Normalized defining polynomial
\( x^{20} + 730 x^{18} + 238720 x^{16} - 1512 x^{15} + 46048400 x^{14} + 551880 x^{13} + 5802206480 x^{12} + 260683920 x^{11} + 498984475640 x^{10} + 35717673600 x^{9} + 29660149388520 x^{8} + 2043573174720 x^{7} + 1203701554412640 x^{6} + 20051039556576 x^{5} + 31870420079405760 x^{4} - 2743310490056640 x^{3} + 498948144686164800 x^{2} - 86249720534725440 x + 3491873768877677712 \)
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: | \(84918716484613018665985720056760042383483090599116800000000000000000000=2^{33}\cdot 3^{16}\cdot 5^{20}\cdot 7^{18}\cdot 31^{10}\cdot 71^{5}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $3519.25$ | 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: | $2, 3, 5, 7, 31, 71$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{6} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{6} a^{7} + \frac{1}{6} a^{5} - \frac{1}{3} a^{3}$, $\frac{1}{6} a^{8} + \frac{1}{3} a^{2}$, $\frac{1}{12} a^{9} - \frac{1}{2} a^{4} - \frac{1}{3} a^{3}$, $\frac{1}{84} a^{10} - \frac{1}{14} a^{8} + \frac{1}{14} a^{6} + \frac{8}{21} a^{4} - \frac{3}{7} a^{2} + \frac{1}{7}$, $\frac{1}{84} a^{11} + \frac{1}{84} a^{9} + \frac{1}{14} a^{7} - \frac{5}{42} a^{5} - \frac{1}{2} a^{4} + \frac{5}{21} a^{3} + \frac{1}{7} a$, $\frac{1}{504} a^{12} + \frac{1}{252} a^{10} - \frac{1}{126} a^{6} - \frac{4}{63} a^{4} + \frac{5}{42} a^{2} - \frac{1}{7}$, $\frac{1}{504} a^{13} + \frac{1}{252} a^{11} - \frac{1}{126} a^{7} - \frac{4}{63} a^{5} + \frac{5}{42} a^{3} - \frac{1}{7} a$, $\frac{1}{504} a^{14} + \frac{1}{252} a^{10} - \frac{5}{63} a^{8} + \frac{1}{42} a^{6} - \frac{47}{126} a^{4} + \frac{4}{21} a^{2} + \frac{3}{7}$, $\frac{1}{3024} a^{15} - \frac{1}{1512} a^{13} + \frac{1}{189} a^{11} - \frac{1}{168} a^{10} - \frac{4}{189} a^{9} + \frac{1}{28} a^{8} + \frac{8}{189} a^{7} + \frac{1}{21} a^{6} - \frac{139}{756} a^{5} + \frac{1}{7} a^{4} - \frac{1}{3} a^{3} + \frac{1}{21} a^{2} - \frac{1}{7} a - \frac{1}{14}$, $\frac{1}{18144} a^{16} + \frac{5}{9072} a^{14} + \frac{1}{4536} a^{12} - \frac{1}{336} a^{11} - \frac{25}{4536} a^{10} + \frac{1}{56} a^{9} - \frac{17}{2268} a^{8} + \frac{1}{42} a^{7} - \frac{145}{4536} a^{6} + \frac{1}{14} a^{5} + \frac{5}{18} a^{4} - \frac{10}{21} a^{3} + \frac{17}{42} a^{2} + \frac{25}{84} a - \frac{1}{2}$, $\frac{1}{18144} a^{17} - \frac{1}{9072} a^{15} - \frac{1}{2268} a^{13} - \frac{1}{1008} a^{12} + \frac{17}{4536} a^{11} - \frac{1}{504} a^{10} - \frac{2}{81} a^{9} + \frac{155}{4536} a^{7} - \frac{5}{63} a^{6} - \frac{11}{378} a^{5} + \frac{25}{126} a^{4} - \frac{5}{21} a^{3} - \frac{5}{84} a^{2} + \frac{3}{14} a - \frac{3}{7}$, $\frac{1}{160655450365417862201168663281450848096} a^{18} - \frac{8132938566204954293607378039767}{546447110086455313609417222045751184} a^{17} - \frac{1830805142788800409229227739565241}{80327725182708931100584331640725424048} a^{16} - \frac{31565916234155744245257892895443}{273223555043227656804708611022875592} a^{15} - \frac{29920801826412765475946862454301981}{40163862591354465550292165820362712024} a^{14} + \frac{413355412458194833035730211298097}{546447110086455313609417222045751184} a^{13} + \frac{244465362726787684402202928674749}{409835332564841485207062916534313388} a^{12} + \frac{507560828048311779552782630869771}{273223555043227656804708611022875592} a^{11} - \frac{126329910004326619218491058297613}{204917666282420742603531458267156694} a^{10} - \frac{14931375809580581344302332304654223}{956282442651296798816480138580064572} a^{9} + \frac{291763891354287064745690780167024373}{5737694655907780792898880831480387432} a^{8} + \frac{49528835234135808229594638783381035}{956282442651296798816480138580064572} a^{7} - \frac{36614669977587729803671832889638263}{478141221325648399408240069290032286} a^{6} + \frac{6201504375354534364093537847809217}{26563401184758244411568892738335127} a^{5} + \frac{9692895869330700740393777381391094}{185943808293307710880982249168345889} a^{4} - \frac{49727053500883993170528750017069111}{106253604739032977646275570953340508} a^{3} + \frac{7898709790147055368814996115566614}{20660423143700856764553583240927321} a^{2} - \frac{561148429535912084008694484403880}{8854467061586081470522964246111709} a + \frac{2354707097647895965181817061649058}{20660423143700856764553583240927321}$, $\frac{1}{15821404882087659004356718811426262998432185308092213806038042819375391375062624} a^{19} - \frac{955437717751550168836337483313949679345}{5273801627362553001452239603808754332810728436030737935346014273125130458354208} a^{18} - \frac{413549564433482507085887115565102735212800950387941076740512620963783580771}{15821404882087659004356718811426262998432185308092213806038042819375391375062624} a^{17} - \frac{113182794133472489012107880428123328018756742400547382829293943092446205997}{5273801627362553001452239603808754332810728436030737935346014273125130458354208} a^{16} + \frac{245981564779558590382144616422128193726465998470097399012193706560180995197}{1977675610260957375544589851428282874804023163511526725754755352421923921882828} a^{15} - \frac{975318677279749683462168067632468132221250462762415132540456602902845424227}{1318450406840638250363059900952188583202682109007684483836503568281282614588552} a^{14} + \frac{295332611978043465552499163655832290525130074979256600758663133759286677299}{1130