Normalized defining polynomial
\( x^{20} - x^{19} + 17 x^{18} + 65 x^{17} + 90 x^{16} + 770 x^{15} + 2069 x^{14} + 3588 x^{13} + 16293 x^{12} + 15488 x^{11} + 105429 x^{10} + 223263 x^{9} + 472533 x^{8} + 888791 x^{7} + 682047 x^{6} + 1592904 x^{5} + 10772482 x^{4} + 8858817 x^{3} + 13596529 x^{2} + 5680831 x + 842147 \)
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: | \(17729933579419947102912400962612224=2^{10}\cdot 11^{5}\cdot 401^{10}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $51.57$ | 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: | $2, 11, 401$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{19} a^{18} + \frac{1}{19} a^{17} - \frac{7}{19} a^{16} + \frac{6}{19} a^{15} - \frac{1}{19} a^{14} + \frac{4}{19} a^{13} - \frac{6}{19} a^{12} - \frac{5}{19} a^{11} + \frac{4}{19} a^{10} + \frac{8}{19} a^{9} + \frac{5}{19} a^{8} + \frac{5}{19} a^{7} - \frac{3}{19} a^{6} + \frac{6}{19} a^{5} - \frac{1}{19} a^{4} - \frac{5}{19} a^{3} - \frac{8}{19} a^{2} - \frac{9}{19} a - \frac{4}{19}$, $\frac{1}{5534342929400489701629631221211182379261645765744038027570378753541276117} a^{19} - \frac{40486065894390762505443610852512424388852020611210766811196129061712955}{5534342929400489701629631221211182379261645765744038027570378753541276117} a^{18} - \frac{2395024369184011653159900926859697558025738571679169003046909616931533483}{5534342929400489701629631221211182379261645765744038027570378753541276117} a^{17} + \frac{2412083887548292339664764766772362910683090221557150850792750845951257564}{5534342929400489701629631221211182379261645765744038027570378753541276117} a^{16} + \frac{191587824371303685995657186425267738717894265755823266795860871841871554}{5534342929400489701629631221211182379261645765744038027570378753541276117} a^{15} + \frac{1759961776126705538378815205267610910537631704514461426435432418107235235}{5534342929400489701629631221211182379261645765744038027570378753541276117} a^{14} - \frac{1509369477575362842520170240495476488629787790535786125160805568115826916}{5534342929400489701629631221211182379261645765744038027570378753541276117} a^{13} + \frac{734867753444279050247710405198303105107419705518598565580719262686781027}{5534342929400489701629631221211182379261645765744038027570378753541276117} a^{12} + \frac{765241741286907497382112606609625830599005630643820863383645987437624329}{5534342929400489701629631221211182379261645765744038027570378753541276117} a^{11} - \frac{1117064553624276445716213836953808963971162108191631265898819524718816417}{5534342929400489701629631221211182379261645765744038027570378753541276117} a^{10} - \frac{1359392198877689750700438529911879777051375316570850041053573054032246020}{5534342929400489701629631221211182379261645765744038027570378753541276117} a^{9} - \frac{1729996325839327300107865618811361021934000625972186706746609678690993459}{5534342929400489701629631221211182379261645765744038027570378753541276117} a^{8} + \frac{2133951762722537258037626898976498741713032111417622807834788798924811577}{5534342929400489701629631221211182379261645765744038027570378753541276117} a^{7} - \frac{2178361658037015285892237408385531728745396986951461579225588625938410536}{5534342929400489701629631221211182379261645765744038027570378753541276117} a^{6} + \frac{965181253622743455620594437669094178133972554247972276932458530971360765}{5534342929400489701629631221211182379261645765744038027570378753541276117} a^{5} - \frac{14120788652953107500219411327447443665598370689892687443712345099222380}{291281206810552089559454274800588546276928724512844106714230460712698743} a^{4} - \frac{957663532884607601249860780656414890690118343755839351922048360631935934}{5534342929400489701629631221211182379261645765744038027570378753541276117} a^{3} + \frac{464178211533232210706984909021604880128906721887178784810343836013709837}{5534342929400489701629631221211182379261645765744038027570378753541276117} a^{2} + \frac{1074487084282527775255196458348574964793871687840871528683234494143059611}{5534342929400489701629631221211182379261645765744038027570378753541276117} a - \frac{1161325623215723520361446086089119337156509864342310347269793004893253545}{5534342929400489701629631221211182379261645765744038027570378753541276117}$
Class group and class number
$C_{382}$, which has order $382$ (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: | \( 795087.603907 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 40 |
| The 13 conjugacy class representatives for $C_5:D_4$ |
| Character table for $C_5:D_4$ |
Intermediate fields
| \(\Q(\sqrt{401}) \), 4.0.7075244.1, 5.5.160801.1 x5, 10.10.10368641602001.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
| Degree 20 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 | R | ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.10.10.7 | $x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$ | $2$ | $5$ | $10$ | $C_{10}$ | $[2]^{5}$ |
| 2.10.0.1 | $x^{10} - x^{3} + 1$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
| $11$ | 11.5.0.1 | $x^{5} + x^{2} - x + 5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |
| 11.5.0.1 | $x^{5} + x^{2} - x + 5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 11.10.5.1 | $x^{10} - 242 x^{6} - 1331 x^{4} - 131769 x^{2} - 4026275$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
| 401 | Data not computed | ||||||