Normalized defining polynomial
\( x^{20} - x^{19} + 19 x^{18} - 15 x^{17} + 148 x^{16} - 88 x^{15} + 618 x^{14} - 266 x^{13} + 1534 x^{12} - 470 x^{11} + 2342 x^{10} - 263 x^{9} + 1906 x^{8} + 1335 x^{7} + 479 x^{6} + 2937 x^{5} - 134 x^{4} + 3709 x^{3} - 1963 x^{2} - 1928 x + 9901 \)
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: | \(3206128490667995866421572265625=3^{10}\cdot 5^{10}\cdot 11^{18}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $33.52$ | 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$$\chi_{165}(1,·)$, $\chi_{165}(131,·)$, $\chi_{165}(136,·)$, $\chi_{165}(139,·)$, $\chi_{165}(14,·)$, $\chi_{165}(79,·)$, $\chi_{165}(16,·)$, $\chi_{165}(19,·)$, $\chi_{165}(89,·)$, $\chi_{165}(91,·)$, $\chi_{165}(94,·)$, $\chi_{165}(31,·)$, $\chi_{165}(161,·)$, $\chi_{165}(101,·)$, $\chi_{165}(104,·)$, $\chi_{165}(41,·)$, $\chi_{165}(109,·)$, $\chi_{165}(116,·)$, $\chi_{165}(119,·)$, $\chi_{165}(59,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{89} a^{11} + \frac{11}{89} a^{9} + \frac{44}{89} a^{7} - \frac{12}{89} a^{5} - \frac{34}{89} a^{3} + \frac{11}{89} a - \frac{34}{89}$, $\frac{1}{89} a^{12} + \frac{11}{89} a^{10} + \frac{44}{89} a^{8} - \frac{12}{89} a^{6} - \frac{34}{89} a^{4} + \frac{11}{89} a^{2} - \frac{34}{89} a$, $\frac{1}{89} a^{13} + \frac{12}{89} a^{9} + \frac{38}{89} a^{7} + \frac{9}{89} a^{5} + \frac{29}{89} a^{3} - \frac{34}{89} a^{2} - \frac{32}{89} a + \frac{18}{89}$, $\frac{1}{89} a^{14} + \frac{12}{89} a^{10} + \frac{38}{89} a^{8} + \frac{9}{89} a^{6} + \frac{29}{89} a^{4} - \frac{34}{89} a^{3} - \frac{32}{89} a^{2} + \frac{18}{89} a$, $\frac{1}{178} a^{15} - \frac{1}{178} a^{13} - \frac{1}{178} a^{12} + \frac{39}{89} a^{10} + \frac{36}{89} a^{9} - \frac{22}{89} a^{8} - \frac{23}{178} a^{7} - \frac{77}{178} a^{6} + \frac{75}{178} a^{5} - \frac{1}{2} a^{4} - \frac{9}{178} a^{3} + \frac{41}{178} a^{2} + \frac{23}{178} a - \frac{55}{178}$, $\frac{1}{3524578} a^{16} - \frac{4875}{3524578} a^{15} - \frac{15235}{3524578} a^{14} - \frac{4161}{1762289} a^{13} - \frac{14975}{3524578} a^{12} - \frac{6966}{1762289} a^{11} - \frac{375627}{1762289} a^{10} - \frac{743912}{1762289} a^{9} + \frac{103415}{3524578} a^{8} - \frac{733163}{1762289} a^{7} - \frac{153595}{1762289} a^{6} + \frac{614519}{1762289} a^{5} - \frac{424790}{1762289} a^{4} - \frac{587334}{1762289} a^{3} - \frac{777252}{1762289} a^{2} + \frac{207847}{1762289} a + \frac{316655}{3524578}$, $\frac{1}{3524578} a^{17} + \frac{141}{3524578} a^{15} + \frac{14405}{3524578} a^{14} - \frac{6238}{1762289} a^{13} - \frac{5385}{1762289} a^{12} + \frac{57}{1762289} a^{11} - \frac{392440}{1762289} a^{10} + \frac{1295377}{3524578} a^{9} + \frac{13613}{3524578} a^{8} - \frac{1178475}{3524578} a^{7} + \frac{375975}{3524578} a^{6} + \frac{927971}{3524578} a^{5} - \frac{1082197}{3524578} a^{4} + \frac{1734565}{3524578} a^{3} + \frac{1457265}{3524578} a^{2} + \frac{66576}{1762289} a + \frac{191692}{1762289}$, $\frac{1}{3524578} a^{18} + \frac{8745}{3524578} a^{15} - \frac{2849}{3524578} a^{14} - \frac{5627}{3524578} a^{13} - \frac{3559}{1762289} a^{12} - \frac{4264}{1762289} a^{11} + \frac{98781}{3524578} a^{10} + \frac{1133859}{3524578} a^{9} + \frac{40405}{1762289} a^{8} + \frac{687875}{1762289} a^{7} - \frac{343354}{1762289} a^{6} + \frac{124232}{1762289} a^{5} - \frac{867940}{1762289} a^{4} - \frac{86797}{1762289} a^{3} - \frac{849103}{3524578} a^{2} + \frac{1232933}{3524578} a + \frac{179659}{1762289}$, $\frac{1}{3524578} a^{19} - \frac{2527}{3524578} a^{15} + \frac{1660}{1762289} a^{14} + \frac{97}{3524578} a^{13} + \frac{1917}{1762289} a^{12} - \frac{437}{3524578} a^{11} - \frac{1577035}{3524578} a^{10} + \frac{367917}{1762289} a^{9} - \frac{656855}{3524578} a^{8} - \frac{734035}{3524578} a^{7} + \frac{1391603}{3524578} a^{6} + \frac{1378765}{3524578} a^{5} - \frac{1218759}{3524578} a^{4} + \frac{16008}{1762289} a^{3} + \frac{724643}{1762289} a^{2} - \frac{945789}{3524578} a + \frac{852480}{1762289}$
Class group and class number
$C_{44}$, which has order $44$ (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: | \( 125582.779517 \) (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
| \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-15}, \sqrt{33})\), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{33})^+\), 10.0.7368586534375.1, 10.0.162778775259375.1 |
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 | ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ | R | R | ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||
| 5 | Data not computed | ||||||
| 11 | Data not computed | ||||||