Normalized defining polynomial
\( x^{20} + 110 x^{18} + 5955 x^{16} - 2 x^{15} + 205640 x^{14} + 160 x^{13} + 4975165 x^{12} + 9830 x^{11} + 87672445 x^{10} + 236110 x^{9} + 1136139650 x^{8} + 2562310 x^{7} + 10673029575 x^{6} + 3591456 x^{5} + 69517358130 x^{4} - 198685110 x^{3} + 283651665155 x^{2} - 1551975480 x + 551416888049 \)
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: | \(84142145903930664062500000000000000000000=2^{20}\cdot 5^{34}\cdot 13^{10}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $111.24$ | 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: | $2, 5, 13$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1300=2^{2}\cdot 5^{2}\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1300}(1,·)$, $\chi_{1300}(1091,·)$, $\chi_{1300}(261,·)$, $\chi_{1300}(519,·)$, $\chi_{1300}(521,·)$, $\chi_{1300}(779,·)$, $\chi_{1300}(781,·)$, $\chi_{1300}(1039,·)$, $\chi_{1300}(209,·)$, $\chi_{1300}(259,·)$, $\chi_{1300}(469,·)$, $\chi_{1300}(729,·)$, $\chi_{1300}(989,·)$, $\chi_{1300}(1249,·)$, $\chi_{1300}(1041,·)$, $\chi_{1300}(1299,·)$, $\chi_{1300}(51,·)$, $\chi_{1300}(311,·)$, $\chi_{1300}(571,·)$, $\chi_{1300}(831,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{7} a^{18} - \frac{3}{7} a^{17} + \frac{2}{7} a^{16} + \frac{2}{7} a^{15} + \frac{3}{7} a^{14} - \frac{1}{7} a^{11} + \frac{2}{7} a^{10} + \frac{1}{7} a^{9} + \frac{1}{7} a^{8} - \frac{1}{7} a^{7} - \frac{2}{7} a^{4} - \frac{1}{7} a^{2} + \frac{2}{7} a - \frac{3}{7}$, $\frac{1}{828735174248368655668597205359479407862387577194885065726400576613821307075051} a^{19} - \frac{3569813111083438538339330176915463268327474839604837636679292854809264463058}{118390739178338379381228172194211343980341082456412152246628653801974472439293} a^{18} + \frac{92033776677661937193288159746315560105084015435610822830321781973422553381}{118390739178338379381228172194211343980341082456412152246628653801974472439293} a^{17} + \frac{325441794341998441916478715734607843689895049722480256719030454059892594551830}{828735174248368655668597205359479407862387577194885065726400576613821307075051} a^{16} - \frac{59074585121818471553293166217665801856695137760003663654117776993496050321209}{828735174248368655668597205359479407862387577194885065726400576613821307075051} a^{15} - \frac{333114977244538603677689192141103164331234400733074167286520934792789199456795}{828735174248368655668597205359479407862387577194885065726400576613821307075051} a^{14} - \frac{55814785251575707357224167768328687086625638523907031385623873783915436446151}{118390739178338379381228172194211343980341082456412152246628653801974472439293} a^{13} + \frac{256800001087062433422431100210170058455424157523039870610611049986761362750360}{828735174248368655668597205359479407862387577194885065726400576613821307075051} a^{12} - \frac{399711530389704250504420736538243700826664901037117816197140194070437288629569}{828735174248368655668597205359479407862387577194885065726400576613821307075051} a^{11} - \frac{32558119881098435981587406783111952096237212021232274830958303242476776464407}{118390739178338379381228172194211343980341082456412152246628653801974472439293} a^{10} + \frac{364089292594153705319492576665921173829295323559391805346904645240826142298106}{828735174248368655668597205359479407862387577194885065726400576613821307075051} a^{9} + \frac{311230281107291662986302096322435136810737270986582223464207424573575398437920}{828735174248368655668597205359479407862387577194885065726400576613821307075051} a^{8} + \frac{277064638014706088321318322726287120071909918425006028658249663630905505369599}{828735174248368655668597205359479407862387577194885065726400576613821307075051} a^{7} + \frac{1680687220075221461319330313543005648357592932522679264246036007654346380996}{118390739178338379381228172194211343980341082456412152246628653801974472439293} a^{6} - \frac{326603589771320189495113381365323404321366770175621325056700725479627620070614}{828735174248368655668597205359479407862387577194885065726400576613821307075051} a^{5} - \frac{209565870676760133808892654309631632554375877165795790726936043880865194041557}{828735174248368655668597205359479407862387577194885065726400576613821307075051} a^{4} - \frac{339883631452141928892694198160704745049713817014461866314860431992345539915842}{828735174248368655668597205359479407862387577194885065726400576613821307075051} a^{3} + \frac{410893483590601061818402343437868932452690545934687920191391925777066747519442}{828735174248368655668597205359479407862387577194885065726400576613821307075051} a^{2} - \frac{380253712166098787422637578207483895324546237193705404905941108281045061413057}{828735174248368655668597205359479407862387577194885065726400576613821307075051} a - \frac{3459849461457797775104550009472122903641612321466938124989992206956613911404}{8205298754934343125431655498608707008538490863315693720063372045681399079951}$
Class group and class number
$C_{6368888}$, which has order $6368888$ (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: | \( 161406.8376411007 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_{10}$ (as 20T3):
| An abelian group of order 20 |
| The 20 conjugacy class representatives for $C_2\times C_{10}$ |
| Character table for $C_2\times C_{10}$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-65}) \), \(\Q(\sqrt{-13}) \), \(\Q(\sqrt{5}, \sqrt{-13})\), 5.5.390625.1, \(\Q(\zeta_{25})^+\), 10.0.290072656250000000000.3, 10.0.58014531250000000000.3 |
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 | ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ | R | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$ |
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 | Data not computed | ||||||
| 5 | Data not computed | ||||||
| 13 | Data not computed | ||||||