335187703}{8280582937} a^{11} - \frac{2301934692}{8280582937} a^{10} + \frac{3605735335}{8280582937} a^{9} - \frac{59632833}{752780267} a^{8} + \frac{264617125}{752780267} a^{7} - \frac{169650750}{8280582937} a^{6} + \frac{3493916703}{8280582937} a^{5} + \frac{559568230}{8280582937} a^{4} - \frac{2650578387}{8280582937} a^{3} - \frac{369921445}{752780267} a^{2} + \frac{3167313914}{8280582937} a + \frac{2342838460}{8280582937}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $11$ | 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: | \( 41483.866986 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 3840 |
| The 36 conjugacy class representatives for t20n279 |
| Character table for t20n279 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 5.5.36497.1, 10.10.4162596903125.1, 10.2.1332031009.1, 10.2.4162596903125.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 10 siblings: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 30 siblings: | data not computed |
| Degree 32 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 | ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 5.12.6.1 | $x^{12} + 500 x^{6} - 3125 x^{2} + 62500$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
| 36497 | Data not computed | ||||||