Normalized defining polynomial
\( x^{22} + 40 x^{20} - 4821 x^{18} - 209496 x^{16} + 5285400 x^{14} + 302815992 x^{12} + 922756032 x^{10} - 93270702624 x^{8} - 1132846722171 x^{6} - 2066536069936 x^{4} + 13681132452239 x^{2} + 35545793447880 \)
Invariants
| Degree: | $22$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[8, 7]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-58418948490087900856198784513049139500595201522971777317979130633702859321491007685912328702163396402872320=-\,2^{71}\cdot 3^{21}\cdot 5\cdot 337^{8}\cdot 1543\cdot 310501^{8}\cdot 21330377\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $71{,}289.43$ | 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: | $2, 3, 5, 337, 1543, 310501, 21330377$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2} a^{17} - \frac{1}{2} a$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{3}$, $\frac{1}{57887320607168480035725746532474501690526662399435613406516} a^{20} - \frac{2950442336895763005424408194123522436283571976756718491245}{57887320607168480035725746532474501690526662399435613406516} a^{18} + \frac{6821522541564658735179692173709612637676064494172081846563}{14471830151792120008931436633118625422631665599858903351629} a^{16} + \frac{5832168576445284563241433903534476907436776464631869840549}{14471830151792120008931436633118625422631665599858903351629} a^{14} + \frac{2016907842336175271833927684427491039808942727709923502565}{14471830151792120008931436633118625422631665599858903351629} a^{12} + \frac{3616447482838561289850993661292656391784625249508377496223}{14471830151792120008931436633118625422631665599858903351629} a^{10} + \frac{6255656531791301601189048680870162671143898472737040777170}{14471830151792120008931436633118625422631665599858903351629} a^{8} + \frac{3372674489032727054476127874717180164365285067760898039858}{14471830151792120008931436633118625422631665599858903351629} a^{6} + \frac{8444471643867987534224861075922752542722607729758487195345}{57887320607168480035725746532474501690526662399435613406516} a^{4} + \frac{14777777014163413047753720120436553972162792194434447510719}{57887320607168480035725746532474501690526662399435613406516} a^{2} - \frac{107757367164295089434167702904843619129541955094180993672}{14471830151792120008931436633118625422631665599858903351629}$, $\frac{1}{173661961821505440107177239597423505071579987198306840219548} a^{21} + \frac{25993217966688477012438465072113728408979759222961088212013}{173661961821505440107177239597423505071579987198306840219548} a^{19} - \frac{276261689554267512857350761899800049093178870504913219501}{28943660303584240017862873266237250845263331199717806703258} a^{17} + \frac{1944056192148428187747144634511492302478925488210623280183}{14471830151792120008931436633118625422631665599858903351629} a^{15} - \frac{4151640769818648245699169649563711460940907624049659949688}{14471830151792120008931436633118625422631665599858903351629} a^{13} + \frac{1205482494279520429950331220430885463928208416502792498741}{14471830151792120008931436633118625422631665599858903351629} a^{11} + \frac{6909162227861140536706828437996262697925188024198648042933}{14471830151792120008931436633118625422631665599858903351629} a^{9} + \frac{5948168213608282354469188169278601862332316889206600463829}{14471830151792120008931436633118625422631665599858903351629} a^{7} + \frac{22110597417012155856650202536132418077749756709731366867287}{57887320607168480035725746532474501690526662399435613406516} a^{5} - \frac{72053203896589307005834899678275198563627201404718972599055}{173661961821505440107177239597423505071579987198306840219548} a^{3} - \frac{14687344886120710187799772038928312660890749510047265338973}{86830980910752720053588619798711752535789993599153420109774} a$
Class group and class number
Not computed
Unit group
| Rank: | $14$ | 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: | Not computed | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | Not computed | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 16220160 |
| The 104 conjugacy class representatives for t22n44 are not computed |
| Character table for t22n44 is not computed |
Intermediate fields
| 11.11.118769262421915560193703211428553337469927424.1 |
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 | R | R | R | $22$ | ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ | $16{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ | ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ | ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ | ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| $3$ | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 3.9.10.1 | $x^{9} + 3 x^{2} + 3$ | $9$ | $1$ | $10$ | $C_3^2:Q_8$ | $[5/4, 5/4]_{4}^{2}$ | |
| 3.9.10.1 | $x^{9} + 3 x^{2} + 3$ | $9$ | $1$ | $10$ | $C_3^2:Q_8$ | $[5/4, 5/4]_{4}^{2}$ | |
| $5$ | 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 5.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 5.8.0.1 | $x^{8} + x^{2} - 2 x + 3$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 5.8.0.1 | $x^{8} + x^{2} - 2 x + 3$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 337 | Data not computed | ||||||
| 1543 | Data not computed | ||||||
| 310501 | Data not computed | ||||||
| 21330377 | Data not computed | ||||||