Normalized defining polynomial
\( x^{20} - 5 x^{19} + 6 x^{18} + 11 x^{17} - 38 x^{16} + 35 x^{15} + 11 x^{14} - 60 x^{13} + 79 x^{12} - 69 x^{11} + 39 x^{10} + 5 x^{9} - 49 x^{8} + 68 x^{7} - 41 x^{6} - 7 x^{5} + 32 x^{4} - 27 x^{3} + 14 x^{2} - 5 x + 1 \)
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: | \(12912557303290376372337=3^{10}\cdot 31^{8}\cdot 256393\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $12.75$ | 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: | $3, 31, 256393$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5} a^{18} - \frac{2}{5} a^{16} + \frac{1}{5} a^{15} - \frac{2}{5} a^{14} + \frac{2}{5} a^{13} + \frac{2}{5} a^{12} - \frac{1}{5} a^{11} - \frac{2}{5} a^{10} - \frac{1}{5} a^{9} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} + \frac{2}{5}$, $\frac{1}{15911965} a^{19} - \frac{136847}{15911965} a^{18} + \frac{1216768}{15911965} a^{17} - \frac{432977}{3182393} a^{16} + \frac{7775941}{15911965} a^{15} - \frac{7212414}{15911965} a^{14} - \frac{749277}{15911965} a^{13} + \frac{445100}{3182393} a^{12} + \frac{718400}{3182393} a^{11} - \frac{3135647}{15911965} a^{10} - \frac{206163}{15911965} a^{9} - \frac{3339087}{15911965} a^{8} - \frac{128747}{3182393} a^{7} - \frac{5018088}{15911965} a^{6} + \frac{7530873}{15911965} a^{5} - \frac{4492241}{15911965} a^{4} + \frac{3481502}{15911965} a^{3} + \frac{7629747}{15911965} a^{2} + \frac{6474087}{15911965} a + \frac{3462046}{15911965}$
Class group and class number
Trivial group, which has order $1$
Unit group
| Rank: | $9$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{20634009}{15911965} a^{19} + \frac{92216443}{15911965} a^{18} - \frac{76063732}{15911965} a^{17} - \frac{52403329}{3182393} a^{16} + \frac{639985441}{15911965} a^{15} - \frac{394952309}{15911965} a^{14} - \frac{399726477}{15911965} a^{13} + \frac{198339279}{3182393} a^{12} - \frac{222805258}{3182393} a^{11} + \frac{897909268}{15911965} a^{10} - \frac{412884583}{15911965} a^{9} - \frac{264095762}{15911965} a^{8} + \frac{169218942}{3182393} a^{7} - \frac{959217043}{15911965} a^{6} + \frac{377794438}{15911965} a^{5} + \frac{287295174}{15911965} a^{4} - \frac{486286058}{15911965} a^{3} + \frac{315645152}{15911965} a^{2} - \frac{156389368}{15911965} a + \frac{51122906}{15911965} \) (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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 1277.33889125 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 51200 |
| The 152 conjugacy class representatives for t20n647 are not computed |
| Character table for t20n647 is not computed |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 10.0.224415603.1 |
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 | ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | $20$ | $20$ |
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 | ||||||
| $31$ | 31.5.0.1 | $x^{5} - x + 10$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |
| 31.5.0.1 | $x^{5} - x + 10$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 31.10.8.2 | $x^{10} - 31 x^{5} + 11532$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ | |
| 256393 | Data not computed | ||||||