Normalized defining polynomial
\( x^{20} - 23 x^{18} + 232 x^{16} - 1354 x^{14} + 5094 x^{12} - 13102 x^{10} + 24113 x^{8} - 33551 x^{6} + 38270 x^{4} - 39480 x^{2} + 39601 \)
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: | \(56933553290160450365440000000000=2^{20}\cdot 5^{10}\cdot 11^{18}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $38.71$ | 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: | \(220=2^{2}\cdot 5\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{220}(1,·)$, $\chi_{220}(71,·)$, $\chi_{220}(201,·)$, $\chi_{220}(139,·)$, $\chi_{220}(141,·)$, $\chi_{220}(79,·)$, $\chi_{220}(81,·)$, $\chi_{220}(19,·)$, $\chi_{220}(149,·)$, $\chi_{220}(91,·)$, $\chi_{220}(29,·)$, $\chi_{220}(31,·)$, $\chi_{220}(219,·)$, $\chi_{220}(39,·)$, $\chi_{220}(129,·)$, $\chi_{220}(109,·)$, $\chi_{220}(111,·)$, $\chi_{220}(181,·)$, $\chi_{220}(189,·)$, $\chi_{220}(191,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{199} a^{11} - \frac{11}{199} a^{9} + \frac{44}{199} a^{7} - \frac{77}{199} a^{5} + \frac{55}{199} a^{3} - \frac{11}{199} a$, $\frac{1}{17711} a^{12} - \frac{6777}{17711} a^{10} - \frac{3140}{17711} a^{8} - \frac{6644}{17711} a^{6} + \frac{1846}{17711} a^{4} + \frac{7949}{17711} a^{2} - \frac{21}{89}$, $\frac{1}{17711} a^{13} - \frac{13}{17711} a^{11} - \frac{6700}{17711} a^{9} + \frac{7596}{17711} a^{7} - \frac{5363}{17711} a^{5} + \frac{8038}{17711} a^{3} - \frac{7739}{17711} a$, $\frac{1}{17711} a^{14} - \frac{6246}{17711} a^{10} + \frac{2198}{17711} a^{8} - \frac{3180}{17711} a^{6} - \frac{3386}{17711} a^{4} + \frac{7043}{17711} a^{2} - \frac{6}{89}$, $\frac{1}{17711} a^{15} - \frac{16}{17711} a^{11} + \frac{4512}{17711} a^{9} + \frac{5275}{17711} a^{7} - \frac{4899}{17711} a^{5} - \frac{4527}{17711} a^{3} + \frac{1120}{17711} a$, $\frac{1}{17711} a^{16} + \frac{2346}{17711} a^{10} + \frac{8168}{17711} a^{8} - \frac{4937}{17711} a^{6} + \frac{82}{199} a^{4} + \frac{4327}{17711} a^{2} + \frac{20}{89}$, $\frac{1}{17711} a^{17} + \frac{32}{17711} a^{11} - \frac{1800}{17711} a^{9} - \frac{487}{17711} a^{7} + \frac{94}{199} a^{5} + \frac{1034}{17711} a^{3} - \frac{5988}{17711} a$, $\frac{1}{17711} a^{18} + \frac{2532}{17711} a^{10} - \frac{6273}{17711} a^{8} + \frac{8442}{17711} a^{6} - \frac{4905}{17711} a^{4} + \frac{5309}{17711} a^{2} - \frac{40}{89}$, $\frac{1}{17711} a^{19} + \frac{40}{17711} a^{11} + \frac{3428}{17711} a^{9} + \frac{5060}{17711} a^{7} - \frac{7842}{17711} a^{5} - \frac{7774}{17711} a^{3} + \frac{1741}{17711} a$
Class group and class number
$C_{4}$, which has order $4$ (assuming GRH)
Unit group
| Rank: | $9$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{1}{199} a^{11} + \frac{11}{199} a^{9} - \frac{44}{199} a^{7} + \frac{77}{199} a^{5} - \frac{55}{199} a^{3} + \frac{11}{199} a \) (order $4$) | 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: | \( 11184526.8933 \) (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
| \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{55}) \), \(\Q(\sqrt{-55}) \), \(\Q(i, \sqrt{55})\), \(\Q(\zeta_{11})^+\), 10.0.219503494144.1, 10.10.7545432611200000.1, 10.0.7368586534375.1 |
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 | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.10.10.7 | $x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$ | $2$ | $5$ | $10$ | $C_{10}$ | $[2]^{5}$ |
| 2.10.10.7 | $x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$ | $2$ | $5$ | $10$ | $C_{10}$ | $[2]^{5}$ | |
| $5$ | 5.10.5.1 | $x^{10} - 50 x^{6} + 625 x^{2} - 12500$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
| 5.10.5.1 | $x^{10} - 50 x^{6} + 625 x^{2} - 12500$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
| 11 | Data not computed | ||||||