Normalized defining polynomial
\( x^{20} - 4 x^{19} - 180 x^{18} + 1861 x^{17} + 41285 x^{16} + 56486 x^{15} - 1770231 x^{14} + 13207675 x^{13} + 539202114 x^{12} + 6259348669 x^{11} + 44724494010 x^{10} + 229585020498 x^{9} + 900899023954 x^{8} + 2784140309493 x^{7} + 6871932187949 x^{6} + 13592479679308 x^{5} + 21415551177601 x^{4} + 26404835212626 x^{3} + 24419514025237 x^{2} + 15370942553698 x + 5232878702531 \)
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: | \(93026920175496071256587866935173423923370361328125=3^{16}\cdot 5^{15}\cdot 11^{12}\cdot 41^{12}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $315.09$ | 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: | $3, 5, 11, 41$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{35027919491189350804384983897579063296526765161434795228168293397879576277710655228185733178967595661510902366779} a^{19} - \frac{5614739102665280906837093586211524628243576269600395180552468345557331082327625082495513789706630635075531911464}{35027919491189350804384983897579063296526765161434795228168293397879576277710655228185733178967595661510902366779} a^{18} - \frac{9110533530891262753358806236361901375375779054400856679473483727740638230162768803827635595441887767076956589744}{35027919491189350804384983897579063296526765161434795228168293397879576277710655228185733178967595661510902366779} a^{17} + \frac{9539006451042211585527036010147493771248243625094698036944124445571611713310458217304880593431402578154878087694}{35027919491189350804384983897579063296526765161434795228168293397879576277710655228185733178967595661510902366779} a^{16} - \frac{314258468172611100423785476107189754182873366322359926984764339402384347060274436393525314077923126527127656151}{35027919491189350804384983897579063296526765161434795228168293397879576277710655228185733178967595661510902366779} a^{15} - \frac{16167379598375278949016197615208490254203834095455137380258948324908840228626793191227210631290609702732833427839}{35027919491189350804384983897579063296526765161434795228168293397879576277710655228185733178967595661510902366779} a^{14} - \frac{12531063419726554375463671566585434398177445951469662632219364566333794050895914595708533987329235909417770812030}{35027919491189350804384983897579063296526765161434795228168293397879576277710655228185733178967595661510902366779} a^{13} - \frac{3696209150637841216484902850507514335356184625863284285558398302199369470112890251084686824486477829206700930655}{35027919491189350804384983897579063296526765161434795228168293397879576277710655228185733178967595661510902366779} a^{12} + \frac{947595153991274773267378991023855008511908312665956180778377558679663296502444622358067812525017992167253554122}{35027919491189350804384983897579063296526765161434795228168293397879576277710655228185733178967595661510902366779} a^{11} - \frac{12081186714662338975696775224300538048753962292402669862866273335388880153981332738875074194320424066459033290335}{35027919491189350804384983897579063296526765161434795228168293397879576277710655228185733178967595661510902366779} a^{10} + \frac{7487658913567713787776514701725169467161204817935994890546768463807958809225414668861060672679789570677583987714}{35027919491189350804384983897579063296526765161434795228168293397879576277710655228185733178967595661510902366779} a^{9} - \frac{6213332201137821852526406545419103372096882667329419015917506536286252410801682683098698046580265956227881411540}{35027919491189350804384983897579063296526765161434795228168293397879576277710655228185733178967595661510902366779} a^{8} + \frac{2630156433984018336987322211221352806298939783040871294553240711619854309317128788777172855543586103120621009836}{35027919491189350804384983897579063296526765161434795228168293397879576277710655228185733178967595661510902366779} a^{7} - \frac{14459969414996906390227459068661720084393827433718312479280231494202421008551612562706735617883299705341228189667}{35027919491189350804384983897579063296526765161434795228168293397879576277710655228185733178967595661510902366779} a^{6} + \frac{2196837229597205867056002638275750690383398265178158169124515928711102951805026573260344353481854979967720203319}{35027919491189350804384983897579063296526765161434795228168293397879576277710655228185733178967595661510902366779} a^{5} + \frac{14252035228686242873709512712015828049987675771564016016461940728844934221725550733430103556497293785969304602234}{35027919491189350804384983897579063296526765161434795228168293397879576277710655228185733178967595661510902366779} a^{4} - \frac{6798908824906738875841928782925339108252751042293417961678442495975004262907640830429659030088783998710178106920}{35027919491189350804384983897579063296526765161434795228168293397879576277710655228185733178967595661510902366779} a^{3} + \frac{15791421351601518036385193655764905226995969439641658310806000974625235276243811540593080475888554870735307297534}{35027919491189350804384983897579063296526765161434795228168293397879576277710655228185733178967595661510902366779} a^{2} - \frac{11208715040585763138629672391349808867761536958636029718033523573226145911334895221113254207942483567501829615362}{35027919491189350804384983897579063296526765161434795228168293397879576277710655228185733178967595661510902366779} a + \frac{8024631228991238210801899069503983022078441615813422455167295979022319363015610163385631639108879731137191282279}{35027919491189350804384983897579063296526765161434795228168293397879576277710655228185733178967595661510902366779}$
Class group and class number
$C_{5}\times C_{5}\times C_{5}\times C_{1665305}$, which has order $208163125$ (assuming GRH)
Unit group
