Normalized defining polynomial
\( x^{20} + 10 x^{18} + 21 x^{16} - 45 x^{14} - 111 x^{12} + 196 x^{10} - 290 x^{8} - 3042 x^{6} - 3445 x^{4} - 845 x^{2} + 169 \)
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: | \(716601923633933608960000000000=2^{20}\cdot 5^{10}\cdot 13^{6}\cdot 347^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $31.10$ | 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: | $2, 5, 13, 347$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{13} a^{16} - \frac{3}{13} a^{14} - \frac{5}{13} a^{12} - \frac{6}{13} a^{10} + \frac{6}{13} a^{8} + \frac{1}{13} a^{6} - \frac{4}{13} a^{4}$, $\frac{1}{13} a^{17} - \frac{3}{13} a^{15} - \frac{5}{13} a^{13} - \frac{6}{13} a^{11} + \frac{6}{13} a^{9} + \frac{1}{13} a^{7} - \frac{4}{13} a^{5}$, $\frac{1}{191734477962599} a^{18} + \frac{37123326933}{1340800545193} a^{16} - \frac{56988154806870}{191734477962599} a^{14} + \frac{85420974185578}{191734477962599} a^{12} - \frac{35594893150413}{191734477962599} a^{10} + \frac{17461607689675}{191734477962599} a^{8} - \frac{70190377256481}{191734477962599} a^{6} + \frac{2172585997622}{191734477962599} a^{4} + \frac{4682600581568}{14748805997123} a^{2} - \frac{6537793985032}{14748805997123}$, $\frac{1}{191734477962599} a^{19} + \frac{37123326933}{1340800545193} a^{17} - \frac{56988154806870}{191734477962599} a^{15} + \frac{85420974185578}{191734477962599} a^{13} - \frac{35594893150413}{191734477962599} a^{11} + \frac{17461607689675}{191734477962599} a^{9} - \frac{70190377256481}{191734477962599} a^{7} + \frac{2172585997622}{191734477962599} a^{5} + \frac{4682600581568}{14748805997123} a^{3} - \frac{6537793985032}{14748805997123} a$
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: | \( 8009363.86951 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 122880 |
| The 138 conjugacy class representatives for t20n794 are not computed |
| Character table for t20n794 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 5.3.4511.1, 10.6.63591003125.1 |
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 | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ | R | ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$ |
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 | ||||||
| 5 | Data not computed | ||||||
| $13$ | 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.4.0.1 | $x^{4} + x^{2} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 13.4.0.1 | $x^{4} + x^{2} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 13.8.6.1 | $x^{8} - 13 x^{4} + 2704$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 347 | Data not computed | ||||||