Normalized defining polynomial
\( x^{20} + 2 x^{18} - 5 x^{16} - 11 x^{14} + 11 x^{12} + 43 x^{10} + 22 x^{8} - 44 x^{6} - 40 x^{4} + 32 x^{2} + 32 \)
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: | \(23455391062151313489920000=2^{25}\cdot 5^{4}\cdot 5783^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $18.56$ | 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: | $2, 5, 5783$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{10} a^{12} + \frac{2}{5} a^{10} + \frac{3}{10} a^{8} - \frac{1}{10} a^{6} + \frac{1}{10} a^{4} + \frac{1}{10} a^{2} - \frac{1}{5}$, $\frac{1}{10} a^{13} + \frac{2}{5} a^{11} + \frac{3}{10} a^{9} - \frac{1}{10} a^{7} + \frac{1}{10} a^{5} + \frac{1}{10} a^{3} - \frac{1}{5} a$, $\frac{1}{20} a^{14} + \frac{7}{20} a^{10} + \frac{7}{20} a^{8} + \frac{1}{4} a^{6} - \frac{3}{20} a^{4} + \frac{1}{5} a^{2} + \frac{2}{5}$, $\frac{1}{20} a^{15} + \frac{7}{20} a^{11} + \frac{7}{20} a^{9} + \frac{1}{4} a^{7} - \frac{3}{20} a^{5} + \frac{1}{5} a^{3} + \frac{2}{5} a$, $\frac{1}{200} a^{16} - \frac{1}{100} a^{14} - \frac{9}{200} a^{12} + \frac{49}{200} a^{10} - \frac{77}{200} a^{8} - \frac{37}{200} a^{6} + \frac{47}{100} a^{4} + \frac{3}{25} a^{2} - \frac{8}{25}$, $\frac{1}{200} a^{17} - \frac{1}{100} a^{15} - \frac{9}{200} a^{13} + \frac{49}{200} a^{11} - \frac{77}{200} a^{9} - \frac{37}{200} a^{7} + \frac{47}{100} a^{5} + \frac{3}{25} a^{3} - \frac{8}{25} a$, $\frac{1}{400} a^{18} + \frac{7}{400} a^{14} - \frac{9}{400} a^{12} + \frac{1}{400} a^{10} + \frac{29}{400} a^{8} - \frac{1}{10} a^{6} + \frac{7}{25} a^{4} + \frac{3}{50} a^{2} + \frac{7}{25}$, $\frac{1}{400} a^{19} + \frac{7}{400} a^{15} - \frac{9}{400} a^{13} + \frac{1}{400} a^{11} + \frac{29}{400} a^{9} - \frac{1}{10} a^{7} + \frac{7}{25} a^{5} + \frac{3}{50} a^{3} + \frac{7}{25} a$
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: | \( 18027.0158831 \) (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 t20n804 are not computed |
| Character table for t20n804 is not computed |
Intermediate fields
| 5.3.5783.1, 10.2.836077225.1 |
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 | ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ | R | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.10.10.12 | $x^{10} - 11 x^{8} + 54 x^{6} - 10 x^{4} + 9 x^{2} - 11$ | $2$ | $5$ | $10$ | $C_2 \times (C_2^4 : C_5)$ | $[2, 2, 2, 2, 2]^{5}$ |
| 2.10.15.14 | $x^{10} - 6 x^{8} + 184 x^{6} - 16 x^{4} - 752 x^{2} + 1184$ | $2$ | $5$ | $15$ | $C_2 \times (C_2^4 : C_5)$ | $[2, 2, 2, 2, 3]^{5}$ | |
| $5$ | 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 5.12.0.1 | $x^{12} - x^{3} - 2 x + 3$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
| 5783 | Data not computed | ||||||