Normalized defining polynomial
\( x^{20} - x^{19} - 39 x^{18} + 43 x^{17} + 579 x^{16} - 664 x^{15} - 4199 x^{14} + 4764 x^{13} + 16062 x^{12} - 17268 x^{11} - 32674 x^{10} + 31502 x^{9} + 34487 x^{8} - 26929 x^{7} - 18180 x^{6} + 9222 x^{5} + 4671 x^{4} - 978 x^{3} - 468 x^{2} - 21 x + 1 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[20, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(82802905234194108120391845703125=3^{10}\cdot 5^{15}\cdot 11^{16}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $39.44$ | 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: | $3, 5, 11$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(165=3\cdot 5\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{165}(64,·)$, $\chi_{165}(1,·)$, $\chi_{165}(4,·)$, $\chi_{165}(136,·)$, $\chi_{165}(137,·)$, $\chi_{165}(16,·)$, $\chi_{165}(23,·)$, $\chi_{165}(152,·)$, $\chi_{165}(91,·)$, $\chi_{165}(92,·)$, $\chi_{165}(158,·)$, $\chi_{165}(31,·)$, $\chi_{165}(34,·)$, $\chi_{165}(38,·)$, $\chi_{165}(49,·)$, $\chi_{165}(47,·)$, $\chi_{165}(113,·)$, $\chi_{165}(53,·)$, $\chi_{165}(122,·)$, $\chi_{165}(124,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{2082669587034804529733814536431} a^{19} - \frac{631040415238238223427867771801}{2082669587034804529733814536431} a^{18} + \frac{152987828715493742049781257249}{2082669587034804529733814536431} a^{17} - \frac{990984620086852314893651586843}{2082669587034804529733814536431} a^{16} - \frac{392771049749764876705225201479}{2082669587034804529733814536431} a^{15} + \frac{450981209062148196235006064378}{2082669587034804529733814536431} a^{14} - \frac{694079443849693900768343536659}{2082669587034804529733814536431} a^{13} + \frac{325365515022304479260318025552}{2082669587034804529733814536431} a^{12} + \frac{232905747903421311913236205052}{2082669587034804529733814536431} a^{11} - \frac{3075027522880656288175878035}{2082669587034804529733814536431} a^{10} - \frac{235570452199100968135958356687}{2082669587034804529733814536431} a^{9} + \frac{133303929826186272482153815804}{2082669587034804529733814536431} a^{8} - \frac{583007030297928898348119760805}{2082669587034804529733814536431} a^{7} + \frac{76782440715765804847410656882}{2082669587034804529733814536431} a^{6} + \frac{697600173347618897379107298608}{2082669587034804529733814536431} a^{5} + \frac{1017821131311634053674797787704}{2082669587034804529733814536431} a^{4} - \frac{378754573027980994274253765017}{2082669587034804529733814536431} a^{3} - \frac{392356911223898611657593156114}{2082669587034804529733814536431} a^{2} - \frac{60337573671184055476426726308}{2082669587034804529733814536431} a + \frac{659223819107552474364194493169}{2082669587034804529733814536431}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $19$ | 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: | \( 2427056473.64 \) (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
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{11})^+\), 10.10.669871503125.1 |
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 | $20$ | R | R | $20$ | R | $20$ | $20$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ | $20$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ | $20$ | $20$ | ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||
| 5 | Data not computed | ||||||
| $11$ | 11.10.8.5 | $x^{10} - 2321 x^{5} + 2033647$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ |
| 11.10.8.5 | $x^{10} - 2321 x^{5} + 2033647$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ | |