Normalized defining polynomial
\( x^{22} - 587 x^{20} + 151074 x^{18} - 22444968 x^{16} + 2131486062 x^{14} - 135179152602 x^{12} + 5798322147168 x^{10} - 166188568269012 x^{8} + 3053662171928622 x^{6} - 32824260622235770 x^{4} + 166011843276804404 x^{2} - 162477537538472376 \)
Invariants
| Degree: | $22$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[22, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(19861680445136995514566695021635141826242824408734364859334714399782912213082611059290887677605650017346912256=2^{71}\cdot 3^{21}\cdot 11^{2}\cdot 337^{8}\cdot 1847\cdot 14387\cdot 17401\cdot 310501^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $92{,}916.24$ | 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: | $2, 3, 11, 337, 1847, 14387, 17401, 310501$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{16}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{16}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{16}$, $\frac{1}{57887320607168480035725746532474501690526662399435613406516} a^{20} - \frac{2950442336895763005424408194123522436283571976756718491815}{57887320607168480035725746532474501690526662399435613406516} a^{18} - \frac{1004302154534722920052499277118458421820145578721665837945}{2631241845780385456169352115112477349569393745428891518478} a^{16} - \frac{643466077911965133265555533337835067610060492455760582310}{14471830151792120008931436633118625422631665599858903351629} a^{14} - \frac{13189478217331914001926526008336601736971377781913671720483}{28943660303584240017862873266237250845263331199717806703258} a^{12} - \frac{9034809236354090268361778797247574508083006775253021989045}{28943660303584240017862873266237250845263331199717806703258} a^{10} - \frac{6166493277942033869131381004148231145943654093755503664616}{14471830151792120008931436633118625422631665599858903351629} a^{8} + \frac{3878884881889995777978015953846472365928244135611399569258}{14471830151792120008931436633118625422631665599858903351629} a^{6} - \frac{6921328388706364494241559168137153061833951129125846707443}{28943660303584240017862873266237250845263331199717806703258} a^{4} + \frac{8917781021277213759491527194457318510806762010851346459499}{28943660303584240017862873266237250845263331199717806703258} a^{2} + \frac{310093089188068675826151051109133234209248850196927082794}{1315620922890192728084676057556238674784696872714445759239}$, $\frac{1}{636760526678853280392983211857219518595793286393791747471676} a^{21} + \frac{54936878270272717030301338338350979254243090422678894914701}{636760526678853280392983211857219518595793286393791747471676} a^{19} + \frac{4258181537026047992286204953106496277318641912136117199011}{28943660303584240017862873266237250845263331199717806703258} a^{17} - \frac{73002616836872565177922738698930962180768388491750277340455}{159190131669713320098245802964304879648948321598447936867919} a^{15} - \frac{13189478217331914001926526008336601736971377781913671720483}{318380263339426640196491605928609759297896643196895873735838} a^{13} - \frac{66922129843522570304087525329722076198609669174688635395561}{318380263339426640196491605928609759297896643196895873735838} a^{11} + \frac{37248997177434326157662928895207645121951342705821206390271}{159190131669713320098245802964304879648948321598447936867919} a^{9} - \frac{25064775421694244239884857312390778479335087064106407134000}{159190131669713320098245802964304879648948321598447936867919} a^{7} - \frac{151639629906627564583555925499323407288150607127714880223733}{318380263339426640196491605928609759297896643196895873735838} a^{5} - \frac{106856860193059746311959965870491684870246562788019880353533}{318380263339426640196491605928609759297896643196895873735838} a^{3} + \frac{4256955857858646860080179223777849258563339468340264360511}{14471830151792120008931436633118625422631665599858903351629} a$
Class group and class number
Not computed
Unit group
| Rank: | $21$ | 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 | ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ | $22$ | R | ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ | ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ | ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ | ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ | $22$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 | ||||||
| $3$ | 3.2.1.1 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 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 | $x^{9} + 3 x^{2} + 3$ | $9$ | $1$ | $10$ | $C_3^2:Q_8$ | $[5/4, 5/4]_{4}^{2}$ | |
| $11$ | 11.2.1.2 | $x^{2} + 33$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 11.2.1.1 | $x^{2} - 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.8.0.1 | $x^{8} + x^{2} - 2 x + 6$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 11.8.0.1 | $x^{8} + x^{2} - 2 x + 6$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 337 | Data not computed | ||||||
| 1847 | Data not computed | ||||||
| 14387 | Data not computed | ||||||
| 17401 | Data not computed | ||||||
| 310501 | Data not computed | ||||||