Normalized defining polynomial
\( x^{20} - 27 x^{18} + 222 x^{16} - 570 x^{14} - 948 x^{12} + 7124 x^{10} - 10809 x^{8} + 1342 x^{6} + 8149 x^{4} - 4600 x^{2} + 125 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[16, 2]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(3576211589862019840319539200000000000=2^{20}\cdot 3^{8}\cdot 5^{11}\cdot 239^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $67.25$ | 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: | $2, 3, 5, 239$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{33} a^{16} - \frac{16}{33} a^{14} + \frac{3}{11} a^{12} - \frac{1}{3} a^{10} - \frac{16}{33} a^{8} - \frac{4}{33} a^{6} + \frac{2}{33} a^{4} - \frac{2}{11} a^{2} - \frac{10}{33}$, $\frac{1}{33} a^{17} - \frac{16}{33} a^{15} + \frac{3}{11} a^{13} - \frac{1}{3} a^{11} - \frac{16}{33} a^{9} - \frac{4}{33} a^{7} + \frac{2}{33} a^{5} - \frac{2}{11} a^{3} - \frac{10}{33} a$, $\frac{1}{15601846590165} a^{18} + \frac{46276211878}{15601846590165} a^{16} + \frac{6372669367117}{15601846590165} a^{14} - \frac{101374021499}{283669938003} a^{12} - \frac{282146446763}{15601846590165} a^{10} - \frac{829378040122}{5200615530055} a^{8} + \frac{329078957677}{5200615530055} a^{6} - \frac{5337814577288}{15601846590165} a^{4} - \frac{2878660632286}{15601846590165} a^{2} + \frac{21367977793}{283669938003}$, $\frac{1}{78009232950825} a^{19} - \frac{426507018127}{78009232950825} a^{17} + \frac{4645733682399}{26003077650275} a^{15} + \frac{7394983903601}{15601846590165} a^{13} - \frac{10683377506873}{78009232950825} a^{11} - \frac{10525449030451}{78009232950825} a^{9} + \frac{18480216383216}{78009232950825} a^{7} + \frac{3106155184289}{26003077650275} a^{5} + \frac{2832884720734}{7091748450075} a^{3} - \frac{129317006820}{1040123106011} a$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $17$ | 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: | \( 673176083601 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 10240 |
| The 100 conjugacy class representatives for t20n426 are not computed |
| Character table for t20n426 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 5.5.12852225.1, 10.10.825898437253125.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
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 | R | $20$ | ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ | $20$ | $20$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ | $20$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ | $20$ | ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$ |
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 | ||||||
| $3$ | 3.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $5$ | 5.4.3.1 | $x^{4} - 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 239 | Data not computed | ||||||