100348720547071739765629387590214173727522006586700431288772812527955361616} a^{13} + \frac{75118486940073952477062118762811809100092279930321439413897963735749152825}{125566705413394119082196181043065579352636391334065188936809863645836439484624} a^{12} - \frac{4130248516412760588743195320288318774505747481860609688292449414373174803023}{1130100348720547071739765629387590214173727522006586700431288772812527955361616} a^{11} + \frac{617591665023611879580442775992155503497924685369176426900709592642558005563}{188350058120091178623294271564598369028954587001097783405214795468754659226936} a^{10} + \frac{6788215676051467667626011623509785199301901387678831743254092446420742148579}{282525087180136767934941407346897553543431880501646675107822193203131988840404} a^{9} - \frac{1687939375521400874950090975828355542216465999834947751679420359951592465177}{188350058120091178623294271564598369028954587001097783405214795468754659226936} a^{8} + \frac{4790893849080438816236176012735716376007053092786161833628954729306337985369}{188350058120091178623294271564598369028954587001097783405214795468754659226936} a^{7} + \frac{7741963389825133605243890990694560729787728742545785660271122567101486568073}{188350058120091178623294271564598369028954587001097783405214795468754659226936} a^{6} - \frac{20819007490306310423681113017213096411946039819713997870669550310518671510621}{219741734473439708393843316825364763867113684834614080639417261380213769098092} a^{5} + \frac{10793535834994255463306076951208297184859357773421421275167621471609159034239}{73247244824479902797947772275121587955704561611538026879805753793404589699364} a^{4} + \frac{980800119132818594669182235164964543578605524600335922205728876221707688707}{8138582758275544755327530252791287550633840179059780764422861532600509966596} a^{3} + \frac{11070214290437645256828032572643603631000866143703048388271704324139405536175}{24415748274826634265982590758373862651901520537179342293268584597801529899788} a^{2} - \frac{5585342841519876021200143223902590608570128422042766140557113596219675431669}{24415748274826634265982590758373862651901520537179342293268584597801529899788} a + \frac{773393003787257107533575259046513930555902481088164669829718504374317247757}{2034645689568886188831882563197821887658460044764945191105715383150127491649}$
Class group and class number
Not computed
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: | Not computed | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | Not computed | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$D_4\times F_5$ (as 20T42):
| A solvable group of order 160 |
| The 25 conjugacy class representatives for $D_4\times F_5$ |
| Character table for $D_4\times F_5$ is not computed |
Intermediate fields
| \(\Q(\sqrt{217}) \), 4.0.106986208.3, 5.1.9724050000.14, 10.2.18949636158212068942500000000.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | R | R | $20$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ | R | ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.10.14.1 | $x^{10} - 2 x^{6} + 2 x^{5} + 2 x^{2} + 2$ | $10$ | $1$ | $14$ | $F_{5}\times C_2$ | $[2]_{5}^{4}$ |
| 2.10.19.1 | $x^{10} - 2$ | $10$ | $1$ | $19$ | $F_{5}\times C_2$ | $[3]_{5}^{4}$ | |
| $3$ | 3.5.4.1 | $x^{5} - 3$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ |
| 3.5.4.1 | $x^{5} - 3$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ | |
| 3.5.4.1 | $x^{5} - 3$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ | |
| 3.5.4.1 | $x^{5} - 3$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ | |
| 5 | Data not computed | ||||||
| $7$ | 7.10.9.2 | $x^{10} + 14$ | $10$ | $1$ | $9$ | $F_{5}\times C_2$ | $[\ ]_{10}^{4}$ |
| 7.10.9.2 | $x^{10} + 14$ | $10$ | $1$ | $9$ | $F_{5}\times C_2$ | $[\ ]_{10}^{4}$ | |
| $31$ | 31.2.1.1 | $x^{2} - 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 31.2.1.1 | $x^{2} - 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 31.2.1.1 | $x^{2} - 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 31.2.1.1 | $x^{2} - 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 31.2.1.1 | $x^{2} - 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 31.2.1.1 | $x^{2} - 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 31.2.1.1 | $x^{2} - 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 31.2.1.1 | $x^{2} - 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 31.2.1.1 | $x^{2} - 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 31.2.1.1 | $x^{2} - 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| $71$ | 71.2.1.2 | $x^{2} + 142$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 71.2.0.1 | $x^{2} - x + 11$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 71.2.0.1 | $x^{2} - x + 11$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 71.2.1.2 | $x^{2} + 142$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.2.0.1 | $x^{2} - x + 11$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 71.2.1.2 | $x^{2} + 142$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.2.1.2 | $x^{2} + 142$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.2.0.1 | $x^{2} - x + 11$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 71.2.1.2 | $x^{2} + 142$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.2.0.1 | $x^{2} - x + 11$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |