Normalized defining polynomial
\( x^{20} + 24 x^{18} + 128 x^{16} - 319 x^{14} - 3032 x^{12} - 548 x^{10} + 12767 x^{8} + 1187 x^{6} - 10844 x^{4} + 3450 x^{2} + 125 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[8, 6]$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{11} a^{12} + \frac{2}{11} a^{10} - \frac{5}{11} a^{8} - \frac{4}{11} a^{4} + \frac{2}{11} a^{2} - \frac{3}{11}$, $\frac{1}{11} a^{13} + \frac{2}{11} a^{11} - \frac{5}{11} a^{9} - \frac{4}{11} a^{5} + \frac{2}{11} a^{3} - \frac{3}{11} a$, $\frac{1}{11} a^{14} + \frac{2}{11} a^{10} - \frac{1}{11} a^{8} - \frac{4}{11} a^{6} - \frac{1}{11} a^{4} + \frac{4}{11} a^{2} - \frac{5}{11}$, $\frac{1}{11} a^{15} + \frac{2}{11} a^{11} - \frac{1}{11} a^{9} - \frac{4}{11} a^{7} - \frac{1}{11} a^{5} + \frac{4}{11} a^{3} - \frac{5}{11} a$, $\frac{1}{33} a^{16} + \frac{1}{33} a^{14} + \frac{1}{33} a^{12} + \frac{10}{33} a^{10} - \frac{16}{33} a^{6} + \frac{7}{33} a^{4} + \frac{8}{33} a^{2} - \frac{2}{33}$, $\frac{1}{33} a^{17} + \frac{1}{33} a^{15} + \frac{1}{33} a^{13} + \frac{10}{33} a^{11} - \frac{16}{33} a^{7} + \frac{7}{33} a^{5} + \frac{8}{33} a^{3} - \frac{2}{33} a$, $\frac{1}{5717820635730555} a^{18} - \frac{24419962253521}{5717820635730555} a^{16} + \frac{88711118956103}{5717820635730555} a^{14} + \frac{15857392173781}{519801875975505} a^{12} - \frac{61041238246388}{300937928196345} a^{10} - \frac{2279921221235728}{5717820635730555} a^{8} - \frac{440637697492286}{1905940211910185} a^{6} + \frac{821109431879139}{1905940211910185} a^{4} - \frac{265904415317703}{1905940211910185} a^{2} - \frac{552700487383780}{1143564127146111}$, $\frac{1}{28589103178652775} a^{19} + \frac{322114621730149}{28589103178652775} a^{17} + \frac{955047578915278}{28589103178652775} a^{15} + \frac{1164021919756}{28589103178652775} a^{13} - \frac{160176823829926}{501563213660575} a^{11} + \frac{11234927554127402}{28589103178652775} a^{9} + \frac{8207787967074067}{28589103178652775} a^{7} - \frac{4987165260011488}{28589103178652775} a^{5} - \frac{2703653457863294}{28589103178652775} a^{3} - \frac{59949004757830}{381188042382037} a$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $13$ | 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: | \( 70146545712.2 \) (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 | $20$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ | $20$ | $20$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ | $20$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ | $20$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 | ||||||
| $3$ | 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 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $5$ | 5.4.3.2 | $x^{4} - 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 239 | Data not computed | ||||||