Normalized defining polynomial
\( x^{20} - 3 x^{19} - x^{18} + 15 x^{17} - 17 x^{16} - 16 x^{15} + 56 x^{14} - 48 x^{13} - 12 x^{12} + 77 x^{11} - 107 x^{10} + 87 x^{9} - 8 x^{8} - 109 x^{7} + 201 x^{6} - 214 x^{5} + 158 x^{4} - 84 x^{3} + 32 x^{2} - 8 x + 1 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 10]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(6092127302465063639061=3^{10}\cdot 73^{2}\cdot 13921^{2}\cdot 99901\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $12.28$ | 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: | $3, 73, 13921, 99901$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{32023} a^{19} + \frac{5317}{32023} a^{18} + \frac{10130}{32023} a^{17} - \frac{3094}{32023} a^{16} - \frac{275}{32023} a^{15} + \frac{10042}{32023} a^{14} + \frac{9132}{32023} a^{13} + \frac{3301}{32023} a^{12} + \frac{12704}{32023} a^{11} - \frac{15196}{32023} a^{10} + \frac{15248}{32023} a^{9} + \frac{5188}{32023} a^{8} - \frac{3674}{32023} a^{7} - \frac{11759}{32023} a^{6} + \frac{15263}{32023} a^{5} - \frac{11382}{32023} a^{4} + \frac{3411}{32023} a^{3} - \frac{10605}{32023} a^{2} + \frac{5958}{32023} a - \frac{6218}{32023}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $9$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{11098}{1033} a^{19} - \frac{26851}{1033} a^{18} - \frac{29640}{1033} a^{17} + \frac{154658}{1033} a^{16} - \frac{87240}{1033} a^{15} - \frac{263537}{1033} a^{14} + \frac{468288}{1033} a^{13} - \frac{183688}{1033} a^{12} - \frac{317144}{1033} a^{11} + \frac{656261}{1033} a^{10} - \frac{718592}{1033} a^{9} + \frac{434996}{1033} a^{8} + \frac{270137}{1033} a^{7} - \frac{1089208}{1033} a^{6} + \frac{1490119}{1033} a^{5} - \frac{1282083}{1033} a^{4} + \frac{771611}{1033} a^{3} - \frac{324830}{1033} a^{2} + \frac{90458}{1033} a - \frac{12261}{1033} \) (order $6$) | 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: | \( 870.501962188 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 29491200 |
| The 702 conjugacy class representatives for t20n1045 are not computed |
| Character table for t20n1045 is not computed |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 10.0.246944619.1 |
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 | $20$ | R | $16{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ | ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ | $20$ | ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ | ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ | ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.8.0.1}{8} }$ | ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$ |
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 | ||||||
| $73$ | $\Q_{73}$ | $x + 5$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{73}$ | $x + 5$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 73.2.0.1 | $x^{2} - x + 11$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 73.4.0.1 | $x^{4} - x + 13$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 73.4.0.1 | $x^{4} - x + 13$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 73.4.0.1 | $x^{4} - x + 13$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 73.4.2.1 | $x^{4} + 1533 x^{2} + 644809$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13921 | Data not computed | ||||||
| 99901 | Data not computed | ||||||