Normalized defining polynomial
\( x^{22} - x^{21} - x^{19} + 3 x^{18} - 3 x^{17} - x^{16} + 5 x^{14} - 3 x^{13} - 2 x^{12} + x^{11} + 6 x^{10} - 4 x^{9} - 2 x^{8} - x^{7} + 5 x^{6} - 2 x^{5} - x^{4} - 2 x^{3} + x^{2} + x + 1 \)
Invariants
| Degree: | $22$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 11]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-6293360649840402636051867=-\,3^{11}\cdot 64661^{2}\cdot 92179^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $13.40$ | 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: | $3, 64661, 92179$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{5981} a^{21} - \frac{1485}{5981} a^{20} + \frac{2732}{5981} a^{19} + \frac{829}{5981} a^{18} + \frac{1853}{5981} a^{17} + \frac{1405}{5981} a^{16} + \frac{2348}{5981} a^{15} + \frac{2491}{5981} a^{14} - \frac{381}{5981} a^{13} - \frac{2794}{5981} a^{12} + \frac{1461}{5981} a^{11} + \frac{2980}{5981} a^{10} - \frac{2355}{5981} a^{9} + \frac{1912}{5981} a^{8} - \frac{2416}{5981} a^{7} + \frac{2724}{5981} a^{6} + \frac{745}{5981} a^{5} + \frac{903}{5981} a^{4} - \frac{309}{5981} a^{3} - \frac{1983}{5981} a^{2} + \frac{121}{5981} a - \frac{133}{5981}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $10$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{975}{5981} a^{21} - \frac{473}{5981} a^{20} + \frac{2155}{5981} a^{19} - \frac{5141}{5981} a^{18} + \frac{6394}{5981} a^{17} - \frac{5755}{5981} a^{16} + \frac{4558}{5981} a^{15} - \frac{11523}{5981} a^{14} + \frac{11309}{5981} a^{13} - \frac{2795}{5981} a^{12} + \frac{997}{5981} a^{11} - \frac{7247}{5981} a^{10} + \frac{12541}{5981} a^{9} - \frac{1872}{5981} a^{8} + \frac{914}{5981} a^{7} - \frac{11626}{5981} a^{6} + \frac{14636}{5981} a^{5} - \frac{4763}{5981} a^{4} + \frac{3756}{5981} a^{3} - \frac{7543}{5981} a^{2} + \frac{4336}{5981} a + \frac{1907}{5981} \) (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: | \( 4402.38521796 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 79833600 |
| The 112 conjugacy class representatives for t22n47 are not computed |
| Character table for t22n47 is not computed |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 11.1.5960386319.1 |
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 | $22$ | R | ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ | $22$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ | $18{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }$ | ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/41.14.0.1}{14} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ | ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ | $22$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.6.3.2 | $x^{6} - 9 x^{2} + 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 64661 | Data not computed | ||||||
| 92179 | Data not computed | ||||||