Normalized defining polynomial
\( x^{20} - 10 x^{19} - 9 x^{18} + 360 x^{17} - 507 x^{16} - 5094 x^{15} + 11686 x^{14} + 36343 x^{13} - 105960 x^{12} - 139153 x^{11} + 502333 x^{10} + 274440 x^{9} - 1331717 x^{8} - 195165 x^{7} + 1954334 x^{6} - 179722 x^{5} - 1451928 x^{4} + 359345 x^{3} + 409455 x^{2} - 140766 x + 1777 \)
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: | \(226737880652372036160972485949760873=19^{8}\cdot 43^{8}\cdot 193^{2}\cdot 313^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $58.58$ | 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: | $19, 43, 193, 313$ | 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}{3213454625121754892846181657940318359525599} a^{19} - \frac{602943621625436299792049747193816742518811}{3213454625121754892846181657940318359525599} a^{18} + \frac{1267448641822151641802239396713840436463831}{3213454625121754892846181657940318359525599} a^{17} - \frac{646201286284444119378853533111669105200523}{3213454625121754892846181657940318359525599} a^{16} + \frac{301871176907926833033497399514383551624141}{3213454625121754892846181657940318359525599} a^{15} - \frac{1522134234113452757201113909678319707189459}{3213454625121754892846181657940318359525599} a^{14} - \frac{197341257683787578758822181611911176006681}{3213454625121754892846181657940318359525599} a^{13} - \frac{1317190719964326676369289346595464967775723}{3213454625121754892846181657940318359525599} a^{12} - \frac{1447301869864256999399402864367023743251831}{3213454625121754892846181657940318359525599} a^{11} + \frac{397383948210629866911541246627033142529049}{3213454625121754892846181657940318359525599} a^{10} + \frac{743416124183370139396513111214431083851234}{3213454625121754892846181657940318359525599} a^{9} + \frac{58003559415245365571181662312421973813643}{3213454625121754892846181657940318359525599} a^{8} - \frac{426557320793851132950008148281459900284161}{3213454625121754892846181657940318359525599} a^{7} + \frac{332683229916930080178523887417951569227995}{3213454625121754892846181657940318359525599} a^{6} - \frac{630705497819550247848307607989746706509271}{3213454625121754892846181657940318359525599} a^{5} - \frac{1535764173473208464553760602570145419122270}{3213454625121754892846181657940318359525599} a^{4} + \frac{689891721490420605084741180231456787856845}{3213454625121754892846181657940318359525599} a^{3} - \frac{643336263468342298632852631647592384515908}{3213454625121754892846181657940318359525599} a^{2} - \frac{890921695934308502908824108708069058955835}{3213454625121754892846181657940318359525599} a - \frac{97808059460332409623491331170613299661291}{3213454625121754892846181657940318359525599}$
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: | \( 129217871434 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 10240 |
| The 208 conjugacy class representatives for t20n418 are not computed |
| Character table for t20n418 is not computed |
Intermediate fields
| 5.5.667489.1, 10.10.139454509882873.1 |
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 | ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ | ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ | $20$ | R | $20$ | ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ | $20$ | ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ | R | $20$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$ |
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 |
|---|---|---|---|---|---|---|---|
| $19$ | 19.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 19.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 19.8.4.1 | $x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 19.8.4.1 | $x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $43$ | 43.2.1.2 | $x^{2} + 387$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 43.2.1.2 | $x^{2} + 387$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 43.2.1.2 | $x^{2} + 387$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 43.2.1.2 | $x^{2} + 387$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 43.4.0.1 | $x^{4} - x + 20$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 43.8.4.1 | $x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $193$ | 193.4.2.2 | $x^{4} - 193 x^{2} + 186245$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 193.8.0.1 | $x^{8} - x + 5$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 193.8.0.1 | $x^{8} - x + 5$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 313 | Data not computed | ||||||