Normalized defining polynomial
\( x^{20} - 6 x^{19} + 31 x^{18} - 126 x^{17} + 585 x^{16} - 2104 x^{15} + 6984 x^{14} - 20704 x^{13} + 60785 x^{12} - 160826 x^{11} + 383571 x^{10} - 820556 x^{9} + 1669105 x^{8} - 3136304 x^{7} + 5344064 x^{6} - 7735664 x^{5} + 9191895 x^{4} - 8447796 x^{3} + 5304891 x^{2} - 2161426 x + 543131 \)
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: | \(10288485539542801387520000000000000=2^{24}\cdot 5^{13}\cdot 3469^{5}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $50.19$ | 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: | $2, 5, 3469$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5} a^{16} - \frac{1}{5} a^{15} + \frac{2}{5} a^{14} - \frac{2}{5} a^{13} - \frac{2}{5} a^{12} - \frac{2}{5} a^{11} - \frac{1}{5} a^{10} + \frac{2}{5} a^{9} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{4} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{17} + \frac{1}{5} a^{15} + \frac{1}{5} a^{13} + \frac{1}{5} a^{12} + \frac{2}{5} a^{11} + \frac{1}{5} a^{10} - \frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{2} - \frac{1}{5}$, $\frac{1}{65} a^{18} + \frac{1}{65} a^{17} + \frac{1}{65} a^{16} + \frac{2}{5} a^{15} - \frac{24}{65} a^{14} - \frac{23}{65} a^{13} + \frac{28}{65} a^{12} - \frac{27}{65} a^{11} + \frac{4}{13} a^{10} + \frac{1}{13} a^{9} - \frac{23}{65} a^{8} - \frac{17}{65} a^{7} + \frac{2}{5} a^{6} + \frac{3}{65} a^{5} + \frac{19}{65} a^{4} + \frac{21}{65} a^{3} + \frac{1}{65} a^{2} + \frac{4}{65} a + \frac{24}{65}$, $\frac{1}{25642085934428640257074518682761121215472685643754384762405} a^{19} - \frac{13353764126733628981978558898693021748895919136329405646}{5128417186885728051414903736552224243094537128750876952481} a^{18} + \frac{153762729762678800069616374540975093145841518951255526961}{5128417186885728051414903736552224243094537128750876952481} a^{17} + \frac{2278691875804345161816044848107092450772131868784946775634}{25642085934428640257074518682761121215472685643754384762405} a^{16} - \frac{8819872266164714037620704789610673063272589657300067663964}{25642085934428640257074518682761121215472685643754384762405} a^{15} + \frac{7804996575338214298064352198399191920930125888884483930664}{25642085934428640257074518682761121215472685643754384762405} a^{14} + \frac{8868949413093366742358934140514220489490064823797158015013}{25642085934428640257074518682761121215472685643754384762405} a^{13} + \frac{2835746398836585241031122631233964252372539532178021608602}{25642085934428640257074518682761121215472685643754384762405} a^{12} - \frac{7215768391106634154924464468055743321828453370875667438906}{25642085934428640257074518682761121215472685643754384762405} a^{11} + \frac{715590454146895963912811869064317626494955826616263775427}{1972468148802203096698039898673932401190206587981106520185} a^{10} - \frac{603549967095052494924144329770794241617204747828369247171}{5128417186885728051414903736552224243094537128750876952481} a^{9} + \frac{8445365068718819963226725560721814347063022513725302483089}{25642085934428640257074518682761121215472685643754384762405} a^{8} + \frac{3034315162722174137499581118279530841407149531468173526809}{25642085934428640257074518682761121215472685643754384762405} a^{7} + \frac{1808487404371344724125518066799585281191080839511372244813}{5128417186885728051414903736552224243094537128750876952481} a^{6} - \frac{348728487544551742247125693466608190588376043535657026639}{25642085934428640257074518682761121215472685643754384762405} a^{5} + \frac{492773356457236813462904523810328869759449263415773217761}{1972468148802203096698039898673932401190206587981106520185} a^{4} + \frac{532346208356099600453147180203949124090160374294802162010}{5128417186885728051414903736552224243094537128750876952481} a^{3} - \frac{5197816071608245134684903846698329834719074394530261591647}{25642085934428640257074518682761121215472685643754384762405} a^{2} - \frac{11160598085525286050897224017166173167715820905782756138296}{25642085934428640257074518682761121215472685643754384762405} a - \frac{9631440232055138462811376683977015068878876897610176825503}{25642085934428640257074518682761121215472685643754384762405}$
Class group and class number
$C_{2}\times C_{2}\times C_{188}$, which has order $752$ (assuming GRH)
Unit group
| Rank: | $9$ | 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: | \( 828338.933858 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 102400 |
| The 130 conjugacy class representatives for t20n771 are not computed |
| Character table for t20n771 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 10.10.9627168800000.1 |
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 | $20$ | R | $20$ | ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ | $20$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | $20$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ | $20$ | ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| $5$ | 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.3.2 | $x^{4} - 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.8.6.2 | $x^{8} + 15 x^{4} + 100$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 3469 | Data not computed | ||||||