Normalized defining polynomial
\( x^{20} - x^{19} + 181 x^{18} + 1028 x^{17} + 33504 x^{16} - 688039 x^{15} + 8748821 x^{14} - 82162506 x^{13} + 815823407 x^{12} - 5666231713 x^{11} + 33297946546 x^{10} - 172227632738 x^{9} + 811531611642 x^{8} - 2411372393964 x^{7} + 6056762926765 x^{6} - 13544719689028 x^{5} + 26340214672016 x^{4} - 37724080355453 x^{3} + 49581034578142 x^{2} - 55463573994491 x + 42348870775201 \)
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: | \(89407661343122988478808492367662580240087921142578125=5^{15}\cdot 11^{16}\cdot 41^{16}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $444.19$ | 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: | $5, 11, 41$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(2255=5\cdot 11\cdot 41\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{2255}(1,·)$, $\chi_{2255}(92,·)$, $\chi_{2255}(201,·)$, $\chi_{2255}(262,·)$, $\chi_{2255}(346,·)$, $\chi_{2255}(452,·)$, $\chi_{2255}(543,·)$, $\chi_{2255}(652,·)$, $\chi_{2255}(713,·)$, $\chi_{2255}(797,·)$, $\chi_{2255}(903,·)$, $\chi_{2255}(994,·)$, $\chi_{2255}(1103,·)$, $\chi_{2255}(1164,·)$, $\chi_{2255}(1248,·)$, $\chi_{2255}(1354,·)$, $\chi_{2255}(1554,·)$, $\chi_{2255}(1699,·)$, $\chi_{2255}(1896,·)$, $\chi_{2255}(2066,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{109} a^{16} - \frac{21}{109} a^{15} - \frac{17}{109} a^{14} + \frac{24}{109} a^{13} - \frac{27}{109} a^{12} - \frac{1}{109} a^{11} + \frac{37}{109} a^{10} - \frac{31}{109} a^{9} - \frac{28}{109} a^{8} + \frac{1}{109} a^{7} - \frac{17}{109} a^{5} + \frac{16}{109} a^{4} + \frac{14}{109} a^{3} + \frac{7}{109} a^{2} - \frac{46}{109} a - \frac{20}{109}$, $\frac{1}{105722565173795516510201573967625411953846239} a^{17} + \frac{240263189465966653178156869838353854311935}{105722565173795516510201573967625411953846239} a^{16} + \frac{42110375622094510032096897710461368006256673}{105722565173795516510201573967625411953846239} a^{15} + \frac{1136998481779487539971338860442325769893016}{105722565173795516510201573967625411953846239} a^{14} - \frac{16736204371203738763187906343828477542878745}{105722565173795516510201573967625411953846239} a^{13} - \frac{3778191394063202626571366859764055564283620}{105722565173795516510201573967625411953846239} a^{12} + \frac{50264755422924907369717169958664748420621566}{105722565173795516510201573967625411953846239} a^{11} - \frac{39393212023970511572113715582518072849748683}{105722565173795516510201573967625411953846239} a^{10} - \frac{15041529560637926286602026866854263450619711}{105722565173795516510201573967625411953846239} a^{9} + \frac{24434287017058463861886355287002017814722300}{105722565173795516510201573967625411953846239} a^{8} - \frac{1663924531071289642701183459505554547838970}{105722565173795516510201573967625411953846239} a^{7} - \frac{2978755306990763998381319967884152350041365}{105722565173795516510201573967625411953846239} a^{6} - \frac{31425712183469900049809145838799152149033513}{105722565173795516510201573967625411953846239} a^{5} - \frac{43672863491593951337432626705580859673529515}{105722565173795516510201573967625411953846239} a^{4} - \frac{24293631075201483012129076328572759794946635}{105722565173795516510201573967625411953846239} a^{3} + \frac{48992124244778036845103054620150550543872264}{105722565173795516510201573967625411953846239} a^{2} + \frac{10439421001563263136665310358594622838731966}{105722565173795516510201573967625411953846239} a - \frac{2808401185199886723696670057522077310284}{41443577096744616428930448438896672659289}$, $\frac{1}{269698263758352362617524215191412425894261755689} a^{18} - \frac{1}{269698263758352362617524215191412425894261755689} a^{17} - \frac{499175748352285768922936441866906514349167759}{269698263758352362617524215191412425894261755689} a^{16} - \frac{2207899222970125385972018270646459568006970753}{269698263758352362617524215191412425894261755689} a^{15} - \frac{87643735480441313020156541265396713014843223915}{269698263758352362617524215191412425894261755689} a^{14} + \frac{27452925213918737526487550686042473439068439235}{269698263758352362617524215191412425894261755689} a^{13} - \frac{40775462009066437309488654115373452549207172858}{269698263758352362617524215191412425894261755689} a^{12} - \frac{7675278082202058639115962838686896639592578893}{269698263758352362617524215191412425894261755689} a^{11} - \frac{40919118888919860178566634101257705793764507500}{269698263758352362617524215191412425894261755689} a^{10} + \frac{30883948476043726570570325679127408441204489024}{269698263758352362617524215191412425894261755689} a^{9} + \frac{74873020458772166242753137776602057490392729624}{269698263758352362617524215191412425894261755689} a^{8} - \frac{46191107391711345002389381113483236469151597756}{269698263758352362617524215191412425894261755689} a^{7} - \frac{85833628336584278047007273105791103948043871739}{269698263758352362617524215191412425894261755689} a^{6} + \frac{82920534407328069237301455079075245191354446358}{269698263758352362617524215191412425894261755689} a^{5} + \frac{42096400348214513953298301466722071126078274910}{269698263758352362617524215191412425894261755689} a^{4} - \frac{35515089332460796629163267698196546393163605337}{269698263758352362617524215191412425894261755689} a^{3} + \frac{20507842484266358000772844625855456161812419849}{269698263758352362617524215191412425894261755689} a^{2} - \frac{34105163845154380644334791998860949005484190}{105722565173795516510201573967625411953846239} a - \frac{9404645087863847225394463172968743455580}{41443577096744616428930448438896672659289}$, $\frac{1}{688000270847556877037304272953293098456261738762639} a^{19} - \frac{1}{688000270847556877037304272953293098456261738762639} a^{18} + \frac{181}{688000270847556877037304272953293098456261738762639} a^{17} - \frac{352150295080009929467684043843274343096935246521}{688000270847556877037304272953293098456261738762639} a^{16} + \frac{64975129534743470544972358155267953157696988063948}{688000270847556877037304272953293098456261738762639} a^{15} - \frac{127061898221057480559522708815975780858154596462519}{688000270847556877037304272953293098456261738762639} a^{14} + \frac{2985725881172828936028125436818437719098913495004}{6311929090344558504929396999571496316112493016171} a^{13} - \frac{170691014307277637707290375818180361896480092763358}{688000270847556877037304272953293098456261738762639} a^{12} + \frac{236964502113792170061655910643334939945150220591963}{688000270847556877037304272953293098456261738762639} a^{11} - \frac{82108132671697455081677671189939068398990843595927}{688000270847556877037304272953293098456261738762639} a^{10} - \frac{216915693335112022943233407630059983173600954755616}{688000270847556877037304272953293098456261738762639} a^{9} + \frac{30193083197955540719629788790906682975151227859956}{688000270847556877037304272953293098456261738762639} a^{8} + \frac{182329999779222879737959780955397453807352584830181}{688000270847556877037304272953293098456261738762639} a^{7} + \frac{117822747900403886022690166379210306498494024570535}{688000270847556877037304272953293098456261738762639} a^{6} - \frac{134322528519807408845734031331504505848543397493255}{688000270847556877037304272953293098456261738762639} a^{5} + \frac{70679371318220814644225130855362200389697070064204}{688000270847556877037304272953293098456261738762639} a^{4} - \frac{6178187922583585825617836168934234257626817161312}{688000270847556877037304272953293098456261738762639} a^{3} - \frac{127074355519349580850739670970213919104426305391}{269698263758352362617524215191412425894261755689} a^{2} - \frac{46718366553841665388029393915251638786305300}{105722565173795516510201573967625411953846239} a - \frac{8314532434852763850289283125661614329888}{41443577096744616428930448438896672659289}$
Class group and class number
$C_{5}\times C_{5}\times C_{5}\times C_{5}\times C_{5}\times C_{1444355}$, which has order $4513609375$ (assuming GRH)
Unit group
| Rank: | $9$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{81454225317810855688111104810102}{2474295997783049198325910231113875466919832621} a^{19} - \frac{114469868841156796474580904671936}{2474295997783049198325910231113875466919832621} a^{18} - \frac{14695227596870610766073260358917568}{2474295997783049198325910231113875466919832621} a^{17} - \frac{119047608032487934715563862961126916}{2474295997783049198325910231113875466919832621} a^{16} - \frac{2957529325139407020558072617788326530}{2474295997783049198325910231113875466919832621} a^{15} + \frac{49325657539908587586220511874847359922}{2474295997783049198325910231113875466919832621} a^{14} - \frac{582837104904026117029906300756687476532}{2474295997783049198325910231113875466919832621} a^{13} + \frac{5079498658408569224014037692753873711168}{2474295997783049198325910231113875466919832621} a^{12} - \frac{51663443250710585826677617155612764951864}{2474295997783049198325910231113875466919832621} a^{11} + \frac{313990287228132194542209849293387182067168}{2474295997783049198325910231113875466919832621} a^{10} - \frac{1724151562982780641183427833144284900625524}{2474295997783049198325910231113875466919832621} a^{9} + \frac{8352447597313368312980154600078389713697762}{2474295997783049198325910231113875466919832621} a^{8} - \frac{37395865613841242864968977331681216818212547}{2474295997783049198325910231113875466919832621} a^{7} + \frac{63182549065941791483543532360739813286545616}{2474295997783049198325910231113875466919832621} a^{6} - \frac{142305380082530779931886560424018033704536554}{2474295997783049198325910231113875466919832621} a^{5} + \frac{277342060133667296613072226137337331441660338}{2474295997783049198325910231113875466919832621} a^{4} - \frac{155960882015914527625748076684698069243438}{969931790585279967983500678602067999576571} a^{3} - \frac{485195043510514017203629089486861473990380403}{2474295997783049198325910231113875466919832621} a^{2} - \frac{90346046576888424721855275095224501558}{380216303639858866320462829714648373021} a + \frac{68691783775258476322264472627212689240}{380216303639858866320462829714648373021} \) (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: | \( 1487816916.7694454 \) (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
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.5.41371966801.1, 10.10.5348873865572019292503125.1 |
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 | $20$ | $20$ | R | $20$ | R | $20$ | $20$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | $20$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ | $20$ | 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 |
|---|---|---|---|---|---|---|---|
| 5 | Data not computed | ||||||
| $11$ | 11.5.4.3 | $x^{5} + 33$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
| 11.5.4.3 | $x^{5} + 33$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
| 11.5.4.3 | $x^{5} + 33$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
| 11.5.4.3 | $x^{5} + 33$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
| $41$ | 41.5.4.4 | $x^{5} + 8856$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
| 41.5.4.4 | $x^{5} + 8856$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
| 41.5.4.4 | $x^{5} + 8856$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
| 41.5.4.4 | $x^{5} + 8856$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |