Normalized defining polynomial
\( x^{20} - 4 x^{19} + 10 x^{18} - 8 x^{17} + 1359 x^{16} - 4588 x^{15} + 47346 x^{14} - 131776 x^{13} + 1059424 x^{12} - 2070016 x^{11} + 29195556 x^{10} - 71901908 x^{9} + 836887947 x^{8} - 1672343968 x^{7} + 14146109838 x^{6} - 20624953016 x^{5} + 133116343639 x^{4} - 113533795584 x^{3} + 724445574450 x^{2} - 170599786492 x + 1755011915551 \)
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: | \(40408914735836677919340723372004250484736000000000000000=2^{55}\cdot 5^{15}\cdot 61^{16}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $603.01$ | 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: | $2, 5, 61$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(4880=2^{4}\cdot 5\cdot 61\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{4880}(1,·)$, $\chi_{4880}(2693,·)$, $\chi_{4880}(9,·)$, $\chi_{4880}(4877,·)$, $\chi_{4880}(81,·)$, $\chi_{4880}(3413,·)$, $\chi_{4880}(569,·)$, $\chi_{4880}(729,·)$, $\chi_{4880}(1437,·)$, $\chi_{4880}(3173,·)$, $\chi_{4880}(4401,·)$, $\chi_{4880}(489,·)$, $\chi_{4880}(4717,·)$, $\chi_{4880}(4637,·)$, $\chi_{4880}(241,·)$, $\chi_{4880}(4853,·)$, $\chi_{4880}(1681,·)$, $\chi_{4880}(2169,·)$, $\chi_{4880}(4157,·)$, $\chi_{4880}(3253,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13} a^{12} - \frac{1}{13}$, $\frac{1}{13} a^{13} - \frac{1}{13} a$, $\frac{1}{13} a^{14} - \frac{1}{13} a^{2}$, $\frac{1}{39} a^{15} + \frac{1}{39} a^{13} + \frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{14}{39} a^{3} + \frac{1}{3} a^{2} + \frac{4}{13} a + \frac{1}{3}$, $\frac{1}{1131} a^{16} + \frac{10}{1131} a^{15} + \frac{1}{87} a^{14} - \frac{1}{39} a^{13} + \frac{1}{87} a^{12} - \frac{11}{87} a^{11} + \frac{4}{87} a^{10} + \frac{4}{87} a^{9} - \frac{37}{87} a^{8} - \frac{1}{29} a^{6} + \frac{7}{29} a^{5} - \frac{547}{1131} a^{4} + \frac{419}{1131} a^{3} - \frac{17}{87} a^{2} - \frac{218}{1131} a + \frac{1}{87}$, $\frac{1}{1131} a^{17} + \frac{5}{377} a^{14} + \frac{14}{377} a^{13} - \frac{4}{377} a^{12} + \frac{9}{29} a^{11} - \frac{12}{29} a^{10} + \frac{10}{87} a^{9} + \frac{22}{87} a^{8} - \frac{1}{29} a^{7} - \frac{12}{29} a^{6} + \frac{4}{39} a^{5} + \frac{6}{29} a^{4} + \frac{2}{87} a^{3} - \frac{148}{377} a^{2} + \frac{64}{377} a - \frac{391}{1131}$, $\frac{1}{163663028677379462750983381983} a^{18} + \frac{24072772908919843834163049}{54554342892459820916994460661} a^{17} - \frac{46388060091618342149846}{144706479820848331344812893} a^{16} - \frac{233444641422195560060115344}{163663028677379462750983381983} a^{15} + \frac{840200109008619401873416799}{54554342892459820916994460661} a^{14} + \frac{64534782851106310901927128}{12589463744413804826998721691} a^{13} + \frac{1473468081113470710924190042}{54554342892459820916994460661} a^{12} + \frac{545358247916376976262759371}{12589463744413804826998721691} a^{11} - \frac{4404662867497578965560991624}{12589463744413804826998721691} a^{10} + \frac{6049515800665756626619660967}{12589463744413804826998721691} a^{9} + \frac{1003299050173288633100348722}{4196487914804601608999573897} a^{8} + \frac{612920351343273782941409177}{12589463744413804826998721691} a^{7} + \frac{20786705091971273225476728312}{54554342892459820916994460661} a^{6} + \frac{10693496049962752467614236268}{163663028677379462750983381983} a^{5} - \frac{736662889946971754943049004}{4196487914804601608999573897} a^{4} - \frac{71853166882561044473858920103}{163663028677379462750983381983} a^{3} + \frac{34276579193874378983410480261}{163663028677379462750983381983} a^{2} - \frac{6241742562617469790944661432}{12589463744413804826998721691} a + \frac{20351231978339577635427461104}{163663028677379462750983381983}$, $\frac{1}{256271930184568698362730418189556488357190735715750068567504007} a^{19} - \frac{348514226669133372030698963595325}{256271930184568698362730418189556488357190735715750068567504007} a^{18} + \frac{19482600131771313834559768144110982501474247228985387198385}{85423976728189566120910139396518829452396911905250022855834669} a^{17} - \frac{44965673686796244487096740028981565285482862525520789720152}{256271930184568698362730418189556488357190735715750068567504007} a^{16} + \frac{185829209794004225503523675060895654287639274257501862362110}{256271930184568698362730418189556488357190735715750068567504007} a^{15} - \frac{1230385350742392116945292725618076418062043821025623651222269}{256271930184568698362730418189556488357190735715750068567504007} a^{14} + \frac{9290179879473716989819142476862396841150807650891539184432623}{256271930184568698362730418189556488357190735715750068567504007} a^{13} + \frac{8156769067556132974492945088010385114078916692566259377892765}{256271930184568698362730418189556488357190735715750068567504007} a^{12} + \frac{2818676947848418425428380132755672151996466968689494441338217}{19713225398812976797133109091504345258245441208903851428269539} a^{11} - \frac{5397000807335960863171305841087813919242760537769562175143970}{19713225398812976797133109091504345258245441208903851428269539} a^{10} - \frac{1075363465313205210279404784475977928657146848797792200198684}{6571075132937658932377703030501448419415147069634617142756513} a^{9} - \frac{8665725478416302838533281679982148437624505300523118952541929}{19713225398812976797133109091504345258245441208903851428269539} a^{8} + \frac{2181012864825305821146630342796965406383619779677205836578636}{8836963109812713736645876489295051322661749507439657536810483} a^{7} - \frac{3962062181057478495369529196347337664919184218989698591288878}{8836963109812713736645876489295051322661749507439657536810483} a^{6} + \frac{12400893926196225779361255902833965043727036695965552751115114}{256271930184568698362730418189556488357190735715750068567504007} a^{5} + \frac{41242255900757528615223176155923154474592271454694669666728929}{85423976728189566120910139396518829452396911905250022855834669} a^{4} - \frac{101652194476996109924329972914981782416141559116392200155925623}{256271930184568698362730418189556488357190735715750068567504007} a^{3} + \frac{24968638803090272278471351463693680266547439579525045188724170}{85423976728189566120910139396518829452396911905250022855834669} a^{2} - \frac{64090599681046017280087511227659287416043736151404720811350847}{256271930184568698362730418189556488357190735715750068567504007} a - \frac{16461667338406991936436282113641345331923740365510280175420907}{256271930184568698362730418189556488357190735715750068567504007}$
Class group and class number
$C_{3}\times C_{3}\times C_{3}\times C_{140940300}$, which has order $3805388100$ (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: | \( 18503259981.639687 \) (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
| \(\Q(\sqrt{10}) \), 4.0.256000.2, 5.5.13845841.1, 10.10.19630828850921574400000.1 |
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 | ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ | R | $20$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/13.1.0.1}{1} }^{20}$ | $20$ | $20$ | $20$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ | $20$ |
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 | ||||||
| 61 | Data not computed | ||||||