Normalized defining polynomial
\( x^{20} - 10 x^{19} + 517 x^{18} - 4368 x^{17} + 110755 x^{16} - 788324 x^{15} + 12810686 x^{14} - 76219200 x^{13} + 875603173 x^{12} - 4297376270 x^{11} + 36355252339 x^{10} - 144109731096 x^{9} + 910187132217 x^{8} - 2820409970036 x^{7} + 13385825582430 x^{6} - 30847128749560 x^{5} + 113890016685799 x^{4} - 179392482504090 x^{3} + 522171308016133 x^{2} - 437186075851096 x + 1102147109671861 \)
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: | \(77020569049466479187232020551557743636491479040000000000=2^{20}\cdot 5^{10}\cdot 241^{18}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $622.77$ | 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: | $2, 5, 241$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(4820=2^{2}\cdot 5\cdot 241\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{4820}(1,·)$, $\chi_{4820}(3651,·)$, $\chi_{4820}(3461,·)$, $\chi_{4820}(391,·)$, $\chi_{4820}(1929,·)$, $\chi_{4820}(2891,·)$, $\chi_{4820}(4429,·)$, $\chi_{4820}(2319,·)$, $\chi_{4820}(1169,·)$, $\chi_{4820}(4819,·)$, $\chi_{4820}(2071,·)$, $\chi_{4820}(1359,·)$, $\chi_{4820}(4061,·)$, $\chi_{4820}(2501,·)$, $\chi_{4820}(4251,·)$, $\chi_{4820}(821,·)$, $\chi_{4820}(759,·)$, $\chi_{4820}(569,·)$, $\chi_{4820}(3999,·)$, $\chi_{4820}(2749,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{64} a^{8} - \frac{1}{16} a^{7} - \frac{1}{16} a^{6} - \frac{3}{32} a^{5} - \frac{11}{64} a^{4} + \frac{3}{32} a^{3} - \frac{1}{2} a^{2} + \frac{9}{32} a + \frac{1}{64}$, $\frac{1}{64} a^{9} - \frac{1}{16} a^{7} - \frac{3}{32} a^{6} - \frac{3}{64} a^{5} + \frac{5}{32} a^{4} - \frac{1}{8} a^{3} - \frac{15}{32} a^{2} - \frac{7}{64} a - \frac{7}{16}$, $\frac{1}{128} a^{10} - \frac{1}{128} a^{9} - \frac{1}{128} a^{8} - \frac{7}{64} a^{7} + \frac{7}{128} a^{6} - \frac{21}{128} a^{5} + \frac{13}{128} a^{4} + \frac{11}{32} a^{3} + \frac{23}{128} a^{2} + \frac{49}{128} a + \frac{47}{128}$, $\frac{1}{1408} a^{11} - \frac{1}{128} a^{8} + \frac{3}{128} a^{7} + \frac{1}{64} a^{6} + \frac{7}{64} a^{5} + \frac{3}{128} a^{4} + \frac{1}{128} a^{3} + \frac{9}{32} a^{2} + \frac{269}{704} a + \frac{17}{128}$, $\frac{1}{45056} a^{12} - \frac{3}{22528} a^{11} - \frac{15}{4096} a^{10} + \frac{1}{256} a^{9} + \frac{15}{2048} a^{8} - \frac{175}{2048} a^{7} + \frac{277}{4096} a^{6} + \frac{29}{128} a^{5} + \frac{95}{2048} a^{4} - \frac{257}{2048} a^{3} + \frac{5675}{45056} a^{2} - \frac{323}{11264} a + \frac{135}{4096}$, $\frac{1}{45056} a^{13} - \frac{9}{45056} a^{11} - \frac{5}{2048} a^{10} - \frac{1}{2048} a^{9} + \frac{11}{2048} a^{8} + \frac{417}{4096} a^{7} - \frac{17}{2048} a^{6} + \frac{255}{2048} a^{5} + \frac{281}{2048} a^{4} - \frac{5017}{45056} a^{3} - \frac{527}{2048} a^{2} - \frac{123}{45056} a - \frac{459}{2048}$, $\frac{1}{1712128} a^{14} - \frac{7}{1712128} a^{13} - \frac{9}{856064} a^{12} + \frac{199}{1712128} a^{11} + \frac{587}{155648} a^{10} - \frac{9}{2048} a^{9} - \frac{199}{155648} a^{8} + \frac{9989}{155648} a^{7} + \frac{373}{8192} a^{6} + \frac{31}{4864} a^{5} - \frac{219869}{1712128} a^{4} + \frac{360939}{1712128} a^{3} - \frac{86367}{214016} a^{2} + \frac{808983}{1712128} a + \frac{2587}{155648}$, $\frac{1}{1712128} a^{15} + \frac{9}{1712128} a^{13} - \frac{3}{1712128} a^{12} + \frac{163}{856064} a^{11} + \frac{157}{155648} a^{10} - \frac{275}{155648} a^{9} - \frac{131}{38912} a^{8} - \frac{445}{77824} a^{7} + \frac{11157}{155648} a^{6} + \frac{1979}{1712128} a^{5} + \frac{773}{9728} a^{4} - \frac{175855}{1712128} a^{3} + \frac{318035}{1712128} a^{2} + \frac{278731}{856064} a + \frac{60897}{155648}$, $\frac{1}{109576192} a^{16} - \frac{1}{13697024} a^{15} - \frac{9}{54788096} a^{14} + \frac{7}{2883584} a^{13} - \frac{5}{720896} a^{12} + \frac{369}{54788096} a^{11} - \frac{369}{622592} a^{10} + \frac{231}{77824} a^{9} + \frac{56555}{9961472} a^{8} - \frac{202017}{4980736} a^{7} - \frac{895231}{13697024} a^{6} - \frac{2047011}{13697024} a^{5} + \frac{863043}{13697024} a^{4} - \frac{14439415}{54788096} a^{3} + \frac{25750577}{54788096} a^{2} + \frac{26180555}{54788096} a - \frac{1697557}{9961472}$, $\frac{1}{109576192} a^{17} - \frac{9}{54788096} a^{15} - \frac{3}{54788096} a^{14} + \frac{51}{13697024} a^{13} - \frac{21}{2883584} a^{12} + \frac{61}{311296} a^{11} - \frac{267}{155648} a^{10} + \frac{17707}{9961472} a^{9} + \frac{29899}{4980736} a^{8} + \frac{210907}{13697024} a^{7} + \frac{99031}{1245184} a^{6} + \frac{2098211}{13697024} a^{5} - \frac{6759255}{54788096} a^{4} + \frac{14707225}{54788096} a^{3} + \frac{14379379}{54788096} a^{2} - \frac{119559}{524288} a + \frac{488627}{1245184}$, $\frac{1}{1434587134338871189243272197233967104} a^{18} - \frac{9}{1434587134338871189243272197233967104} a^{17} + \frac{4508149342270268846828062671}{1434587134338871189243272197233967104} a^{16} - \frac{9016298684540537693656125291}{358646783584717797310818049308491776} a^{15} + \frac{2230870402006241762932590255}{358646783584717797310818049308491776} a^{14} + \frac{142169134165415717298454060629}{358646783584717797310818049308491776} a^{13} + \frac{6486451381814864606926579244059}{717293567169435594621636098616983552} a^{12} - \frac{3948689905407562292858362585533}{65208506106312326783785099874271232} a^{11} + \frac{21984972639406158693250475586585}{6864053274348665977240536828870656} a^{10} + \frac{2109900192446436759412503487961}{11856092019329513960688199977140224} a^{9} + \frac{3222170240568940051598720551518479}{1434587134338871189243272197233967104} a^{8} + \frac{14711913003109997358341977044495549}{717293567169435594621636098616983552} a^{7} - \frac{16344224017106285949923722938028257}{179323391792358898655409024654245888} a^{6} + \frac{64973684032896749481151094410736721}{717293567169435594621636098616983552} a^{5} + \frac{2354808932307936858687534198489493}{179323391792358898655409024654245888} a^{4} + \frac{281803442123614660170482234715313467}{717293567169435594621636098616983552} a^{3} + \frac{259105516153419886935990488346391189}{1434587134338871189243272197233967104} a^{2} + \frac{20017138701873996619747855948468551}{130417012212624653567570199748542464} a - \frac{217755954519579032900216966980413}{624004843122605997930957893533696}$, $\frac{1}{4218939170616568418910457929427099180074008576} a^{19} + \frac{367609175}{1054734792654142104727614482356774795018502144} a^{18} + \frac{7662559211211706873438454021575750053}{2109469585308284209455228964713549590037004288} a^{17} - \frac{8127555545128287822544554609562852227}{4218939170616568418910457929427099180074008576} a^{16} - \frac{1971296650764185622437599322614004757}{131841849081767763090951810294596849377312768} a^{15} + \frac{269166951797947111431873072015182771325}{1054734792654142104727614482356774795018502144} a^{14} + \frac{5097823839875105159868393558650165127491}{2109469585308284209455228964713549590037004288} a^{13} - \frac{935661502540339060741088405003293871407}{263683698163535526181903620589193698754625536} a^{12} + \frac{1025150189409727527742907494727880206975}{20186311821131906310576353729316264019492864} a^{11} - \frac{445034097405946802224074729043588261185619}{191769962300753109950475360428504508185182208} a^{10} - \frac{1022043898654067190361379035672886625819901}{1054734792654142104727614482356774795018502144} a^{9} + \frac{22609052664826551980558941766733909242042307}{4218939170616568418910457929427099180074008576} a^{8} + \frac{153727178630983310433495167237532303460443779}{2109469585308284209455228964713549590037004288} a^{7} + \frac{88905446104452376161239525107205034510764693}{2109469585308284209455228964713549590037004288} a^{6} + \frac{446218710807067795096932175825360072154138049}{2109469585308284209455228964713549590037004288} a^{5} + \frac{330010987823209758202356327579257928951709903}{2109469585308284209455228964713549590037004288} a^{4} - \frac{532982098495496362891974779351140083744369393}{4218939170616568418910457929427099180074008576} a^{3} + \frac{115966810650526447788058336209888483629646241}{2109469585308284209455228964713549590037004288} a^{2} - \frac{20359365889794718180395273599618434188678203}{191769962300753109950475360428504508185182208} a + \frac{4001584292480989072560550978202072352680891}{34867265872864201809177338259728092397305856}$
Class group and class number
$C_{2}\times C_{4}\times C_{4}\times C_{4}\times C_{4}\times C_{6624600}$, which has order $3391795200$ (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: | \( 431310293438.41327 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_{10}$ (as 20T3):
| An abelian group of order 20 |
| The 20 conjugacy class representatives for $C_2\times C_{10}$ |
| Character table for $C_2\times C_{10}$ |
Intermediate fields
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 | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/11.1.0.1}{1} }^{20}$ | ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/19.1.0.1}{1} }^{20}$ | ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.4.4.1 | $x^{4} + 8 x^{2} + 4$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ |
| 2.4.4.1 | $x^{4} + 8 x^{2} + 4$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.4.4.1 | $x^{4} + 8 x^{2} + 4$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.4.4.1 | $x^{4} + 8 x^{2} + 4$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.4.4.1 | $x^{4} + 8 x^{2} + 4$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| $5$ | 5.10.5.1 | $x^{10} - 50 x^{6} + 625 x^{2} - 12500$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
| 5.10.5.1 | $x^{10} - 50 x^{6} + 625 x^{2} - 12500$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
| 241 | Data not computed | ||||||