Normalized defining polynomial
\( x^{20} + 2 x^{18} - 15 x^{16} - 18 x^{14} + 63 x^{12} + 33 x^{10} - 71 x^{8} - 6 x^{6} - 12 x^{4} - x^{2} + 1 \)
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: | \(24868623129665465017517056=2^{10}\cdot 11^{16}\cdot 727^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $18.61$ | 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: | $2, 11, 727$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{11} a^{12} - \frac{1}{11} a^{10} - \frac{2}{11} a^{8} - \frac{5}{11} a^{6} + \frac{4}{11} a^{4} - \frac{4}{11} a^{2} + \frac{3}{11}$, $\frac{1}{11} a^{13} - \frac{1}{11} a^{11} - \frac{2}{11} a^{9} - \frac{5}{11} a^{7} + \frac{4}{11} a^{5} - \frac{4}{11} a^{3} + \frac{3}{11} a$, $\frac{1}{11} a^{14} - \frac{3}{11} a^{10} + \frac{4}{11} a^{8} - \frac{1}{11} a^{6} - \frac{1}{11} a^{2} + \frac{3}{11}$, $\frac{1}{22} a^{15} - \frac{1}{22} a^{12} - \frac{3}{22} a^{11} + \frac{1}{22} a^{10} - \frac{7}{22} a^{9} - \frac{9}{22} a^{8} - \frac{1}{22} a^{7} - \frac{3}{11} a^{6} - \frac{1}{2} a^{5} - \frac{2}{11} a^{4} + \frac{5}{11} a^{3} + \frac{2}{11} a^{2} - \frac{4}{11} a - \frac{3}{22}$, $\frac{1}{22} a^{16} - \frac{1}{22} a^{13} - \frac{1}{22} a^{12} + \frac{1}{22} a^{11} - \frac{9}{22} a^{10} - \frac{9}{22} a^{9} - \frac{5}{22} a^{8} - \frac{3}{11} a^{7} + \frac{1}{22} a^{6} - \frac{2}{11} a^{5} - \frac{2}{11} a^{4} + \frac{2}{11} a^{3} + \frac{3}{11} a^{2} - \frac{3}{22} a + \frac{3}{11}$, $\frac{1}{22} a^{17} - \frac{1}{22} a^{14} - \frac{1}{22} a^{13} - \frac{1}{22} a^{12} - \frac{9}{22} a^{11} - \frac{7}{22} a^{10} - \frac{5}{22} a^{9} - \frac{1}{11} a^{8} + \frac{1}{22} a^{7} + \frac{3}{11} a^{6} - \frac{2}{11} a^{5} - \frac{2}{11} a^{4} + \frac{3}{11} a^{3} + \frac{5}{22} a^{2} + \frac{3}{11} a - \frac{3}{11}$, $\frac{1}{2882} a^{18} + \frac{26}{1441} a^{16} - \frac{35}{2882} a^{14} - \frac{1}{22} a^{13} + \frac{3}{131} a^{12} - \frac{5}{11} a^{11} - \frac{480}{1441} a^{10} - \frac{9}{22} a^{9} + \frac{5}{131} a^{8} + \frac{5}{22} a^{7} + \frac{422}{1441} a^{6} + \frac{7}{22} a^{5} - \frac{518}{1441} a^{4} - \frac{7}{22} a^{3} + \frac{32}{1441} a^{2} + \frac{4}{11} a - \frac{993}{2882}$, $\frac{1}{2882} a^{19} + \frac{26}{1441} a^{17} - \frac{35}{2882} a^{15} - \frac{1}{22} a^{14} + \frac{3}{131} a^{13} - \frac{480}{1441} a^{11} + \frac{3}{22} a^{10} + \frac{5}{131} a^{9} + \frac{7}{22} a^{8} + \frac{422}{1441} a^{7} + \frac{1}{22} a^{6} - \frac{518}{1441} a^{5} - \frac{1}{2} a^{4} + \frac{32}{1441} a^{3} - \frac{5}{11} a^{2} - \frac{993}{2882} a + \frac{4}{11}$
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: | \( 109024.27006 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 81920 |
| The 332 conjugacy class representatives for t20n751 are not computed |
| Character table for t20n751 is not computed |
Intermediate fields
| \(\Q(\zeta_{11})^+\), 10.8.155838906487.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 20 siblings: | 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 | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.10.0.1 | $x^{10} - x^{3} + 1$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |
| 2.10.10.4 | $x^{10} - 5 x^{8} + 14 x^{6} - 22 x^{4} + 17 x^{2} - 37$ | $2$ | $5$ | $10$ | $C_2 \times (C_2^4 : C_5)$ | $[2, 2, 2, 2]^{10}$ | |
| $11$ | 11.10.8.5 | $x^{10} - 2321 x^{5} + 2033647$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ |
| 11.10.8.5 | $x^{10} - 2321 x^{5} + 2033647$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ | |
| 727 | Data not computed | ||||||