Normalized defining polynomial
\( x^{20} - 2 x^{19} - 2 x^{17} + 9 x^{16} + 5 x^{15} + 6 x^{14} - 9 x^{13} - 14 x^{12} - 71 x^{11} - 3 x^{10} + 91 x^{9} + 202 x^{8} + 302 x^{7} + 146 x^{6} - 20 x^{5} + 2 x^{4} - 150 x^{3} + 40 x^{2} + 207 x + 83 \)
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: | \(1536632364115844947237225=5^{2}\cdot 23^{2}\cdot 47^{12}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $16.19$ | 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: | $5, 23, 47$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{11} a^{17} - \frac{5}{11} a^{16} + \frac{3}{11} a^{15} - \frac{5}{11} a^{14} + \frac{5}{11} a^{13} - \frac{2}{11} a^{12} + \frac{2}{11} a^{11} + \frac{3}{11} a^{10} - \frac{5}{11} a^{9} - \frac{2}{11} a^{8} + \frac{2}{11} a^{5} - \frac{1}{11} a^{4} + \frac{4}{11} a^{3} + \frac{4}{11} a^{2} - \frac{4}{11} a + \frac{5}{11}$, $\frac{1}{517} a^{18} + \frac{15}{517} a^{17} - \frac{64}{517} a^{16} - \frac{161}{517} a^{14} - \frac{100}{517} a^{13} + \frac{127}{517} a^{12} - \frac{166}{517} a^{11} + \frac{20}{47} a^{10} + \frac{129}{517} a^{9} - \frac{216}{517} a^{8} - \frac{16}{47} a^{7} + \frac{79}{517} a^{6} - \frac{115}{517} a^{5} - \frac{126}{517} a^{4} - \frac{257}{517} a^{3} - \frac{34}{517} a^{2} - \frac{196}{517} a + \frac{34}{517}$, $\frac{1}{3102275586952317189618985} a^{19} + \frac{569279121619737581575}{620455117390463437923797} a^{18} + \frac{4152926966599524013147}{620455117390463437923797} a^{17} - \frac{94097754304574153925907}{3102275586952317189618985} a^{16} - \frac{82072321013137166188164}{620455117390463437923797} a^{15} - \frac{24239645439986990389688}{620455117390463437923797} a^{14} - \frac{844873043117334919956264}{3102275586952317189618985} a^{13} - \frac{1223495759765530881758427}{3102275586952317189618985} a^{12} - \frac{134920848650728918474533}{3102275586952317189618985} a^{11} - \frac{1299512764857699954144257}{3102275586952317189618985} a^{10} - \frac{678426099900725474106082}{3102275586952317189618985} a^{9} + \frac{1380629077893454841837277}{3102275586952317189618985} a^{8} - \frac{742431926940726284062634}{3102275586952317189618985} a^{7} + \frac{1259642713831526497902179}{3102275586952317189618985} a^{6} - \frac{102499510825852300657626}{282025053359301562692635} a^{5} + \frac{1252155167277768383856393}{3102275586952317189618985} a^{4} - \frac{45707991870664533448282}{282025053359301562692635} a^{3} - \frac{860675277245946805102324}{3102275586952317189618985} a^{2} - \frac{18769112396633890664737}{163277662471174588927315} a - \frac{358755213994569546510724}{3102275586952317189618985}$
Class group and class number
Trivial group, which has order $1$ (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: | \( 4956.93830859 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 5120 |
| The 104 conjugacy class representatives for t20n347 are not computed |
| Character table for t20n347 is not computed |
Intermediate fields
| \(\Q(\sqrt{-47}) \), 5.1.2209.1 x5, 10.0.229345007.1 |
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 | ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ | R | ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 5.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 5.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 5.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.0.1 | $x^{4} + x^{2} - 2 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 5.4.0.1 | $x^{4} + x^{2} - 2 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| $23$ | 23.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 23.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 23.4.2.2 | $x^{4} - 23 x^{2} + 3703$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 23.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 23.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| $47$ | 47.2.1.2 | $x^{2} + 94$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 47.2.1.2 | $x^{2} + 94$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.2 | $x^{2} + 94$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.2.1.2 | $x^{2} + 94$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 47.4.3.1 | $x^{4} + 94$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 47.4.2.1 | $x^{4} + 1175 x^{2} + 373321$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 47.4.3.1 | $x^{4} + 94$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ |