Normalized defining polynomial
\( x^{22} - 2 x^{21} + 7 x^{20} - 20 x^{19} + 38 x^{18} - 74 x^{17} + 107 x^{16} - 167 x^{15} + 185 x^{14} - 284 x^{13} + 308 x^{12} - 417 x^{11} + 466 x^{10} - 462 x^{9} + 446 x^{8} - 292 x^{7} + 170 x^{6} - 55 x^{5} - 7 x^{4} + 11 x^{3} - 8 x^{2} + x + 1 \)
Invariants
| Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
| Signature: | $[4, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
| Discriminant: | \(-1533107758369291702785569375\)\(\medspace = -\,5^{4}\cdot 7^{4}\cdot 83^{5}\cdot 127^{4}\cdot 997\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $17.21$ | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
| |
| Ramified primes: | $5, 7, 83, 127, 997$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
| |
| $|\Aut(K/\Q)|$: | $2$ | ||
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{49167961677028904155} a^{21} - \frac{2461309068973386848}{49167961677028904155} a^{20} + \frac{4533706425681003074}{9833592335405780831} a^{19} + \frac{171370386362851268}{9833592335405780831} a^{18} - \frac{13398709126581984782}{49167961677028904155} a^{17} - \frac{16445471766014570587}{49167961677028904155} a^{16} + \frac{4208835173944438599}{49167961677028904155} a^{15} - \frac{17596714550259048316}{49167961677028904155} a^{14} + \frac{7733008968402678216}{49167961677028904155} a^{13} - \frac{3945031426006518546}{9833592335405780831} a^{12} + \frac{6332201538439407758}{49167961677028904155} a^{11} + \frac{819246935174649052}{9833592335405780831} a^{10} + \frac{10718725144472687101}{49167961677028904155} a^{9} + \frac{7836926667780161432}{49167961677028904155} a^{8} - \frac{18072120242673022621}{49167961677028904155} a^{7} - \frac{23104295534366604726}{49167961677028904155} a^{6} - \frac{2476415384533957679}{49167961677028904155} a^{5} - \frac{11774754108422869416}{49167961677028904155} a^{4} + \frac{24116653514121259529}{49167961677028904155} a^{3} - \frac{4163388242535411363}{49167961677028904155} a^{2} - \frac{3811034455186373428}{9833592335405780831} a + \frac{9090439958337064706}{49167961677028904155}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $12$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
| |
| Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
| |
| 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) | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
| |
| Regulator: | \( 56909.7191275 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
| A non-solvable group of order 40874803200 |
| The 400 conjugacy class representatives for t22n52 are not computed |
| Character table for t22n52 is not computed |
Intermediate fields
| 11.3.136113034225.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 44 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 | ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ | R | R | $22$ | $22$ | ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ | ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.11.0.1}{11} }^{2}$ | $22$ | ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/41.14.0.1}{14} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ | ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ | $16{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 5.12.0.1 | $x^{12} - x^{3} - 2 x + 3$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
| 7 | Data not computed | ||||||
| 83 | Data not computed | ||||||
| 127 | Data not computed | ||||||
| 997 | Data not computed | ||||||