Normalized defining polynomial
\( x^{20} + 220 x^{18} + 19250 x^{16} + 869000 x^{14} + 22220000 x^{12} + 336600000 x^{10} + 3070375000 x^{8} + 16637500000 x^{6} + 50668750000 x^{4} + 75625000000 x^{2} + 37812500000 \)
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: | \(1956221995410498663297568389201920000000000=2^{55}\cdot 5^{10}\cdot 11^{18}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $130.19$ | 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: | $2, 5, 11$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(880=2^{4}\cdot 5\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{880}(1,·)$, $\chi_{880}(641,·)$, $\chi_{880}(521,·)$, $\chi_{880}(469,·)$, $\chi_{880}(589,·)$, $\chi_{880}(81,·)$, $\chi_{880}(149,·)$, $\chi_{880}(201,·)$, $\chi_{880}(349,·)$, $\chi_{880}(801,·)$, $\chi_{880}(549,·)$, $\chi_{880}(401,·)$, $\chi_{880}(361,·)$, $\chi_{880}(109,·)$, $\chi_{880}(29,·)$, $\chi_{880}(629,·)$, $\chi_{880}(841,·)$, $\chi_{880}(441,·)$, $\chi_{880}(189,·)$, $\chi_{880}(789,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{5} a^{2}$, $\frac{1}{5} a^{3}$, $\frac{1}{50} a^{4}$, $\frac{1}{50} a^{5}$, $\frac{1}{250} a^{6}$, $\frac{1}{250} a^{7}$, $\frac{1}{2500} a^{8}$, $\frac{1}{2500} a^{9}$, $\frac{1}{137500} a^{10}$, $\frac{1}{137500} a^{11}$, $\frac{1}{1375000} a^{12}$, $\frac{1}{1375000} a^{13}$, $\frac{1}{6875000} a^{14}$, $\frac{1}{6875000} a^{15}$, $\frac{1}{1581250000} a^{16} + \frac{1}{158125000} a^{14} + \frac{1}{6325000} a^{12} + \frac{7}{3162500} a^{10} - \frac{2}{14375} a^{8} + \frac{3}{2875} a^{6} - \frac{7}{1150} a^{4} - \frac{4}{115} a^{2} + \frac{5}{23}$, $\frac{1}{1581250000} a^{17} + \frac{1}{158125000} a^{15} + \frac{1}{6325000} a^{13} + \frac{7}{3162500} a^{11} - \frac{2}{14375} a^{9} + \frac{3}{2875} a^{7} - \frac{7}{1150} a^{5} - \frac{4}{115} a^{3} + \frac{5}{23} a$, $\frac{1}{1573343750000} a^{18} - \frac{9}{62933750000} a^{16} + \frac{189}{3146687500} a^{14} - \frac{87}{572125000} a^{12} - \frac{68}{31466875} a^{10} + \frac{396}{2860625} a^{8} - \frac{197}{1144250} a^{6} + \frac{126}{114425} a^{4} - \frac{129}{22885} a^{2} + \frac{1950}{4577}$, $\frac{1}{1573343750000} a^{19} - \frac{9}{62933750000} a^{17} + \frac{189}{3146687500} a^{15} - \frac{87}{572125000} a^{13} - \frac{68}{31466875} a^{11} + \frac{396}{2860625} a^{9} - \frac{197}{1144250} a^{7} + \frac{126}{114425} a^{5} - \frac{129}{22885} a^{3} + \frac{1950}{4577} a$
Class group and class number
$C_{186}\times C_{74586}$, which has order $13872996$ (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: | \( 530208.2507325789 \) (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
| \(\Q(\sqrt{2}) \), 4.0.6195200.5, \(\Q(\zeta_{11})^+\), 10.10.7024111812608.1 |
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 | R | $20$ | R | ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ | R | $20$ | ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ | $20$ | ${\href{/LocalNumberField/23.1.0.1}{1} }^{20}$ | $20$ | ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ | $20$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | $20$ | $20$ |
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 | ||||||
| 11 | Data not computed | ||||||