Normalized defining polynomial
\( x^{20} + 17 x^{18} + 124 x^{16} + 502 x^{14} + 1230 x^{12} + 1858 x^{10} + 1721 x^{8} + 833 x^{6} + 554 x^{4} - 1560 x^{2} + 7921 \)
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$$\chi_{220}(1,·)$, $\chi_{220}(131,·)$, $\chi_{220}(51,·)$, $\chi_{220}(129,·)$, $\chi_{220}(201,·)$, $\chi_{220}(141,·)$, $\chi_{220}(81,·)$, $\chi_{220}(211,·)$, $\chi_{220}(149,·)$, $\chi_{220}(151,·)$, $\chi_{220}(29,·)$, $\chi_{220}(199,·)$, $\chi_{220}(159,·)$, $\chi_{220}(171,·)$, $\chi_{220}(109,·)$, $\chi_{220}(179,·)$, $\chi_{220}(181,·)$, $\chi_{220}(119,·)$, $\chi_{220}(59,·)$, $\chi_{220}(189,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{89} a^{11} + \frac{11}{89} a^{9} + \frac{44}{89} a^{7} - \frac{12}{89} a^{5} - \frac{34}{89} a^{3} + \frac{11}{89} a$, $\frac{1}{17711} a^{12} - \frac{6753}{17711} a^{10} + \frac{3248}{17711} a^{8} - \frac{6420}{17711} a^{6} - \frac{1636}{17711} a^{4} + \frac{8021}{17711} a^{2} + \frac{47}{199}$, $\frac{1}{17711} a^{13} + \frac{13}{17711} a^{11} + \frac{6830}{17711} a^{9} + \frac{7908}{17711} a^{7} + \frac{5727}{17711} a^{5} + \frac{8220}{17711} a^{3} + \frac{7765}{17711} a$, $\frac{1}{17711} a^{14} + \frac{6064}{17711} a^{10} + \frac{1106}{17711} a^{8} + \frac{632}{17711} a^{6} - \frac{5934}{17711} a^{4} - \frac{7953}{17711} a^{2} - \frac{14}{199}$, $\frac{1}{17711} a^{15} + \frac{94}{17711} a^{11} + \frac{6280}{17711} a^{9} + \frac{3617}{17711} a^{7} - \frac{5138}{17711} a^{5} + \frac{206}{17711} a^{3} + \frac{3928}{17711} a$, $\frac{1}{17711} a^{16} + \frac{3466}{17711} a^{10} - \frac{608}{17711} a^{8} - \frac{3832}{17711} a^{6} - \frac{5409}{17711} a^{4} - \frac{6184}{17711} a^{2} - \frac{40}{199}$, $\frac{1}{17711} a^{17} + \frac{83}{17711} a^{11} - \frac{2399}{17711} a^{9} + \frac{6715}{17711} a^{7} - \frac{235}{17711} a^{5} + \frac{2572}{17711} a^{3} - \frac{5351}{17711} a$, $\frac{1}{17711} a^{18} - \frac{8652}{17711} a^{10} + \frac{2796}{17711} a^{8} + \frac{1295}{17711} a^{6} - \frac{3328}{17711} a^{4} + \frac{1924}{17711} a^{2} + \frac{79}{199}$, $\frac{1}{17711} a^{19} - \frac{95}{17711} a^{11} + \frac{8368}{17711} a^{9} + \frac{5872}{17711} a^{7} + \frac{254}{17711} a^{5} - \frac{5638}{17711} a^{3} - \frac{5108}{17711} a$
Class group and class number
$C_{124}$, which has order $124$ (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: | \( 281202.490766 \) (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
| \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{-5}, \sqrt{11})\), \(\Q(\zeta_{11})^+\), 10.0.685948419200000.1, 10.0.7368586534375.1, \(\Q(\zeta_{44})^+\) |
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 | ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ | R | ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/43.1.0.1}{1} }^{20}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.10.10.11 | $x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$ | $2$ | $5$ | $10$ | $C_{10}$ | $[2]^{5}$ |
| 2.10.10.11 | $x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$ | $2$ | $5$ | $10$ | $C_{10}$ | $[2]^{5}$ | |
| $5$ | 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 | ||||||