| Rank: | $9$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{42303238874979219836303026774172710812333178567149601517903870364690737960648049410372227401038}{1524594877465621599381742185642401736745400521171464836552385192465728691812093433497392612895495834688709} a^{19} + \frac{174618103603280132803603410889346478709233347947088987601999282925560672230440072098390619106852}{1524594877465621599381742185642401736745400521171464836552385192465728691812093433497392612895495834688709} a^{18} + \frac{8315730474786797686850111162322929054397739453318711051589455314300209371243713398243724156543384}{1524594877465621599381742185642401736745400521171464836552385192465728691812093433497392612895495834688709} a^{17} - \frac{88801124219961622242243857212207837486914702686794648050165392644273845965438651893410365234270824}{1524594877465621599381742185642401736745400521171464836552385192465728691812093433497392612895495834688709} a^{16} - \frac{1802176503682311453077483912359202422507758244778452042362414117600764010309595669960358137306721705}{1524594877465621599381742185642401736745400521171464836552385192465728691812093433497392612895495834688709} a^{15} - \frac{83297793941073630775621182581237938506168911436421528086734476959213607896950921242693449080393425}{1524594877465621599381742185642401736745400521171464836552385192465728691812093433497392612895495834688709} a^{14} + \frac{88410258172767626211942702673734873922963381235718353083055338725518708021375006487473161187460846791}{1524594877465621599381742185642401736745400521171464836552385192465728691812093433497392612895495834688709} a^{13} - \frac{680866409810784250780431312178432739807763134172170163080140716394313242191921322168281878928567954756}{1524594877465621599381742185642401736745400521171464836552385192465728691812093433497392612895495834688709} a^{12} - \frac{23308425876810749081125914713060561536404840347889457366607530685348872143204896707352465640676751940907}{1524594877465621599381742185642401736745400521171464836552385192465728691812093433497392612895495834688709} a^{11} - \frac{245303777135146882103970865123857916805036368731340465165057363868763172185898928240632831014490717381917}{1524594877465621599381742185642401736745400521171464836552385192465728691812093433497392612895495834688709} a^{10} - \frac{1599712090219776296077404561197774675594193440579298853177497782137375336735867505081458517455354113843236}{1524594877465621599381742185642401736745400521171464836552385192465728691812093433497392612895495834688709} a^{9} - \frac{7483804929629837061023550828415933476745609266902346206580325921880839995386691962766432705638965720296001}{1524594877465621599381742185642401736745400521171464836552385192465728691812093433497392612895495834688709} a^{8} - \frac{26605278174158463270405647714572720460558071194702578119821767111897815385037826382098677171213951164807489}{1524594877465621599381742185642401736745400521171464836552385192465728691812093433497392612895495834688709} a^{7} - \frac{73825243876828983427040557850865935650258755269073224604621747724688397923077176405792465901074071715941278}{1524594877465621599381742185642401736745400521171464836552385192465728691812093433497392612895495834688709} a^{6} - \frac{161639642642177247367398012746812903221135829047645880413738714605144025022129169652785837400450924344614979}{1524594877465621599381742185642401736745400521171464836552385192465728691812093433497392612895495834688709} a^{5} - \frac{278744366874267998279834748030754642378368978859357341413181497965898005296562333050047559199095355606820830}{1524594877465621599381742185642401736745400521171464836552385192465728691812093433497392612895495834688709} a^{4} - \frac{373773104064746051669602902852369256540700092991215325728086427591547029620772146022753753132874983850347969}{1524594877465621599381742185642401736745400521171464836552385192465728691812093433497392612895495834688709} a^{3} - \frac{376660180397389870482922869197816646958283782488689401541968048994129721281476197004727098937746019348439956}{1524594877465621599381742185642401736745400521171464836552385192465728691812093433497392612895495834688709} a^{2} - \frac{257866734925543276859560439021477360083252795636968810846209403745132299658703306005310647329067752096438606}{1524594877465621599381742185642401736745400521171464836552385192465728691812093433497392612895495834688709} a - \frac{97054425798148038663374797106665350881214973835652217178311278293013159122136431066336427931635134616590215}{1524594877465621599381742185642401736745400521171464836552385192465728691812093433497392612895495834688709} \) (order $10$) | 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: | \( 963337255.3397985 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_5\times F_5$ (as 20T29):
| A solvable group of order 100 |
| The 25 conjugacy class representatives for $C_5\times F_5$ |
| Character table for $C_5\times F_5$ is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 25 sibling: | data not computed |
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 | R | $20$ | R | $20$ | $20$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | $20$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{5}$ | $20$ | R | $20$ | $20$ | $20$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 | ||||||
| 5 | Data not computed | ||||||
| $11$ | 11.5.4.4 | $x^{5} - 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
| 11.5.4.5 | $x^{5} - 99$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
| 11.5.0.1 | $x^{5} + x^{2} - x + 5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 11.5.4.2 | $x^{5} - 891$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
| $41$ | 41.5.0.1 | $x^{5} - x + 7$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |
| 41.5.4.2 | $x^{5} + 246$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
| 41.5.4.5 | $x^{5} - 53136$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
| 41.5.4.1 | $x^{5} - 41$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |