Normalized defining polynomial
\( x^{20} - 10 x^{19} + 23 x^{18} + 78 x^{17} - 374 x^{16} - 68 x^{15} + 2157 x^{14} - 1261 x^{13} - 6828 x^{12} + 6089 x^{11} + 13699 x^{10} - 13072 x^{9} - 18747 x^{8} + 14875 x^{7} + 17766 x^{6} - 8040 x^{5} - 10656 x^{4} + 587 x^{3} + 2930 x^{2} + 851 x + 59 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[20, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(1098857225002750228881347147041=23\cdot 31^{2}\cdot 83\cdot 709\cdot 1877^{2}\cdot 1933^{2}\cdot 8011^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $31.77$ | 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: | $23, 31, 83, 709, 1877, 1933, 8011$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{23} a^{19} + \frac{2}{23} a^{18} + \frac{1}{23} a^{17} - \frac{2}{23} a^{16} - \frac{7}{23} a^{15} + \frac{9}{23} a^{14} + \frac{11}{23} a^{13} - \frac{2}{23} a^{12} + \frac{2}{23} a^{11} - \frac{5}{23} a^{10} - \frac{8}{23} a^{8} - \frac{6}{23} a^{7} - \frac{9}{23} a^{6} - \frac{6}{23} a^{5} + \frac{7}{23} a^{4} + \frac{8}{23} a^{3} - \frac{7}{23} a^{2} - \frac{6}{23} a - \frac{3}{23}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $19$ | 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: | \( 304104672.061 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 3715891200 |
| The 481 conjugacy class representatives for t20n1110 are not computed |
| Character table for t20n1110 is not computed |
Intermediate fields
| 10.10.901040998181.1 |
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 | ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ | $16{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ | ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | R | ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ | R | ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ | $20$ | ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $23$ | 23.2.1.1 | $x^{2} - 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 23.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 23.6.0.1 | $x^{6} - x + 15$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 23.6.0.1 | $x^{6} - x + 15$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| $31$ | 31.4.2.1 | $x^{4} + 713 x^{2} + 138384$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 31.4.0.1 | $x^{4} - 2 x + 17$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 31.4.0.1 | $x^{4} - 2 x + 17$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 31.8.0.1 | $x^{8} - x + 22$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| $83$ | 83.2.1.2 | $x^{2} + 249$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 83.9.0.1 | $x^{9} - x + 7$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
| 83.9.0.1 | $x^{9} - x + 7$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
| 709 | Data not computed | ||||||
| 1877 | Data not computed | ||||||
| 1933 | Data not computed | ||||||
| 8011 | Data not computed | ||||||