Normalized defining polynomial
\( x^{20} - 22 x^{18} + 308 x^{16} - 2640 x^{14} + 16544 x^{12} - 70048 x^{10} + 216832 x^{8} - 418176 x^{6} + 557568 x^{4} - 309760 x^{2} + 123904 \)
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: | \(352517555563337816067682238201856=2^{30}\cdot 3^{10}\cdot 11^{18}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $42.40$ | 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: | $2, 3, 11$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(264=2^{3}\cdot 3\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{264}(1,·)$, $\chi_{264}(197,·)$, $\chi_{264}(257,·)$, $\chi_{264}(137,·)$, $\chi_{264}(13,·)$, $\chi_{264}(173,·)$, $\chi_{264}(205,·)$, $\chi_{264}(149,·)$, $\chi_{264}(25,·)$, $\chi_{264}(89,·)$, $\chi_{264}(29,·)$, $\chi_{264}(97,·)$, $\chi_{264}(101,·)$, $\chi_{264}(49,·)$, $\chi_{264}(169,·)$, $\chi_{264}(109,·)$, $\chi_{264}(113,·)$, $\chi_{264}(185,·)$, $\chi_{264}(61,·)$, $\chi_{264}(85,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{352} a^{10}$, $\frac{1}{352} a^{11}$, $\frac{1}{704} a^{12}$, $\frac{1}{704} a^{13}$, $\frac{1}{1408} a^{14}$, $\frac{1}{1408} a^{15}$, $\frac{1}{121088} a^{16} + \frac{7}{60544} a^{14} + \frac{3}{15136} a^{12} - \frac{5}{7568} a^{10} + \frac{21}{688} a^{8} + \frac{9}{172} a^{6} + \frac{21}{172} a^{4} + \frac{9}{43} a^{2} - \frac{3}{43}$, $\frac{1}{121088} a^{17} + \frac{7}{60544} a^{15} + \frac{3}{15136} a^{13} - \frac{5}{7568} a^{11} + \frac{21}{688} a^{9} + \frac{9}{172} a^{7} + \frac{21}{172} a^{5} + \frac{9}{43} a^{3} - \frac{3}{43} a$, $\frac{1}{1768611328} a^{18} - \frac{133}{110538208} a^{16} - \frac{62617}{442152832} a^{14} - \frac{11597}{20097856} a^{12} + \frac{6103}{13817276} a^{10} + \frac{55701}{5024464} a^{8} + \frac{34063}{2512232} a^{6} - \frac{5453}{628058} a^{4} + \frac{142657}{628058} a^{2} - \frac{53590}{314029}$, $\frac{1}{1768611328} a^{19} - \frac{133}{110538208} a^{17} - \frac{62617}{442152832} a^{15} - \frac{11597}{20097856} a^{13} + \frac{6103}{13817276} a^{11} + \frac{55701}{5024464} a^{9} + \frac{34063}{2512232} a^{7} - \frac{5453}{628058} a^{5} + \frac{142657}{628058} a^{3} - \frac{53590}{314029} a$
Class group and class number
$C_{22}$, which has order $22$ (assuming GRH)
Unit group
| Rank: | $9$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{10843}{1768611328} a^{18} - \frac{2615}{20097856} a^{16} + \frac{792503}{442152832} a^{14} - \frac{3292525}{221076416} a^{12} + \frac{5041151}{55269104} a^{10} - \frac{927211}{2512232} a^{8} + \frac{2778327}{2512232} a^{6} - \frac{2435481}{1256116} a^{4} + \frac{856116}{314029} a^{2} - \frac{159235}{314029} \) (order $6$) | 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: | \( 11866284.7339 \) (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
| \(\Q(\sqrt{66}) \), \(\Q(\sqrt{-22}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{-22})\), \(\Q(\zeta_{11})^+\), 10.10.18775450875101184.1, 10.0.77265229938688.1, 10.0.52089208083.1 |
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 | R | ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/43.1.0.1}{1} }^{20}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| 3 | Data not computed | ||||||
| 11 | Data not computed | ||||||