Normalized defining polynomial
\( x^{20} - 10 x^{19} + 52 x^{18} - 183 x^{17} + 482 x^{16} - 1000 x^{15} + 1682 x^{14} - 2332 x^{13} + 2684 x^{12} - 2558 x^{11} + 1984 x^{10} - 1192 x^{9} + 468 x^{8} + 2 x^{7} - 199 x^{6} + 212 x^{5} - 141 x^{4} + 63 x^{3} - 14 x^{2} - x + 1 \)
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: | \(2120282838067894597861328125=5^{10}\cdot 11\cdot 5471^{4}\cdot 22031\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $23.24$ | 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: | $5, 11, 5471, 22031$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{1129} a^{18} - \frac{9}{1129} a^{17} + \frac{80}{1129} a^{16} - \frac{436}{1129} a^{15} - \frac{381}{1129} a^{14} + \frac{551}{1129} a^{13} + \frac{555}{1129} a^{12} + \frac{546}{1129} a^{11} + \frac{56}{1129} a^{10} - \frac{364}{1129} a^{9} + \frac{305}{1129} a^{8} + \frac{322}{1129} a^{7} - \frac{344}{1129} a^{6} + \frac{282}{1129} a^{5} - \frac{226}{1129} a^{4} + \frac{259}{1129} a^{3} - \frac{341}{1129} a^{2} + \frac{273}{1129} a + \frac{61}{1129}$, $\frac{1}{1129} a^{19} - \frac{1}{1129} a^{17} + \frac{284}{1129} a^{16} + \frac{211}{1129} a^{15} + \frac{509}{1129} a^{14} - \frac{131}{1129} a^{13} - \frac{104}{1129} a^{12} + \frac{454}{1129} a^{11} + \frac{140}{1129} a^{10} + \frac{416}{1129} a^{9} - \frac{320}{1129} a^{8} + \frac{296}{1129} a^{7} - \frac{556}{1129} a^{6} + \frac{54}{1129} a^{5} + \frac{483}{1129} a^{4} - \frac{268}{1129} a^{3} - \frac{538}{1129} a^{2} + \frac{260}{1129} a + \frac{549}{1129}$
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: | \( 404828.553737 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 204800 |
| The 230 conjugacy class representatives for t20n876 are not computed |
| Character table for t20n876 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 10.6.93537003125.1 |
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 | $20$ | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ | R | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ | ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $11$ | 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.2.1.1 | $x^{2} - 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.5.0.1 | $x^{5} + x^{2} - x + 5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 11.5.0.1 | $x^{5} + x^{2} - x + 5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 5471 | Data not computed | ||||||
| 22031 | Data not computed | ||||||