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Properties

Label 22.10.2107507630...0000.1
Degree $22$
Signature $[10, 6]$
Discriminant $2^{72}\cdot 3^{22}\cdot 5^{10}\cdot 113^{10}\cdot 337^{8}\cdot 310501^{8}\cdot 11155561^{10}$
Root discriminant $680{,}610{,}426.61$
Ramified primes $2, 3, 5, 113, 337, 310501, 11155561$
Class number Not computed
Class group Not computed
Galois group 22T43

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1495722903681177175207908834306985386050749026948136139791810294784018587329257867815333630665502500000000, 0, -593268499597857685637766137050940784311360565636847362570893397339081071862088416668321250000000, 0, 5695692754570027594526557200758139973678062841883039454997024866872864512936560150000000, 0, 880101110859184651857903730474356017770939048207067136486977359706942040000000, 0, -6532269446805556223473454007587119713902772265719196261716243847500000, 0, 495161089736201224478016513692993953268704949758149081450000, 0, 2211187005788657175999069028194179567572695016920000, 0, -447129861314250111523595651626294337559000, 0, -125802676272554404269617160982350, 0, 20767967363363151457665, 0, 1521110995119, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 1521110995119*x^20 + 20767967363363151457665*x^18 - 125802676272554404269617160982350*x^16 - 447129861314250111523595651626294337559000*x^14 + 2211187005788657175999069028194179567572695016920000*x^12 + 495161089736201224478016513692993953268704949758149081450000*x^10 - 6532269446805556223473454007587119713902772265719196261716243847500000*x^8 + 880101110859184651857903730474356017770939048207067136486977359706942040000000*x^6 + 5695692754570027594526557200758139973678062841883039454997024866872864512936560150000000*x^4 - 593268499597857685637766137050940784311360565636847362570893397339081071862088416668321250000000*x^2 - 1495722903681177175207908834306985386050749026948136139791810294784018587329257867815333630665502500000000)
 
gp: K = bnfinit(x^22 + 1521110995119*x^20 + 20767967363363151457665*x^18 - 125802676272554404269617160982350*x^16 - 447129861314250111523595651626294337559000*x^14 + 2211187005788657175999069028194179567572695016920000*x^12 + 495161089736201224478016513692993953268704949758149081450000*x^10 - 6532269446805556223473454007587119713902772265719196261716243847500000*x^8 + 880101110859184651857903730474356017770939048207067136486977359706942040000000*x^6 + 5695692754570027594526557200758139973678062841883039454997024866872864512936560150000000*x^4 - 593268499597857685637766137050940784311360565636847362570893397339081071862088416668321250000000*x^2 - 1495722903681177175207908834306985386050749026948136139791810294784018587329257867815333630665502500000000, 1)
 

Normalized defining polynomial

\( x^{22} + 1521110995119 x^{20} + 20767967363363151457665 x^{18} - 125802676272554404269617160982350 x^{16} - 447129861314250111523595651626294337559000 x^{14} + 2211187005788657175999069028194179567572695016920000 x^{12} + 495161089736201224478016513692993953268704949758149081450000 x^{10} - 6532269446805556223473454007587119713902772265719196261716243847500000 x^{8} + 880101110859184651857903730474356017770939048207067136486977359706942040000000 x^{6} + 5695692754570027594526557200758139973678062841883039454997024866872864512936560150000000 x^{4} - 593268499597857685637766137050940784311360565636847362570893397339081071862088416668321250000000 x^{2} - 1495722903681177175207908834306985386050749026948136139791810294784018587329257867815333630665502500000000 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(210750763041141022987845301990873183151664528018075542264081718322671288099187925306758388876751268561732119068641101552222738530664734338197323950475428186387011150565189796236823101440000000000=2^{72}\cdot 3^{22}\cdot 5^{10}\cdot 113^{10}\cdot 337^{8}\cdot 310501^{8}\cdot 11155561^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $680{,}610{,}426.61$
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:  $2, 3, 5, 113, 337, 310501, 11155561$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{18908675895} a^{4} + \frac{2805641173}{6302891965} a^{2}$, $\frac{1}{56726027685} a^{5} + \frac{2805641173}{18908675895} a^{3}$, $\frac{1}{357538024102154051025} a^{6} + \frac{2805641173}{119179341367384683675} a^{4} + \frac{2995107039}{6302891965} a^{2}$, $\frac{1}{2145228144612924306150} a^{7} + \frac{2805641173}{715076048204308102050} a^{5} + \frac{998369013}{12605783930} a^{3}$, $\frac{1}{13521141235772658644386035084750} a^{8} + \frac{2805641173}{4507047078590886214795345028250} a^{6} + \frac{998369013}{79452894244923122450} a^{4} - \frac{1313928607}{6302891965} a^{2}$, $\frac{1}{81126847414635951866316210508500} a^{9} + \frac{2805641173}{27042282471545317288772070169500} a^{7} - \frac{9610676891}{1430152096408616204100} a^{5} + \frac{436664437}{7563470358} a^{3}$, $\frac{1}{511333754715489964418331197363273214202500} a^{10} + \frac{2805641173}{170444584905163321472777065787757738067500} a^{8} - \frac{9610676891}{9014094157181772429590690056500} a^{6} + \frac{436664437}{47671736546953873470} a^{4} + \frac{1136207753}{6302891965} a^{2}$, $\frac{1}{1022667509430979928836662394726546428405000} a^{11} - \frac{4188860411}{1022667509430979928836662394726546428405000} a^{9} - \frac{9231745159}{54084564943090634577544140339000} a^{7} + \frac{4583070931}{715076048204308102050} a^{5} + \frac{47158898}{1260578393} a^{3}$, $\frac{1}{19337288484177255346622613615418823567379966797475000} a^{12} + \frac{2805641173}{6445762828059085115540871205139607855793322265825000} a^{10} - \frac{9610676891}{340889169810326642945554131575515476135000} a^{8} - \frac{231610261}{200313203492928276213126445700} a^{6} - \frac{166943342}{23835868273476936735} a^{4} - \frac{1991861411}{6302891965} a^{2}$, $\frac{1}{19337288484177255346622613615418823567379966797475000} a^{13} + \frac{2805641173}{6445762828059085115540871205139607855793322265825000} a^{11} - \frac{3620462813}{1022667509430979928836662394726546428405000} a^{9} - \frac{49392781}{3004698052393924143196896685500} a^{7} + \frac{143923565}{28603041928172324082} a^{5} + \frac{669495388}{6302891965} a^{3}$, $\frac{1}{731285041270847114159885767463537419035550972978631758730250000} a^{14} + \frac{2805641173}{243761680423615704719961922487845806345183657659543919576750000} a^{12} - \frac{9610676891}{12891525656118170231081742410279215711586644531650000} a^{10} - \frac{231610261}{7575314884673925398790091812789232803000} a^{8} - \frac{83471671}{450704707859088621479534502825} a^{6} + \frac{2155515277}{119179341367384683675} a^{4} + \frac{814727692}{6302891965} a^{2}$, $\frac{1}{731285041270847114159885767463537419035550972978631758730250000} a^{15} + \frac{2805641173}{243761680423615704719961922487845806345183657659543919576750000} a^{13} + \frac{332789671}{1432391739568685581231304712253246190176293836850000} a^{11} + \frac{295138268}{127833438678872491104582799340818303550625} a^{9} - \frac{2414498641}{54084564943090634577544140339000} a^{7} + \frac{6200294513}{1430152096408616204100} a^{5} + \frac{247315717}{37817351790} a^{3}$, $\frac{1}{27655263664504293987490690374382730573496073661609701372768267964647500000} a^{16} + \frac{2805641173}{9218421221501431329163563458127576857832024553869900457589422654882500000} a^{14} - \frac{9610676891}{487523360847231409439923844975691612690367315319087839153500000} a^{12} - \frac{231610261}{286478347913737116246260942450649238035258767370000} a^{10} - \frac{83471671}{17044458490516332147277706578775773806750} a^{8} + \frac{2155515277}{4507047078590886214795345028250} a^{6} - \frac{1829388091}{79452894244923122450} a^{4} + \frac{2485191236}{6302891965} a^{2}$, $\frac{1}{27655263664504293987490690374382730573496073661609701372768267964647500000} a^{17} + \frac{2805641173}{9218421221501431329163563458127576857832024553869900457589422654882500000} a^{15} - \frac{9610676891}{487523360847231409439923844975691612690367315319087839153500000} a^{13} + \frac{436664437}{2578305131223634046216348482055843142317328906330000} a^{11} + \frac{1136207753}{340889169810326642945554131575515476135000} a^{9} - \frac{988615783}{18028188314363544859181380113000} a^{7} + \frac{3812777569}{476717365469538734700} a^{5} + \frac{762563167}{18908675895} a^{3}$, $\frac{1}{1045848834845763421690517297237998526198689467844847494490823515768061790844025000000} a^{18} + \frac{2805641173}{348616278281921140563505765745999508732896489281615831496941171922687263614675000000} a^{16} - \frac{9610676891}{18436842443002862658327126916255153715664049107739800915178845309765000000} a^{14} - \frac{231610261}{10833852463271809098664974332793146948674829229313063092300000} a^{12} - \frac{83471671}{644576282805908511554087120513960785579332226582500} a^{10} + \frac{2155515277}{170444584905163321472777065787757738067500} a^{8} + \frac{7117619657}{9014094157181772429590690056500} a^{6} + \frac{1763610803}{79452894244923122450} a^{4} + \frac{1823931483}{6302891965} a^{2}$, $\frac{1}{1045848834845763421690517297237998526198689467844847494490823515768061790844025000000} a^{19} + \frac{2805641173}{348616278281921140563505765745999508732896489281615831496941171922687263614675000000} a^{17} - \frac{9610676891}{18436842443002862658327126916255153715664049107739800915178845309765000000} a^{15} - \frac{231610261}{10833852463271809098664974332793146948674829229313063092300000} a^{13} - \frac{83471671}{644576282805908511554087120513960785579332226582500} a^{11} + \frac{81826933}{255666877357744982209165598681636607101250} a^{9} + \frac{495119489}{2253523539295443107397672514125} a^{7} + \frac{10532821139}{1430152096408616204100} a^{5} + \frac{76938664}{18908675895} a^{3}$, $\frac{1}{572378730705916538614659912466107933741207951023914926772226656289287422337629359881941875143609770578455316236911598681858384789442516083592187750000000} a^{20} - \frac{2861278186130768943428024285448357298179588292917867843958653852919}{63597636745101837623851101385123103749023105669323880752469628476587491370847706653549097238178863397606146248545733186873153865493612898176909750000000} a^{18} + \frac{52999118260265784041121004286932952679500892137600993442859710122201}{3363410378297239179785047629064726943378766166604913440051833332065660823515199748657445441828430229043605184103015859898906679393037050000000} a^{16} - \frac{11857183397352135596387537931706417532155744734531378473167006341529}{26681485237048951859870944409125112962163600773418249760939963470131133871617929188616875188883474929143936105075028522235718500000} a^{14} - \frac{10197886201189971957556997375867766776680997087774398165538636844073}{783928363690221491661723207942298823368409718082373241389546555617450508001843386630362910362243383454944044911783500000} a^{12} + \frac{3351001359446897548960928206981431724335972757609943884878269630443}{12437597979520841932465571110932072327982694814956633028526716237590636031183240924078934460401200664637190000} a^{10} + \frac{125924011255019044178547421804244600167768833042153637083680585095973}{4933290166080469194772867692224545501940352080490020357078544752428830513511443680630711262873915000} a^{8} - \frac{12652722691163744031981036029077656506756488183730946024461719337}{34982686944219992431876296405286622146851112267235709513769218338193780222483983206500} a^{6} - \frac{14068094408192639198241518084243992845458216379765059885185625953197}{574914395290414524723945991985750191638893873855967148957459276319872486985525} a^{4} + \frac{78926629272707600018232914127023979376328286343317716785895267974676}{273643145947725850663829040572370111204449562791752761206081981282955} a^{2} + \frac{2950442336895763005424408194123522436283571976756718491280}{14471830151792120008931436633118625422631665599858903351629}$, $\frac{1}{572378730705916538614659912466107933741207951023914926772226656289287422337629359881941875143609770578455316236911598681858384789442516083592187750000000} a^{21} - \frac{2861278186130768943428024285448357298179588292917867843958653852919}{63597636745101837623851101385123103749023105669323880752469628476587491370847706653549097238178863397606146248545733186873153865493612898176909750000000} a^{19} + \frac{52999118260265784041121004286932952679500892137600993442859710122201}{3363410378297239179785047629064726943378766166604913440051833332065660823515199748657445441828430229043605184103015859898906679393037050000000} a^{17} - \frac{11857183397352135596387537931706417532155744734531378473167006341529}{26681485237048951859870944409125112962163600773418249760939963470131133871617929188616875188883474929143936105075028522235718500000} a^{15} - \frac{10197886201189971957556997375867766776680997087774398165538636844073}{783928363690221491661723207942298823368409718082373241389546555617450508001843386630362910362243383454944044911783500000} a^{13} + \frac{3351001359446897548960928206981431724335972757609943884878269630443}{12437597979520841932465571110932072327982694814956633028526716237590636031183240924078934460401200664637190000} a^{11} + \frac{12914505834755998317203544649573652230707082070790562976265780243979}{14799870498241407584318603076673636505821056241470061071235634257286491540534331041892133788621745000} a^{9} - \frac{1221902115938177595290388556955996695835908449231111621399433477051}{11859130874090577434406064481392164907782527058592905525167765016647691495422070307003500} a^{7} - \frac{817098369051457154762178750123663796981963453664839782191999673681}{114982879058082904944789198397150038327778774771193429791491855263974497397105} a^{5} - \frac{44424160720809656216635050889315261102188117474114734845077644734817}{547286291895451701327658081144740222408899125583505522412163962565910} a^{3} + \frac{2950442336895763005424408194123522436283571976756718491280}{14471830151792120008931436633118625422631665599858903351629} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $15$
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

22T43:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 8110080
The 52 conjugacy class representatives for t22n43 are not computed
Character table for t22n43 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

Degree 22 sibling: data not computed
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 R R R ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ $16{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ $16{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
3Data not computed
5Data not computed
$113$$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
113.2.1.2$x^{2} + 339$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.1.2$x^{2} + 339$$2$$1$$1$$C_2$$[\ ]_{2}$
113.8.4.1$x^{8} + 127690 x^{4} - 1442897 x^{2} + 4076184025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
113.8.4.1$x^{8} + 127690 x^{4} - 1442897 x^{2} + 4076184025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
337Data not computed
310501Data not computed
11155561Data not computed