Normalized defining polynomial
\( x^{20} - 8 x^{18} - 5 x^{17} + 24 x^{16} + 14 x^{15} - 54 x^{14} - 7 x^{13} + 138 x^{12} + 57 x^{11} - 201 x^{10} - 171 x^{9} + 159 x^{8} + 207 x^{7} - 95 x^{6} - 166 x^{5} + 24 x^{4} + 77 x^{3} + 14 x^{2} - 6 x - 1 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[8, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(1208667589881678916015625=5^{10}\cdot 109\cdot 149\cdot 2760559^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $16.00$ | 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: | $5, 109, 149, 2760559$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{4141302483137641} a^{19} + \frac{461333921777832}{4141302483137641} a^{18} - \frac{1443785371713197}{4141302483137641} a^{17} - \frac{1281138126602287}{4141302483137641} a^{16} - \frac{422819427328842}{4141302483137641} a^{15} - \frac{1196669566402068}{4141302483137641} a^{14} + \frac{125086354400282}{4141302483137641} a^{13} + \frac{1411033618607257}{4141302483137641} a^{12} - \frac{1466130063842565}{4141302483137641} a^{11} - \frac{1447092271065089}{4141302483137641} a^{10} - \frac{1187593959302539}{4141302483137641} a^{9} - \frac{625438836777904}{4141302483137641} a^{8} - \frac{1435214469209986}{4141302483137641} a^{7} - \frac{311668986880095}{4141302483137641} a^{6} - \frac{63669008244728}{4141302483137641} a^{5} + \frac{918330889090873}{4141302483137641} a^{4} + \frac{413460950698340}{4141302483137641} a^{3} + \frac{322419848010008}{4141302483137641} a^{2} - \frac{97763500605406}{4141302483137641} a - \frac{1868064099036498}{4141302483137641}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $13$ | 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: | \( 20311.8930394 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 29491200 |
| The 702 conjugacy class representatives for t20n1045 are not computed |
| Character table for t20n1045 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 10.4.8626746875.1 |
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 | ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ | R | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ | $16{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ | ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ | $20$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.10.5.1 | $x^{10} - 50 x^{6} + 625 x^{2} - 12500$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
| 5.10.5.1 | $x^{10} - 50 x^{6} + 625 x^{2} - 12500$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
| 109 | Data not computed | ||||||
| $149$ | $\Q_{149}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{149}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 149.2.1.2 | $x^{2} + 298$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 149.6.0.1 | $x^{6} - x + 14$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 149.10.0.1 | $x^{10} - x + 71$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
| 2760559 | Data not computed | ||||||