Normalized defining polynomial
\( x^{20} + 580 x^{18} + 139780 x^{16} + 18228675 x^{14} + 1399760400 x^{12} + 64198067195 x^{10} + 1686964570175 x^{8} + 22404681693200 x^{6} + 106563464376375 x^{4} + 44555856193975 x^{2} + 2276840094745 \)
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: | \(26334194176019849978482055664062500000000000000000000=2^{20}\cdot 5^{35}\cdot 29^{15}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $417.86$ | 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: | $2, 5, 29$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(2900=2^{2}\cdot 5^{2}\cdot 29\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{2900}(1,·)$, $\chi_{2900}(1027,·)$, $\chi_{2900}(581,·)$, $\chi_{2900}(1607,·)$, $\chi_{2900}(1161,·)$, $\chi_{2900}(2187,·)$, $\chi_{2900}(1741,·)$, $\chi_{2900}(2767,·)$, $\chi_{2900}(2321,·)$, $\chi_{2900}(289,·)$, $\chi_{2900}(869,·)$, $\chi_{2900}(423,·)$, $\chi_{2900}(1449,·)$, $\chi_{2900}(1003,·)$, $\chi_{2900}(2029,·)$, $\chi_{2900}(1583,·)$, $\chi_{2900}(2609,·)$, $\chi_{2900}(2163,·)$, $\chi_{2900}(2743,·)$, $\chi_{2900}(447,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{29} a^{4}$, $\frac{1}{29} a^{5}$, $\frac{1}{29} a^{6}$, $\frac{1}{29} a^{7}$, $\frac{1}{841} a^{8}$, $\frac{1}{841} a^{9}$, $\frac{1}{841} a^{10}$, $\frac{1}{841} a^{11}$, $\frac{1}{24389} a^{12}$, $\frac{1}{24389} a^{13}$, $\frac{1}{24389} a^{14}$, $\frac{1}{24389} a^{15}$, $\frac{1}{338570463733} a^{16} + \frac{123837}{11674843577} a^{14} + \frac{13858}{11674843577} a^{12} - \frac{143431}{402580813} a^{10} + \frac{123557}{402580813} a^{8} - \frac{57612}{13882097} a^{6} - \frac{214214}{13882097} a^{4} + \frac{178735}{478693} a^{2} - \frac{231494}{478693}$, $\frac{1}{338570463733} a^{17} + \frac{123837}{11674843577} a^{15} + \frac{13858}{11674843577} a^{13} - \frac{143431}{402580813} a^{11} + \frac{123557}{402580813} a^{9} - \frac{57612}{13882097} a^{7} - \frac{214214}{13882097} a^{5} + \frac{178735}{478693} a^{3} - \frac{231494}{478693} a$, $\frac{1}{66097511806088232065566185109183} a^{18} + \frac{46971137167028939318}{66097511806088232065566185109183} a^{16} - \frac{38466181378408589429345909}{2279224545037525243640213279627} a^{14} + \frac{28192965726812137214009672}{2279224545037525243640213279627} a^{12} - \frac{40287738062947853590071833}{78593949828880180815179768263} a^{10} + \frac{20406754424699195464263446}{78593949828880180815179768263} a^{8} + \frac{35930656221453039678450019}{2710136200995868303971716147} a^{6} + \frac{37687582640041402492434589}{2710136200995868303971716147} a^{4} - \frac{44890411201284935573067742}{93452972448133389792128143} a^{2} - \frac{42877380651083708085844743}{93452972448133389792128143}$, $\frac{1}{9848529259107146577769361581268267} a^{19} + \frac{14493641469012012769492}{9848529259107146577769361581268267} a^{17} - \frac{959570068489589630823804418}{339604457210591261302391778664423} a^{15} - \frac{6874030982872617701069177904}{339604457210591261302391778664423} a^{13} + \frac{410842145669795795051701034}{11710498524503146941461785471187} a^{11} + \frac{6104230733230605793006966942}{11710498524503146941461785471187} a^{9} + \frac{3222106900332935592315975680}{403810293948384377291785705903} a^{7} + \frac{2643639863510336813615358076}{403810293948384377291785705903} a^{5} + \frac{574722789150227063678391145}{13924492894771875079027093307} a^{3} + \frac{724735905266662755985493593}{13924492894771875079027093307} a$
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{441166888}$, which has order $3529335104$ (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: | \( 257696579.12792215 \) (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
| \(\Q(\sqrt{145}) \), 4.0.48778000.2, 5.5.390625.1, 10.10.15648764801025390625.1 |
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 | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ | $20$ | $20$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | $20$ | $20$ | R | $20$ | ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ | $20$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | $20$ | ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.10.10.11 | $x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$ | $2$ | $5$ | $10$ | $C_{10}$ | $[2]^{5}$ |
| 2.10.10.11 | $x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$ | $2$ | $5$ | $10$ | $C_{10}$ | $[2]^{5}$ | |
| 5 | Data not computed | ||||||
| 29 | Data not computed | ||||||