Normalized defining polynomial
\( x^{20} - 10 x^{19} + 31 x^{18} + 6 x^{17} - 212 x^{16} + 268 x^{15} + 486 x^{14} - 1156 x^{13} - 336 x^{12} + 2276 x^{11} - 592 x^{10} - 2546 x^{9} + 1756 x^{8} + 1626 x^{7} - 2150 x^{6} - 475 x^{5} + 1385 x^{4} - 43 x^{3} - 400 x^{2} + 85 x + 71 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[8, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(6329352282423861537868725791015625=3^{2}\cdot 5^{10}\cdot 19^{8}\cdot 29^{6}\cdot 7128529\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $48.99$ | 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: | $3, 5, 19, 29, 7128529$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{939} a^{18} - \frac{3}{313} a^{17} - \frac{52}{313} a^{16} - \frac{142}{313} a^{15} - \frac{101}{939} a^{14} - \frac{64}{939} a^{13} - \frac{380}{939} a^{12} + \frac{466}{939} a^{11} + \frac{54}{313} a^{10} + \frac{244}{939} a^{9} - \frac{25}{313} a^{8} - \frac{14}{313} a^{7} + \frac{40}{939} a^{6} - \frac{248}{939} a^{5} - \frac{128}{939} a^{4} + \frac{347}{939} a^{3} + \frac{34}{313} a^{2} + \frac{89}{313} a - \frac{448}{939}$, $\frac{1}{939} a^{19} - \frac{79}{313} a^{17} + \frac{16}{313} a^{16} - \frac{179}{939} a^{15} - \frac{34}{939} a^{14} - \frac{17}{939} a^{13} - \frac{137}{939} a^{12} - \frac{113}{313} a^{11} - \frac{176}{939} a^{10} + \frac{81}{313} a^{9} + \frac{74}{313} a^{8} - \frac{338}{939} a^{7} + \frac{112}{939} a^{6} + \frac{457}{939} a^{5} + \frac{134}{939} a^{4} + \frac{136}{313} a^{3} + \frac{82}{313} a^{2} + \frac{77}{939} a - \frac{92}{313}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $13$ | 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: | \( 2621433624.36 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 819200 |
| The 275 conjugacy class representatives for t20n955 are not computed |
| Character table for t20n955 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 10.6.9932496465625.1 |
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 | $16{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }$ | R | R | ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ | $16{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ | R | ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ | ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||
| 5 | Data not computed | ||||||
| $19$ | $\Q_{19}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{19}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 19.5.4.1 | $x^{5} - 19$ | $5$ | $1$ | $4$ | $D_{5}$ | $[\ ]_{5}^{2}$ | |
| 19.5.4.1 | $x^{5} - 19$ | $5$ | $1$ | $4$ | $D_{5}$ | $[\ ]_{5}^{2}$ | |
| 19.8.0.1 | $x^{8} - x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| $29$ | $\Q_{29}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{29}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 29.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.4.0.1 | $x^{4} - x + 19$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.4.0.1 | $x^{4} - x + 19$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.8.6.2 | $x^{8} + 145 x^{4} + 7569$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 7128529 | Data not computed | ||||||