Normalized defining polynomial
\( x^{20} - 22 x^{15} - 127 x^{10} - 22 x^{5} + 1 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[4, 8]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(2560000000000000000000000=2^{30}\cdot 5^{22}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $16.61$ | 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: | $2, 5$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} - \frac{1}{5} a^{5} + \frac{1}{5}$, $\frac{1}{5} a^{11} - \frac{1}{5} a^{6} + \frac{1}{5} a$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{7} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{13} - \frac{1}{5} a^{8} + \frac{1}{5} a^{3}$, $\frac{1}{5} a^{14} - \frac{1}{5} a^{9} + \frac{1}{5} a^{4}$, $\frac{1}{585} a^{15} + \frac{31}{585} a^{10} + \frac{229}{585} a^{5} - \frac{53}{585}$, $\frac{1}{2925} a^{16} + \frac{2}{2925} a^{15} - \frac{2}{25} a^{14} - \frac{1}{25} a^{13} - \frac{2}{25} a^{12} + \frac{31}{2925} a^{11} + \frac{179}{2925} a^{10} - \frac{3}{25} a^{9} + \frac{11}{25} a^{8} - \frac{3}{25} a^{7} + \frac{814}{2925} a^{6} + \frac{341}{2925} a^{5} - \frac{2}{25} a^{4} - \frac{1}{25} a^{3} - \frac{2}{25} a^{2} + \frac{1117}{2925} a - \frac{574}{2925}$, $\frac{1}{2925} a^{17} + \frac{2}{2925} a^{15} - \frac{2}{25} a^{14} - \frac{86}{2925} a^{12} + \frac{1}{25} a^{11} - \frac{289}{2925} a^{10} - \frac{3}{25} a^{9} - \frac{824}{2925} a^{7} - \frac{11}{25} a^{6} - \frac{361}{2925} a^{5} - \frac{2}{25} a^{4} + \frac{40}{117} a^{2} + \frac{1}{25} a - \frac{1042}{2925}$, $\frac{1}{2925} a^{18} + \frac{2}{2925} a^{15} - \frac{1}{25} a^{14} + \frac{148}{2925} a^{13} + \frac{2}{25} a^{11} - \frac{289}{2925} a^{10} + \frac{11}{25} a^{9} - \frac{473}{2925} a^{8} + \frac{3}{25} a^{6} - \frac{361}{2925} a^{5} - \frac{1}{25} a^{4} + \frac{1234}{2925} a^{3} + \frac{2}{25} a - \frac{1042}{2925}$, $\frac{1}{2925} a^{19} - \frac{1}{2925} a^{15} + \frac{31}{2925} a^{14} + \frac{2}{25} a^{13} + \frac{1}{25} a^{12} + \frac{2}{25} a^{11} - \frac{31}{2925} a^{10} + \frac{814}{2925} a^{9} + \frac{3}{25} a^{8} - \frac{11}{25} a^{7} + \frac{3}{25} a^{6} - \frac{814}{2925} a^{5} + \frac{1117}{2925} a^{4} + \frac{2}{25} a^{3} + \frac{1}{25} a^{2} + \frac{2}{25} a - \frac{1117}{2925}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $11$ | 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: | \( 22120.756709 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times F_5$ (as 20T13):
| A solvable group of order 40 |
| The 10 conjugacy class representatives for $C_2\times F_5$ |
| Character table for $C_2\times F_5$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 5.1.200000.1, 10.2.320000000000.1, 10.2.1600000000000.1, 10.2.200000000000.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
| Degree 10 siblings: | data not computed |
| Degree 20 sibling: | data not computed |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.4.6.1 | $x^{4} - 6 x^{2} + 4$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ |
| 2.8.12.1 | $x^{8} + 6 x^{6} + 8 x^{5} + 16$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ | |
| 2.8.12.1 | $x^{8} + 6 x^{6} + 8 x^{5} + 16$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ | |
| 5 | Data not computed | ||||||