Normalized defining polynomial
\( x^{20} + 24 x^{18} + 234 x^{16} + 1198 x^{14} + 3467 x^{12} + 5684 x^{10} + 5047 x^{8} + 2260 x^{6} + 467 x^{4} + 38 x^{2} + 1 \)
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: | \(39983356593077948293054464=2^{20}\cdot 3^{10}\cdot 71^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $19.06$ | 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: | $2, 3, 71$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{2} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{14} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{15} + \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{4} a^{16} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{17} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{4} a$, $\frac{1}{5356} a^{18} + \frac{285}{5356} a^{16} - \frac{365}{5356} a^{14} - \frac{337}{5356} a^{12} - \frac{133}{5356} a^{10} + \frac{33}{412} a^{8} - \frac{2155}{5356} a^{6} - \frac{1}{2} a^{5} + \frac{423}{2678} a^{4} - \frac{45}{103} a^{2} - \frac{1}{2} a + \frac{1221}{5356}$, $\frac{1}{5356} a^{19} + \frac{285}{5356} a^{17} - \frac{365}{5356} a^{15} - \frac{337}{5356} a^{13} - \frac{133}{5356} a^{11} + \frac{33}{412} a^{9} - \frac{2155}{5356} a^{7} - \frac{1}{2} a^{6} + \frac{423}{2678} a^{5} - \frac{45}{103} a^{3} - \frac{1}{2} a^{2} + \frac{1221}{5356} a$
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: | \( -\frac{487}{5356} a^{19} - \frac{93}{412} a^{18} - \frac{5795}{2678} a^{17} - \frac{2197}{412} a^{16} - \frac{111469}{5356} a^{15} - \frac{10477}{206} a^{14} - \frac{278801}{2678} a^{13} - \frac{51949}{206} a^{12} - \frac{1547385}{5356} a^{11} - \frac{286125}{412} a^{10} - \frac{44883}{103} a^{9} - \frac{431297}{412} a^{8} - \frac{1711533}{5356} a^{7} - \frac{163421}{206} a^{6} - \frac{434765}{5356} a^{5} - \frac{105561}{412} a^{4} + \frac{2891}{412} a^{3} - \frac{5211}{206} a^{2} + \frac{7309}{2678} a + \frac{14}{103} \) (order $12$) | 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: | \( 264102.546071 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_{10}\times D_5$ (as 20T24):
| A solvable group of order 100 |
| The 40 conjugacy class representatives for $C_{10}\times D_5$ |
| Character table for $C_{10}\times D_5$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\zeta_{12})\), 10.10.6323239406592.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 20 sibling: | data not computed |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{10}$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| 3 | Data not computed | ||||||
| $71$ | 71.10.0.1 | $x^{10} - x + 22$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |
| 71.10.8.2 | $x^{10} - 71 x^{5} + 55451$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ | |