Normalized defining polynomial
\( x^{20} - 220 x^{18} - 2420 x^{17} + 21450 x^{16} + 583836 x^{15} + 3502950 x^{14} - 33013640 x^{13} - 752736160 x^{12} - 5543554500 x^{11} + 2279418570 x^{10} + 495397268300 x^{9} + 6227682310245 x^{8} + 48294891164740 x^{7} + 272661356669900 x^{6} + 1169108267232268 x^{5} + 3836312578497410 x^{4} + 9520797872018840 x^{3} + 17430947729479900 x^{2} + 21572531775270080 x + 15613369350198257 \)
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: | \(4663996685529944093936844800000000000000000000000000000000=2^{55}\cdot 5^{32}\cdot 11^{18}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $764.61$ | 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: | $2, 5, 11$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(4400=2^{4}\cdot 5^{2}\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{4400}(1,·)$, $\chi_{4400}(901,·)$, $\chi_{4400}(2721,·)$, $\chi_{4400}(521,·)$, $\chi_{4400}(3021,·)$, $\chi_{4400}(941,·)$, $\chi_{4400}(3281,·)$, $\chi_{4400}(3781,·)$, $\chi_{4400}(2201,·)$, $\chi_{4400}(4061,·)$, $\chi_{4400}(1861,·)$, $\chi_{4400}(3041,·)$, $\chi_{4400}(361,·)$, $\chi_{4400}(1581,·)$, $\chi_{4400}(3101,·)$, $\chi_{4400}(2561,·)$, $\chi_{4400}(821,·)$, $\chi_{4400}(841,·)$, $\chi_{4400}(1081,·)$, $\chi_{4400}(3141,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{11} a^{5}$, $\frac{1}{11} a^{6}$, $\frac{1}{11} a^{7}$, $\frac{1}{11} a^{8}$, $\frac{1}{121} a^{9} + \frac{3}{11} a^{4}$, $\frac{1}{121} a^{10}$, $\frac{1}{121} a^{11}$, $\frac{1}{1331} a^{12} - \frac{4}{121} a^{8} - \frac{5}{121} a^{7} + \frac{4}{11} a^{4} - \frac{1}{11} a^{3} - \frac{2}{11} a^{2}$, $\frac{1}{1331} a^{13} - \frac{5}{121} a^{8} - \frac{2}{11} a^{3}$, $\frac{1}{1331} a^{14} + \frac{2}{11} a^{4}$, $\frac{1}{1331} a^{15}$, $\frac{1}{629563} a^{16} + \frac{15}{57233} a^{15} - \frac{9}{57233} a^{14} + \frac{17}{57233} a^{13} - \frac{16}{57233} a^{12} - \frac{90}{57233} a^{11} - \frac{6}{5203} a^{10} - \frac{6}{5203} a^{9} - \frac{13}{473} a^{8} + \frac{36}{5203} a^{7} - \frac{50}{5203} a^{6} - \frac{19}{473} a^{5} + \frac{136}{473} a^{4} + \frac{67}{473} a^{3} + \frac{109}{473} a^{2} + \frac{148}{473} a + \frac{15}{43}$, $\frac{1}{629563} a^{17} + \frac{10}{57233} a^{15} - \frac{3}{57233} a^{14} + \frac{17}{57233} a^{13} + \frac{13}{57233} a^{12} + \frac{1}{473} a^{11} - \frac{5}{5203} a^{10} - \frac{13}{5203} a^{9} + \frac{196}{5203} a^{8} + \frac{73}{5203} a^{7} + \frac{3}{473} a^{5} + \frac{202}{473} a^{4} + \frac{105}{473} a^{3} + \frac{8}{473} a^{2} - \frac{12}{43} a + \frac{19}{43}$, $\frac{1}{6963222605638808419096678852609711627} a^{18} + \frac{2455319228713175646182772255792}{6963222605638808419096678852609711627} a^{17} + \frac{2636928644611212155067148516546}{6963222605638808419096678852609711627} a^{16} + \frac{162569104485718611636658677529316}{633020236876255310826970804782701057} a^{15} + \frac{17975812309487509541708987028923}{633020236876255310826970804782701057} a^{14} + \frac{3863563855870442741659623886864}{14721400857587332809929553599597699} a^{13} - \frac{2574016107123297703547013869840}{57547294261477755529724618616609187} a^{12} - \frac{1611896748335200210266961298895342}{633020236876255310826970804782701057} a^{11} - \frac{15383176651095518297408254478423}{57547294261477755529724618616609187} a^{10} + \frac{18496899902380200802557295945795}{5231572205588886866338601692419017} a^{9} + \frac{63477042090586514652743834054810}{57547294261477755529724618616609187} a^{8} + \frac{1062424126141965887010222299860238}{57547294261477755529724618616609187} a^{7} - \frac{388131418142801040584371625861091}{57547294261477755529724618616609187} a^{6} + \frac{214153127458269525401270737090689}{5231572205588886866338601692419017} a^{5} - \frac{46376136288317564446915593443286}{121664469897415973635781434707419} a^{4} - \frac{96002314580584013505568222486749}{5231572205588886866338601692419017} a^{3} - \frac{866026103516592949270999228479479}{5231572205588886866338601692419017} a^{2} - \frac{2344949403445780947099274422667827}{5231572205588886866338601692419017} a - \frac{1128569740170898480816821866727}{15341853975333979080171852470437}$, $\frac{1}{171359291134025051558527639532138215881360424094589002908985279235071083} a^{19} - \frac{6715000834457782279412881325392941}{171359291134025051558527639532138215881360424094589002908985279235071083} a^{18} + \frac{75831530981203829661078920153705147547078678463609178743938541389}{171359291134025051558527639532138215881360424094589002908985279235071083} a^{17} + \frac{103987604654323973233653231516209502214951599403412134849010697246}{171359291134025051558527639532138215881360424094589002908985279235071083} a^{16} + \frac{3708297903233422905084744680769304974242420014007409521608012601674}{15578117375820459232593421775648928716487311281326272991725934475915553} a^{15} + \frac{2477667298655673227329129082411071396554739276706685766483544233465}{15578117375820459232593421775648928716487311281326272991725934475915553} a^{14} - \frac{5560103171729291648263781214658663196281901723829533183815263342823}{15578117375820459232593421775648928716487311281326272991725934475915553} a^{13} - \frac{1570889351629952381124237540726699284914002468805354049599486508750}{15578117375820459232593421775648928716487311281326272991725934475915553} a^{12} - \frac{7993953623453615098329620237141432901135280089969995582602600455349}{15578117375820459232593421775648928716487311281326272991725934475915553} a^{11} + \frac{242917975292590107666618623535193780921661102017255679312833845379}{1416192488710950839326674706877175337862482843756933908338721315992323} a^{10} - \frac{3688534553350726041274017465629750137379283637143189407153461357047}{1416192488710950839326674706877175337862482843756933908338721315992323} a^{9} - \frac{6829374909594858025808120947083033938139702101398669968119948283145}{1416192488710950839326674706877175337862482843756933908338721315992323} a^{8} + \frac{408137473328790568908747515381357500943252316697688507235932404513}{34541280212462215593333529436028666777133727896510583130212715024203} a^{7} + \frac{28827549668614823292505597246351122225423696719105993200310134645233}{1416192488710950839326674706877175337862482843756933908338721315992323} a^{6} - \frac{1015420510826205656358353266234614561373562583913457858477297439483}{128744771700995530847879518807015939805680258523357628030792846908393} a^{5} - \frac{44148154139559679859314885438844763639255345108350562163695557483015}{128744771700995530847879518807015939805680258523357628030792846908393} a^{4} - \frac{355277370378494973130353806029530148312053774191149679064596976764}{3140116382951110508484866312366242434284884354228234830019337729473} a^{3} + \frac{3219755118529749843976097777212489344535123547226498204746209386553}{11704070154635957349807228982455994527789114411214329820981167900763} a^{2} - \frac{26346782052866362868474352777494235950847942589549730257882099597093}{128744771700995530847879518807015939805680258523357628030792846908393} a - \frac{6364749318364959518498077958342695888003179262424337709576758108}{377550650149547011284104160724386920251261755200462252289715093573}$
Class group and class number
$C_{5}\times C_{2621610050}$, which has order $13108050250$ (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: | \( 62756100051.98372 \) (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
| \(\Q(\sqrt{2}) \), 4.0.247808.2, 5.5.5719140625.1, 10.10.1071794405000000000000000.2 |
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 | $20$ | R | ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ | R | $20$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | $20$ | ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ | $20$ | ${\href{/LocalNumberField/31.1.0.1}{1} }^{20}$ | $20$ | ${\href{/LocalNumberField/41.1.0.1}{1} }^{20}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ | $20$ | $20$ |
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 | ||||||
| 11 | Data not computed | ||||||