Normalized defining polynomial
\( x^{20} + 205 x^{18} + 16400 x^{16} + 661125 x^{14} + 14478125 x^{12} + 174378125 x^{10} + 1117890625 x^{8} + 3526640625 x^{6} + 5125000000 x^{4} + 2962890625 x^{2} + 400390625 \)
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: | \(44998002377408544344171356675696640000000000=2^{20}\cdot 5^{10}\cdot 41^{19}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $152.29$ | 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: | $2, 5, 41$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(820=2^{2}\cdot 5\cdot 41\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{820}(799,·)$, $\chi_{820}(1,·)$, $\chi_{820}(579,·)$, $\chi_{820}(441,·)$, $\chi_{820}(201,·)$, $\chi_{820}(459,·)$, $\chi_{820}(141,·)$, $\chi_{820}(461,·)$, $\chi_{820}(81,·)$, $\chi_{820}(279,·)$, $\chi_{820}(221,·)$, $\chi_{820}(159,·)$, $\chi_{820}(419,·)$, $\chi_{820}(39,·)$, $\chi_{820}(681,·)$, $\chi_{820}(759,·)$, $\chi_{820}(761,·)$, $\chi_{820}(699,·)$, $\chi_{820}(701,·)$, $\chi_{820}(319,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{5} a^{2}$, $\frac{1}{5} a^{3}$, $\frac{1}{25} a^{4}$, $\frac{1}{25} a^{5}$, $\frac{1}{125} a^{6}$, $\frac{1}{125} a^{7}$, $\frac{1}{1875} a^{8} + \frac{1}{3}$, $\frac{1}{1875} a^{9} + \frac{1}{3} a$, $\frac{1}{9375} a^{10} + \frac{1}{15} a^{2}$, $\frac{1}{9375} a^{11} + \frac{1}{15} a^{3}$, $\frac{1}{46875} a^{12} + \frac{1}{75} a^{4}$, $\frac{1}{46875} a^{13} + \frac{1}{75} a^{5}$, $\frac{1}{234375} a^{14} + \frac{1}{375} a^{6}$, $\frac{1}{234375} a^{15} + \frac{1}{375} a^{7}$, $\frac{1}{256640625} a^{16} + \frac{2}{5703125} a^{14} + \frac{7}{684375} a^{12} - \frac{2}{136875} a^{10} + \frac{53}{410625} a^{8} - \frac{34}{5475} a^{4} - \frac{17}{219} a^{2} - \frac{110}{657}$, $\frac{1}{256640625} a^{17} + \frac{2}{5703125} a^{15} + \frac{7}{684375} a^{13} - \frac{2}{136875} a^{11} + \frac{53}{410625} a^{9} - \frac{34}{5475} a^{5} - \frac{17}{219} a^{3} - \frac{110}{657} a$, $\frac{1}{7536035091796875} a^{18} + \frac{1762651}{1507207018359375} a^{16} + \frac{18641914}{20096093578125} a^{14} - \frac{9177052}{6698697859375} a^{12} - \frac{3830878}{165173371875} a^{10} + \frac{67638688}{482306245875} a^{8} - \frac{112461062}{32153749725} a^{6} + \frac{75059731}{10717916575} a^{4} - \frac{1443614141}{19292249835} a^{2} + \frac{653515540}{3858449967}$, $\frac{1}{7536035091796875} a^{19} + \frac{1762651}{1507207018359375} a^{17} + \frac{18641914}{20096093578125} a^{15} - \frac{9177052}{6698697859375} a^{13} - \frac{3830878}{165173371875} a^{11} + \frac{67638688}{482306245875} a^{9} - \frac{112461062}{32153749725} a^{7} + \frac{75059731}{10717916575} a^{5} - \frac{1443614141}{19292249835} a^{3} + \frac{653515540}{3858449967} a$
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{843604}$, which has order $6748832$ (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: | \( 5104264.636551031 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 20 |
| The 20 conjugacy class representatives for $C_{20}$ |
| Character table for $C_{20}$ |
Intermediate fields
| \(\Q(\sqrt{41}) \), 4.0.27568400.4, 5.5.2825761.1, 10.10.327381934393961.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ | R | $20$ | $20$ | $20$ | $20$ | $20$ | ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ | $20$ | ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ | $20$ | $20$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.10.10.7 | $x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$ | $2$ | $5$ | $10$ | $C_{10}$ | $[2]^{5}$ |
| 2.10.10.7 | $x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$ | $2$ | $5$ | $10$ | $C_{10}$ | $[2]^{5}$ | |
| $5$ | 5.10.5.2 | $x^{10} - 625 x^{2} + 6250$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
| 5.10.5.2 | $x^{10} - 625 x^{2} + 6250$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
| 41 | Data not computed | ||||||