Normalized defining polynomial
\( x^{20} - x^{19} - 4 x^{18} + 4 x^{17} + 3 x^{16} - 6 x^{15} + 13 x^{14} + 36 x^{13} - 4 x^{12} - 82 x^{11} - 75 x^{10} + 71 x^{9} + 199 x^{8} + 28 x^{7} - 72 x^{6} - 113 x^{5} - 58 x^{4} + 96 x^{3} + 87 x^{2} + 57 x + 31 \)
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: | \(378138159793919664573793=47^{10}\cdot 193^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $15.10$ | 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: | $47, 193$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{1337700524681450645320861} a^{19} - \frac{326274457004985072151392}{1337700524681450645320861} a^{18} - \frac{621841188206914342403455}{1337700524681450645320861} a^{17} + \frac{636611353853490513163580}{1337700524681450645320861} a^{16} + \frac{59115110611826181210490}{1337700524681450645320861} a^{15} - \frac{585364951095555406705890}{1337700524681450645320861} a^{14} + \frac{326943271369975198575219}{1337700524681450645320861} a^{13} - \frac{93542884880239647904075}{1337700524681450645320861} a^{12} + \frac{117108008075567828135990}{1337700524681450645320861} a^{11} - \frac{321970942793957999569107}{1337700524681450645320861} a^{10} - \frac{446052631940353860926339}{1337700524681450645320861} a^{9} - \frac{419167204113087872468952}{1337700524681450645320861} a^{8} + \frac{458020091669292132867415}{1337700524681450645320861} a^{7} + \frac{316122190635797858462719}{1337700524681450645320861} a^{6} + \frac{313069748741198612577204}{1337700524681450645320861} a^{5} + \frac{571349285666705274489256}{1337700524681450645320861} a^{4} - \frac{70223369872002889234457}{1337700524681450645320861} a^{3} + \frac{567119900371467862406677}{1337700524681450645320861} a^{2} - \frac{50923882098275907554242}{1337700524681450645320861} a + \frac{152289856197062997010909}{1337700524681450645320861}$
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: | \( 2098.65426229 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 10240 |
| The 208 conjugacy class representatives for t20n418 are not computed |
| Character table for t20n418 is not computed |
Intermediate fields
| 5.1.2209.1, 10.2.941778433.1 |
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 | ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ | ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ | $20$ | ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ | ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ | $20$ | ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ | R | $20$ | ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $47$ | 47.4.0.1 | $x^{4} - x + 39$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 47.4.2.1 | $x^{4} + 1175 x^{2} + 373321$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 47.4.2.1 | $x^{4} + 1175 x^{2} + 373321$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 47.8.6.2 | $x^{8} + 1175 x^{4} + 373321$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ | |
| $193$ | 193.2.0.1 | $x^{2} - x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 193.2.0.1 | $x^{2} - x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 193.2.0.1 | $x^{2} - x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 193.2.0.1 | $x^{2} - x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 193.4.3.3 | $x^{4} + 965$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 193.8.0.1 | $x^{8} - x + 5$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |