Normalized defining polynomial
\( x^{20} - 6 x^{19} + 19 x^{18} - 31 x^{17} + 40 x^{16} - 286 x^{15} + 410 x^{14} - 326 x^{13} - 7337 x^{12} + 3281 x^{11} - 381 x^{10} + 7101 x^{9} + 40779 x^{8} + 16602 x^{7} - 46928 x^{6} - 216915 x^{5} + 92851 x^{4} + 347098 x^{3} - 172510 x^{2} - 139152 x + 77411 \)
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: | \(94952935569355941136243902587890625=5^{14}\cdot 11^{5}\cdot 9931^{5}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $56.09$ | 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: | $5, 11, 9931$ | 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}{2085863228620200181373718132917563929866568844734691999823} a^{19} + \frac{890531469824810596513602287459426439933386136545465713860}{2085863228620200181373718132917563929866568844734691999823} a^{18} + \frac{5119818652248871136420368535132486020490071990499752943}{2085863228620200181373718132917563929866568844734691999823} a^{17} + \frac{1026752665749908305521064848738954568385674900910136606054}{2085863228620200181373718132917563929866568844734691999823} a^{16} + \frac{33429508505080540839421881999966948244860148353758779412}{2085863228620200181373718132917563929866568844734691999823} a^{15} - \frac{928897178213710888818214821950852669667819637646464524084}{2085863228620200181373718132917563929866568844734691999823} a^{14} - \frac{454461456843182797327007366810586293959063055679312278928}{2085863228620200181373718132917563929866568844734691999823} a^{13} + \frac{967431826048876410022833349659793919394652012278605906312}{2085863228620200181373718132917563929866568844734691999823} a^{12} + \frac{825237292297393462019561731076635781508490853659171304716}{2085863228620200181373718132917563929866568844734691999823} a^{11} - \frac{330531355079032220153664941323730002839764330804687072086}{2085863228620200181373718132917563929866568844734691999823} a^{10} - \frac{768466061211554183975560061284148721524163162405254686095}{2085863228620200181373718132917563929866568844734691999823} a^{9} - \frac{359134062613499500271147361372022490012146194084905879735}{2085863228620200181373718132917563929866568844734691999823} a^{8} + \frac{951623582802592049069509448460772151513683076984098579039}{2085863228620200181373718132917563929866568844734691999823} a^{7} + \frac{600907945492935649616225224336362400073859435101013033800}{2085863228620200181373718132917563929866568844734691999823} a^{6} + \frac{785105411544429723541151774616814517408370066991651326744}{2085863228620200181373718132917563929866568844734691999823} a^{5} - \frac{768226509443210477302327674577985311795033506808619135532}{2085863228620200181373718132917563929866568844734691999823} a^{4} - \frac{522949426656426490995549185242365719689717181866303023818}{2085863228620200181373718132917563929866568844734691999823} a^{3} + \frac{939152082962218660967936348621381889520676346735585827629}{2085863228620200181373718132917563929866568844734691999823} a^{2} + \frac{671090227277699947402141855392377945841582670976583856942}{2085863228620200181373718132917563929866568844734691999823} a - \frac{290955343187668151740848574562949831749185943463538869137}{2085863228620200181373718132917563929866568844734691999823}$
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: | \( 8666325568.83 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 7372800 |
| The 324 conjugacy class representatives for t20n1023 are not computed |
| Character table for t20n1023 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 10.10.932312193828125.1 |
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 | ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ | $20$ | R | $20$ | R | $20$ | ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | $20$ | ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ | ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | $20$ | ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 5.12.10.1 | $x^{12} + 6 x^{11} + 27 x^{10} + 80 x^{9} + 195 x^{8} + 366 x^{7} + 571 x^{6} + 702 x^{5} + 1005 x^{4} + 1140 x^{3} + 357 x^{2} - 138 x + 44$ | $6$ | $2$ | $10$ | $D_6$ | $[\ ]_{6}^{2}$ | |
| $11$ | 11.2.1.2 | $x^{2} + 33$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 11.8.4.1 | $x^{8} + 484 x^{4} - 1331 x^{2} + 58564$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 11.10.0.1 | $x^{10} + x^{2} - x + 6$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
| 9931 | Data not computed | ||||||