Normalized defining polynomial
\( x^{20} - x^{19} + 206 x^{18} - 213 x^{17} + 14756 x^{16} - 16247 x^{15} + 409213 x^{14} - 522942 x^{13} + 2482133 x^{12} - 6142727 x^{11} - 32065694 x^{10} - 206931350 x^{9} + 249229907 x^{8} + 2124172560 x^{7} + 930449339 x^{6} - 12719714095 x^{5} + 9464090843 x^{4} - 1873109137 x^{3} + 84463551288 x^{2} - 21510890052 x + 606400602193 \)
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: | \(2833027880450684991730528545126418023295128981=3^{10}\cdot 11^{18}\cdot 29^{15}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $187.33$ | 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: | $3, 11, 29$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(957=3\cdot 11\cdot 29\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{957}(128,·)$, $\chi_{957}(1,·)$, $\chi_{957}(262,·)$, $\chi_{957}(202,·)$, $\chi_{957}(463,·)$, $\chi_{957}(784,·)$, $\chi_{957}(17,·)$, $\chi_{957}(724,·)$, $\chi_{957}(563,·)$, $\chi_{957}(215,·)$, $\chi_{957}(800,·)$, $\chi_{957}(289,·)$, $\chi_{957}(610,·)$, $\chi_{957}(41,·)$, $\chi_{957}(365,·)$, $\chi_{957}(626,·)$, $\chi_{957}(115,·)$, $\chi_{957}(887,·)$, $\chi_{957}(824,·)$, $\chi_{957}(697,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{265201436257} a^{16} + \frac{39066927753}{265201436257} a^{15} + \frac{850078764}{265201436257} a^{14} + \frac{49376657160}{265201436257} a^{13} + \frac{90717497558}{265201436257} a^{12} - \frac{106098776661}{265201436257} a^{11} - \frac{114064942528}{265201436257} a^{10} + \frac{100315213114}{265201436257} a^{9} - \frac{72170604007}{265201436257} a^{8} + \frac{56356376665}{265201436257} a^{7} + \frac{65797853784}{265201436257} a^{6} - \frac{28367769400}{265201436257} a^{5} - \frac{12144899730}{265201436257} a^{4} - \frac{104758447658}{265201436257} a^{3} - \frac{107910087688}{265201436257} a^{2} + \frac{88731103542}{265201436257} a + \frac{120424597062}{265201436257}$, $\frac{1}{265201436257} a^{17} + \frac{55538983573}{265201436257} a^{15} - \frac{125061847080}{265201436257} a^{14} - \frac{86234878088}{265201436257} a^{13} + \frac{40384330079}{265201436257} a^{12} - \frac{41208702071}{265201436257} a^{11} - \frac{43312652746}{265201436257} a^{10} + \frac{131279556058}{265201436257} a^{9} - \frac{9811456003}{265201436257} a^{8} + \frac{128629386184}{265201436257} a^{7} - \frac{33487592936}{265201436257} a^{6} - \frac{36726369808}{265201436257} a^{5} - \frac{89658418118}{265201436257} a^{4} - \frac{21215246746}{265201436257} a^{3} - \frac{125579273250}{265201436257} a^{2} - \frac{50302230538}{265201436257} a - \frac{26175060043}{265201436257}$, $\frac{1}{28906956552013} a^{18} - \frac{45}{28906956552013} a^{17} - \frac{17}{28906956552013} a^{16} + \frac{11486602739416}{28906956552013} a^{15} + \frac{6166535063424}{28906956552013} a^{14} + \frac{3801044877780}{28906956552013} a^{13} - \frac{9096029387523}{28906956552013} a^{12} + \frac{5843761722232}{28906956552013} a^{11} + \frac{13586018259314}{28906956552013} a^{10} + \frac{7854467924261}{28906956552013} a^{9} - \frac{7499881430456}{28906956552013} a^{8} - \frac{816033800883}{28906956552013} a^{7} - \frac{9817721212970}{28906956552013} a^{6} + \frac{7077956739271}{28906956552013} a^{5} - \frac{12318386222878}{28906956552013} a^{4} - \frac{7547256181949}{28906956552013} a^{3} + \frac{13906481904903}{28906956552013} a^{2} + \frac{11436259418297}{28906956552013} a + \frac{24018699220}{265201436257}$, $\frac{1}{116411440095741728118226668770330303461746878524241109366103988391886725784026069377} a^{19} + \frac{1477193790576317415018475956859478989723409647984156074327723932562812}{116411440095741728118226668770330303461746878524241109366103988391886725784026069377} a^{18} - \frac{31089323590651737138232979159431727691739885574483252253191444019983268}{116411440095741728118226668770330303461746878524241109366103988391886725784026069377} a^{17} - \frac{37058235016937940853600334984971819671385572887432156178499506492894273}{116411440095741728118226668770330303461746878524241109366103988391886725784026069377} a^{16} + \frac{53748926413150852261758306829444631951440065879749446348188329948916149107542329604}{116411440095741728118226668770330303461746878524241109366103988391886725784026069377} a^{15} - \frac{55402117397135108008730990694555784586895544322203458791581791143830409489680850828}{116411440095741728118226668770330303461746878524241109366103988391886725784026069377} a^{14} - \frac{6654925640139518972292406804851816524795611637556046256573020507069758229739035377}{116411440095741728118226668770330303461746878524241109366103988391886725784026069377} a^{13} + \frac{2648797537212655625202221504199783656086718183674227111113827909663586423420847884}{116411440095741728118226668770330303461746878524241109366103988391886725784026069377} a^{12} + \frac{11684242261319946656894047219769729195510816612602789211790183576316442410235894736}{116411440095741728118226668770330303461746878524241109366103988391886725784026069377} a^{11} - \frac{38490988158193194488235451120976349054673052491826219772681988703530226225803374929}{116411440095741728118226668770330303461746878524241109366103988391886725784026069377} a^{10} + \frac{20321230113364748163839376961623403885951167212562772853602264421939291288131781106}{116411440095741728118226668770330303461746878524241109366103988391886725784026069377} a^{9} + \frac{38292318035639036904889535880028814075541772837022254203578778675838530822437494053}{116411440095741728118226668770330303461746878524241109366103988391886725784026069377} a^{8} + \frac{52916835844440157142817543343994218397564597312270995716394918085371568955027284431}{116411440095741728118226668770330303461746878524241109366103988391886725784026069377} a^{7} + \frac{30874901031301722134706798636844813821762817892083261961432209613286952081416009340}{116411440095741728118226668770330303461746878524241109366103988391886725784026069377} a^{6} + \frac{39332459098827615216323455730870938584387919325777565859230661389673568336978997258}{116411440095741728118226668770330303461746878524241109366103988391886725784026069377} a^{5} + \frac{24630918480908094302832096315073940131353936849734090996177802454110384847910409781}{116411440095741728118226668770330303461746878524241109366103988391886725784026069377} a^{4} + \frac{48149785291900728020993833153136536422081141388233611351740377768050780919603129904}{116411440095741728118226668770330303461746878524241109366103988391886725784026069377} a^{3} + \frac{55065965549447836100052488180526628281283149269666522580188637186643617237489671841}{116411440095741728118226668770330303461746878524241109366103988391886725784026069377} a^{2} + \frac{9389606071082923146057302060311766859324218028944124533702943287999901932148308597}{116411440095741728118226668770330303461746878524241109366103988391886725784026069377} a + \frac{154579505079304618075234444792803946948439994298923532671380850037877472850387310}{1067994863263685579066299713489268839098595215818725774000954021943914915449780453}$
Class group and class number
$C_{2}\times C_{6124066}$, which has order $12248132$ (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: | \( 15197121.675070647 \) (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
| \(\Q(\sqrt{29}) \), 4.0.26559621.1, \(\Q(\zeta_{11})^+\), 10.10.4396746947664269.1 |
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 | $20$ | R | ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ | $20$ | $20$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ | R | $20$ | $20$ | $20$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ | $20$ | ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||
| 11 | Data not computed | ||||||
| 29 | Data not computed | ||||||