Normalized defining polynomial
\( x^{20} + 620 x^{18} + 163370 x^{16} + 23832800 x^{14} + 2101010275 x^{12} + 114636279655 x^{10} + 3808990097300 x^{8} + 72806825933425 x^{6} + 711319770305875 x^{4} + 2763074514531375 x^{2} + 1976877420853005 \)
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: | \(2133342346188491938901276370239257812500000000000000000000=2^{20}\cdot 5^{35}\cdot 31^{18}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $735.28$ | 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: | $2, 5, 31$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(3100=2^{2}\cdot 5^{2}\cdot 31\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{3100}(1,·)$, $\chi_{3100}(2627,·)$, $\chi_{3100}(841,·)$, $\chi_{3100}(2447,·)$, $\chi_{3100}(529,·)$, $\chi_{3100}(23,·)$, $\chi_{3100}(2867,·)$, $\chi_{3100}(1887,·)$, $\chi_{3100}(481,·)$, $\chi_{3100}(1763,·)$, $\chi_{3100}(1521,·)$, $\chi_{3100}(1961,·)$, $\chi_{3100}(1703,·)$, $\chi_{3100}(1709,·)$, $\chi_{3100}(743,·)$, $\chi_{3100}(1969,·)$, $\chi_{3100}(883,·)$, $\chi_{3100}(1589,·)$, $\chi_{3100}(249,·)$, $\chi_{3100}(2107,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{33} a^{5} - \frac{10}{33} a^{3} - \frac{13}{33} a$, $\frac{1}{33} a^{6} - \frac{10}{33} a^{4} - \frac{13}{33} a^{2}$, $\frac{1}{33} a^{7} - \frac{14}{33} a^{3} + \frac{2}{33} a$, $\frac{1}{33} a^{8} - \frac{14}{33} a^{4} + \frac{2}{33} a^{2}$, $\frac{1}{33} a^{9} - \frac{2}{11} a^{3} + \frac{16}{33} a$, $\frac{1}{35818299} a^{10} + \frac{10}{1155429} a^{8} + \frac{1085}{1155429} a^{6} + \frac{48050}{1155429} a^{4} - \frac{410654}{1155429} a^{2} + \frac{181}{1061}$, $\frac{1}{680547681} a^{11} + \frac{315127}{21953151} a^{9} + \frac{106124}{21953151} a^{7} - \frac{162028}{21953151} a^{5} + \frac{5261452}{21953151} a^{3} - \frac{92438}{221749} a$, $\frac{1}{680547681} a^{12} - \frac{8}{680547681} a^{10} + \frac{207983}{21953151} a^{8} - \frac{86902}{21953151} a^{6} - \frac{204142}{1995741} a^{4} + \frac{73868}{7317717} a^{2} + \frac{377}{1061}$, $\frac{1}{680547681} a^{13} + \frac{68011}{21953151} a^{9} + \frac{1699}{385143} a^{7} - \frac{215551}{21953151} a^{5} - \frac{9576046}{21953151} a^{3} + \frac{33767}{665247} a$, $\frac{1}{21096978111} a^{14} - \frac{1}{75616409} a^{10} - \frac{14176}{1155429} a^{8} - \frac{10205}{7317717} a^{6} + \frac{61754722}{226849227} a^{4} + \frac{1789241}{21953151} a^{2} + \frac{291}{1061}$, $\frac{1}{696200277663} a^{15} + \frac{5}{7486024491} a^{13} - \frac{5}{7486024491} a^{11} - \frac{526970}{65859453} a^{9} + \frac{3366248}{241484661} a^{7} + \frac{79635367}{7486024491} a^{5} - \frac{293829868}{724453983} a^{3} + \frac{9154}{35013} a$, $\frac{1}{1419552366154857} a^{16} - \frac{2207}{157728040683873} a^{14} - \frac{74}{267789542757} a^{12} - \frac{2642}{219100534983} a^{10} + \frac{3860809145}{492387223779} a^{8} + \frac{228353697449}{15264003937149} a^{6} + \frac{9806627404325}{45792011811447} a^{4} + \frac{4561335154}{44762474889} a^{2} + \frac{543470}{2163379}$, $\frac{1}{44006123350800567} a^{17} + \frac{17}{1419552366154857} a^{15} - \frac{3359}{15264003937149} a^{13} + \frac{9221}{45792011811447} a^{11} - \frac{4691310347}{1477161671337} a^{9} + \frac{158131911292}{24904427476401} a^{7} + \frac{194862600941}{45792011811447} a^{5} - \frac{666347883557}{1477161671337} a^{3} - \frac{200702386}{1356438633} a$, $\frac{1}{44006123350800567} a^{18} + \frac{8428}{473184122051619} a^{14} + \frac{22478}{45792011811447} a^{12} + \frac{191516}{15264003937149} a^{10} + \frac{5160410238611}{473184122051619} a^{8} - \frac{1592283854}{4162910164677} a^{6} - \frac{3931575747152}{15264003937149} a^{4} - \frac{7906236466}{44762474889} a^{2} + \frac{419753}{2163379}$, $\frac{1}{44006123350800567} a^{19} + \frac{272}{473184122051619} a^{15} - \frac{67}{378446378607} a^{13} - \frac{50}{5088001312383} a^{11} + \frac{5847948311173}{473184122051619} a^{9} - \frac{53105834770}{45792011811447} a^{7} + \frac{1881901102}{803368628271} a^{5} + \frac{150465975494}{492387223779} a^{3} - \frac{417855142}{1356438633} a$
Class group and class number
$C_{2}\times C_{10}\times C_{979089050}$, which has order $19581781000$ (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: | \( 212853377248.3273 \) (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
| \(\Q(\sqrt{5}) \), 4.0.1922000.2, 5.5.360750390625.2, 10.10.650704221680450439453125.2 |
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 | ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ | R | $20$ | ${\href{/LocalNumberField/11.1.0.1}{1} }^{20}$ | $20$ | $20$ | ${\href{/LocalNumberField/19.1.0.1}{1} }^{20}$ | $20$ | ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ | R | $20$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ | $20$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ | $20$ | ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$ |
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 | Data not computed | ||||||
| 5 | Data not computed | ||||||
| $31$ | 31.10.9.2 | $x^{10} - 1519$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |
| 31.10.9.2 | $x^{10} - 1519$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ | |