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D.15.12.2 modrWalk

Procedure from library modwalk.lib (see modwalk_lib).

Return:
a standard basis of I

Note:
The procedure computes a standard basis of I (over the rational numbers) by using modular methods.

Example:
 
LIB "modwalk.lib";
ring R1 = 0, (x,y,z,t), dp;
ideal I = 3x3+x2+1, 11y5+y3+2, 5z4+z2+4;
I = std(I);
ring R2 = 0, (x,y,z,t), lp;
ideal I = fetch(R1, I);
int radius = 2;
ideal J = modrWalk(I,radius);
J;
==> J[1]=x3+1/3x2+1/3
==> J[2]=z4+1/5z2+4/5
==> J[3]=y5+1/11y3+2/11
ring S1 = 0, (a,b,c,d), Dp;
ideal I = 5b2, ac2+9d3+3a2+5b, 2a2c+7abd+bcd+4a2, 2ad2+6b2d+7c3+8ad+4c;
I = std(I);
ring S2 = 0, (c,d,b,a), lp;
ideal I = fetch(S1,I);
// I is assumed to be a Dp-Groebner basis.
// We compute a lp-Groebner basis.
ideal J = modrWalk(I,radius,"Dp");
J;
==> J[1]=a25+16a24+96a23+256a22+256a21+256/9a20+1024/3a19+2048a18+65536/9a17+\
   32768/3a16+16384/81a15+131072/81a14+1048576/81a13+1048576/27a12+1048576/9\
   a11
==> J[2]=ba11+1522867351997104938459/91668001658017308797687087104a24+4293036\
   9782248629690765/91668001658017308797687087104a23+80925218629630777478637\
   /22917000414504327199421771776a22+7108535670237178684767/2864625051813040\
   899927721472a21-3255817194541612658349/89519532869157528122741296a20+5380\
   8965391546362724459/358078131476630112490965184a19+1534729815590907963215\
   01/358078131476630112490965184a18-260815719913165309506063/44759766434578\
   764061370648a17-1485276141860757031491027/89519532869157528122741296a16-4\
   92332725360316960775/22379883217289382030685324a15+7423992361030571232440\
   /16784912412967036523013993a14-17640364913371983121693/167849124129670365\
   23013993a13-37723213977586186442564/5594970804322345507671331a12+92047580\
   41857159721472414/5594970804322345507671331a11
==> J[3]=b2a6-63/2ba10+41087306587333357057895823883/924013456712814472680685\
   83800832a24+93915562116924232413944264677/1320019223875449246686694054297\
   6a23+1314662746341964624103002499857/30800448557093815756022861266944a22+\
   2628268795042931967685617407557/23100336417820361817017145950208a21+32861\
   7969148352577032114618159/2887542052227545227127143243776a20+175631724829\
   284757906915538269/12993939235023953522072144596992a19+303812212296039043\
   29090032467/206253003730538944794795945984a18+581332950565269518541458030\
   9/6445406366579342024837373312a17+1392577308804410648719627124495/4060606\
   01094498547564754518656a16+728489619081836101063651608791/135353533698166\
   182521584839552a15+80182830319998353431517995037/913636352462621732020697\
   666976a14+22597043807001043905240513127/32629869730807919000739202392a13+\
   2699689128255025271541196980091/456818176231310866010348833488a12+6291773\
   289016174025274735397/352483160672307766983293853a11+4/7a6
==> J[4]=b3a5+4/7ba5-5398327059462849163101023479/739210765370251578144548670\
   406656a24+3500067651845908053488406611/92401345671281447268068583800832a2\
   3+81936880336538263803253409291/46200672835640723634034291900416a22+85587\
   506267677081700930548967/6600096119377246233433470271488a21+4349147060430\
   84846670081811347/11550168208910180908508572975104a20+1021713731964491721\
   016405426709/25987878470047907044144289193984a19+189087309367688338214840\
   3875/812121202188997095129509037312a18+3873208423822126582196454031/10312\
   6501865269472397397972992a17+104243440049859097996976327663/4060606010944\
   98547564754518656a16+52530075262606982469983545909/5075757513681231844559\
   4314832a15+6154556265260978917193662164647/365454540985048692808279066790\
   4a14+138400848395446486358821423/5639730570756924271732701648a13+68577767\
   53456717192310999393/38068181352609238834195736124a12+3439955547980759942\
   57860510037/228409088115655433005174416744a11+9/2a10+18a9
==> J[5]=b4-63/2b3a4+4/7b2+3859043113737/128ba10-61261515/8ba9-525086793/32ba\
   8+3969/16ba7-19845/4ba6+9/2ba5-317530772199391516703685862633925319890715\
   5/9853890705461273608546303647569412096a24-889000025950798002413919214249\
   17407047623059/17244308734557228814956031383246471168a23-4443358152931308\
   1841158459544460024461570519/1437025727879769067913002615270539264a22-443\
   94201884268758058296712245149507940575423/5388846479549134004673759807264\
   52224a21-88794706503699382133468723144783169844707611/1077769295909826800\
   934751961452904448a20-1763218978464215084468461907410740543335425/1732129\
   22556936450150227993804931072a19-4436403904696107325808495191699219229446\
   85/4210036312147760941151374849425408a18-21791665871565292166347154127869\
   734967876143/33680290497182087529210998795403264a17-382244791239220889094\
   021792022832016196677083/151561307237319393881449494579314688a16-25381669\
   887869257097552146559952295821945667/6315054468221641411727062274138112a1\
   5-3081988985408645474889243822837753592680251/487161344691383766047516232\
   57636864a14-42991086686519981772739427292780808208480959/8525323532099215\
   9058315340700864512a13-44791324614077928834488691100822888689474979/10656\
   654415124019882289417587608064a12-250969820105414481651193596059252574282\
   0133/197345452131926294116470696066816a11+771901337907/64a10-1701/4a9+437\
   53689/4a8-19845/8a7-19845/2a6
==> J[6]=da-1323/800b3a4+63/20b2a4+2701701074331/1280ba10-61269453/320ba9+306\
   291699/160ba8-27783/16ba6-189/200ba4-685883099436497069901143509465258298\
   0621641/123173633818265920106828795594617651200a24-5486790468650205133145\
   8584531975294116209709/61586816909132960053414397797308825600a23-60954530\
   98260341081175657932145681344303369/1140496609428388149137303662913126400\
   a22-54839253809480536502430419967744127237862429/384917605682081000333839\
   9862331801600a21-6855741428155530311932580353074238311128479/481147007102\
   601250417299982791475200a20-2878029541644786739384025698911163129301521/1\
   732129225569364501502279938049310720a19-539637460347386518746735121876349\
   5466944867/288688204261560750250379989674885120a18-1360851075145362976080\
   2370233473935440353147/120286751775650312604324995697868800a17-2261317453\
   3404843047583509526614944559001629/54129038299042640671946248064040960a16\
   -19899168362053216404911620136404443756991/309379505595808417192193918976\
   00a15-3443170203569948104100043669593479708510001/30447584043211485377969\
   7645360230400a14-27203224432640761938740031635681832222207391/30447584043\
   2114853779697645360230400a13-11007706754707314979975768772869745826795967\
   3/152237920216057426889848822680115200a12-1380727908059280736173048219943\
   2331439121223/6343246675669059453743700945004800a11-1929592445193/640a10+\
   437570343/1600a9+164090367/100a8+3969/8a6+9/5a4
==> J[7]=db2+4/7d+81/160b3a4-189/200b3a3+9/5b2a3+250047/32ba10+4750893/320ba9\
   -567/40ba8-567/10ba7+81/280ba4-27/50ba3-627430952592078879073202920720376\
   153131/5173292620367168644486809414973941350400a24-2906859466655459274817\
   4299538370243493/18476045072739888016024319339192647680a23-69048240138740\
   571289193103124930690109/11975214398998075565941688460587827200a22+488085\
   08639174871490992992380443024047/11547528170462430010015199586995404800a2\
   1+5093069077149293873574182379116672811143/808326971932370100701063971089\
   67833600a20+3286117544936595300844164893345508176887/36374713736956654531\
   547878699035525120a19-57124126734867585149642894515011901057/189451634046\
   6492423518118682241433600a18-296407013643924816106633846582502767991/2526\
   021787288656564690824909655244800a17-193831832299525900077669479610197531\
   4379/11367098042798954541108712093448601600a16+49045157132492520042165362\
   0398209599/433131307834131784069071486566400a15+9565278519988924726047202\
   0815175898220317/25575970596297647717494602210259353600a14-51800242230268\
   556716933919846651112061/456713760648172280669546468040345600a13-15498981\
   08498974500694541229046869490537/1598498162268602982343412638141209600a12\
   -700479417253540849801700529315888259/26641636037810049705723543969020160\
   a11+20611017/1600a10+21504771/320a9+243/50a8-1701/100a7+36/35a3
==> J[8]=d2
==> J[9]=ca-933058125616695839233135894945/1072490027019586483846883328d2b2+1\
   81798447786251225329012915026263433/135133743404467896964707299328d2ba-93\
   3273424783513215621888049885/536245013509793241923441664d2b-1303085419745\
   782218197463665/4189414168045259702526888d2a2+123559947629567300589706290\
   55/268122506754896620961720832d2a-1199840232441193816221375102415/2413102\
   560794069588655487488d2+2491745382424941244889399/13406125337744831048086\
   0416db4-3555609453357215927937645400587721691/150148603782742107738563665\
   92db3a+4335598656140076483779831/44687084459149436826953472db3+4878432589\
   3233437517048449/234607193410534543341505728db2a2+55721010090835588135100\
   5/22343542229574718413476736db2a+209832422953387424512351/279294277869683\
   9801684592db2+3555609187498321955011643845587212179/367864079267718163959\
   480981504dba3+3555609913276024211483765490498708131/551796118901577245939\
   221472256dba2-145779987568990819629896861928862547827/5517961189015772459\
   39221472256dba+619371236591439497682833/11171771114787359206738368db-7/25\
   6da5+3960875342878629886012627/7447847409858239471158912da4+7771229957205\
   37769623969/67030626688724155240430208da3-1777804694090865761119737579475\
   5517375/275898059450788622969610736128da2-1777804726575333846228895680123\
   3949435/68974514862697155742402684032da+308860778482849028860165/83788283\
   36090519405053776d-8662493913356167095523081/89374168918298873653906944b6\
   a+3555609478199473823814428733192201719/157656033971879213125491849216b6-\
   691947094320446395469857/3723923704929119735579456b5a+3555609453353845049\
   444286489768089771/78828016985939606562745924608b5+3124231237895906145355\
   25/14895694819716478942317824b4a3+585027110547429460971921/14895694819716\
   478942317824b4a2-156660231874261273783321/5585885557393679603369184b4a+35\
   55609503988303295791145181547167843/275898059450788622969610736128b4+7251\
   4685191899110463387/7447847409858239471158912b3a4+18620705341639481373493\
   9/1861961852464559867789728b3a3+7/128b3a2-605003325032464298356105/186196\
   1852464559867789728b3a+3555609574059246059164309038972786827/137949029725\
   394311484805368064b3+5598273092605420181626917/14895694819716478942317824\
   b2a5+345539722766148179457993/7447847409858239471158912b2a4+2385184823515\
   36506954653/1861961852464559867789728b2a3+83575301506775637281703/3723923\
   704929119735579456b2a2+29088485856305111765919/1861961852464559867789728b\
   2a+99452502321473754949153/1861961852464559867789728b2+268425929241471187\
   8543/3723923704929119735579456ba6-3555609361116578937489533571638679851/6\
   1310679877953027326580163584ba5-3555609452325367082052038930932292483/153\
   27669969488256831645040896ba4+26601007630913544819277/4654904631161399669\
   47432ba3+1/32ba2-1/8ba+1/2b+114250471277661636359733/74478474098582394711\
   58912a7+12679503922384077256353/58186307889517495868429a6-355560938818173\
   1522239475158951103475/30655339938976513663290081792a5-355560945315066769\
   2457791360246789887/7663834984744128415822520448a4+6177215569656980577203\
   3/930980926232279933894864a3
==> J[10]=cb-51883786357160728310192955375/119165558557731831538542592d2b2+48\
   1386418930770946162004881943819/714993351346390989231255552d2ba-518957552\
   11944011767335954675/59582779278865915769271296d2b-1449222196664147064041\
   05875/930980926232279933894864d2a2+686849362653815991337907745/2979138963\
   9432957884635648d2a-7413168888332102018906216665/297913896394329578846356\
   48d2-26167188897135995571014271/3723923704929119735579456db4-282448310179\
   81993161819308533755843/238331117115463663077085184db3a-73175592066975551\
   63387487/3723923704929119735579456db3+5828856071765676222590037/148956948\
   19716478942317824db2a2+1514279630896063963667001/744784740985823947115891\
   2db2a-7272026608629444225731923/930980926232279933894864db2+4034975482007\
   866250015055868745901/834158909904122820769798144dba3+4034975571974411908\
   603340014247845/1251238364856184231154697216dba2-165434014380325235914269\
   294689579645/1251238364856184231154697216dba-9191448705489243016349701/93\
   0980926232279933894864db+1982200932715552522311051/7447847409858239471158\
   912da4-1509246788223390533808813/7447847409858239471158912da3-20174879971\
   725315644732071092598465/625619182428092115577348608da2-20174879904431593\
   887186862336251865/156404795607023028894337152da-252418340441531775603155\
   /116372615779034991736858d+456202226050467425828019/297913896394329578846\
   35648b6a+4034975979152509491033174446023465/357496675673195494615627776b6\
   +98473672424910027857535/1861961852464559867789728b5a+4034975963999940661\
   474358666763413/178748337836597747307813888b5+156350945449711844463807/14\
   895694819716478942317824b4a3+44456950855317828247245/74478474098582394711\
   58912b4a2+287293351674361156855479/3723923704929119735579456b4a+403497587\
   1473609837678772215952349/625619182428092115577348608b4+36289524490360625\
   623293/7447847409858239471158912b3a4+93186002055131135410743/186196185246\
   4559867789728b3a3+14067667489272861122505/465490463116139966947432b3a+403\
   4975963999940661474358666763413/312809591214046057788674304b3+28007879038\
   85023179576435/14895694819716478942317824b2a5-3166195896743065087591305/1\
   861961852464559867789728b2a4-25421432867156444145874623/37239237049291197\
   35579456b2a3+6350992979331118321035/1861961852464559867789728b2a2+7277356\
   5252869158574463/1861961852464559867789728b2a-183127380362847622840257/18\
   61961852464559867789728b2-609777990697319804055717/3723923704929119735579\
   456ba6-12104927812697071560185256446107551/417079454952061410384899072ba5\
   -12104928260203911237239845023297399/104269863738015352596224768ba4-73181\
   62508071328602934805/465490463116139966947432ba3+57158936813980064889315/\
   7447847409858239471158912a7-495460117069359674277087/18619618524645598677\
   89728a6-12104927983035189386839242655559079/208539727476030705192449536a5\
   -12104927942658956332312117401751119/52134931869007676298112384a4-4543530\
   12794757196085679/116372615779034991736858a3
==> J[11]=cd-1920728475331697593613813414134769/40479832696557566640893158405\
   760d2b2+5764250385873/695693440d2ba2+170343769105112001960355183524143521\
   3637/22314507773977358610792353571175200d2ba-9550514870995836288455565940\
   87971/10119958174139391660223289601440d2b-5134471018910672985557392164732\
   31/30359874522418174980669868804320d2a2+578418649527504905240134080527951\
   /212519121656927224864689081630240d2a-12103339010832893399408191559722552\
   7/4462901554795471722158470714235040d2-7/192db5+1147683237034990994983583\
   601133/50599790870696958301116448007200db4-205020774988302452161558087787\
   41633596977/1525778309331785204156742124524800db3a-1247540565954706731437\
   06177862071/531297804142318062161722704075600db3+45245274457/13111145600d\
   b2a3+19609715263921901913882415977783/1239694876332075478377352976176400d\
   b2a2-600479018859110558176461236909/44274817011859838513476892006300db2a+\
   207177810818429459822193583049/12649947717674239575279112001800db2+750069\
   /434808400dba4+20502370680999266417773030012226074920153/3738156857862873\
   7501840182050857600dba3+266527052187652797085694382527598245524237/728940\
   587283260381285883549991723200dba2-10927607308402564393464140298984404810\
   503013/728940587283260381285883549991723200dba-94737793618754761262780795\
   71708/77480929770754717398584561011025db+141/160da5+314705839803364227080\
   348103279561/88549634023719677026953784012600da4+487721587861887922936009\
   5347211139/2479389752664150956754705952352800da3-266527002462028938884091\
   409627316476299109/72894058728326038128588354999172320da2-266527007565508\
   754871920250344407518887293/18223514682081509532147088749793080da+3025614\
   3869401522377177148021/15496185954150943479716912202205d-2774246800694075\
   97699203193603/50599790870696958301116448007200b6a+2665270082831643532436\
   64810929874296778193/208268739223788680367395299997635200b6-1363025389640\
   91843759370200177/12649947717674239575279112001800b5a+2665270074245399349\
   68190180907757621618989/104134369611894340183697649998817600b5+1157639083\
   985432496588843519/1011995817413939166022328960144b4a3-199072043416803091\
   67106115383333/25299895435348479150558224003600b4a2-165013423602324137714\
   4920979/11068704252964959628369223001575b4a+26652700796488534220913288043\
   7847897363253/364470293641630190642941774995861600b4+67173200729089839223\
   22872143/12649947717674239575279112001800b3a4+237446067261032194148820974\
   007/44274817011859838513476892006300b3a3-63/40b3a2-1363025389640918437593\
   70200177/22137408505929919256738446003150b3a+2665270074245399349681901809\
   07757621618989/182235146820815095321470887497930800b3+5185943801834721646\
   92309614313/25299895435348479150558224003600b2a5-770602149619515685695972\
   47049/3162486929418559893819778000450b2a4+10758795425866054064618407863/1\
   581243464709279946909889000225b2a3-19907204341680309167106115383333/44274\
   817011859838513476892006300b2a2+37746229454460238097434832253/22137408505\
   929919256738446003150b2a-2209350347317311748521632649/4427481701185983851\
   347689200630b2+6028969243696296971626938816/11068704252964959628369223001\
   575ba6-799580999051237938343509597947149307658791/24298019576108679376196\
   1183330574400ba5-61506232752761366658040613260219086724539/46726960723285\
   92187730022756357200ba4+237446067261032194148820974007/774809297707547173\
   98584561011025ba3-9/10ba2+74084911454781737813187087759/88549634023719677\
   026953784012600a7+86142849942174331756440965391/1106870425296495962836922\
   3001575a6-799581007386086816652274228881949428897327/12149009788054339688\
   0980591665287200a5-799581022696526264615760751033222556661879/30372524470\
   135849220245147916321800a4+272305294824613701394594332189/774809297707547\
   17398584561011025a3
==> J[12]=c2+1/3cb2+5/3d+3a3
intvec w = 3,2,1,2;
ring S3 = 0, (c,d,b,a), (a(w),lp);
ideal I = fetch(S1,I);
// I is assumed to be a Dp-Groebner basis.
// We compute a (a(w),lp)-Groebner basis.
ideal J = modrWalk(I,radius,"Dp",w);
J;
==> J[1]=d2
==> J[2]=c2+3a3+1/3cb2+5/3d
==> J[3]=ca2+4ca+7/2b3+2b
==> J[4]=cda-6/7ba3+4/7c2-2/21cb3+1/7dba-10/21db
==> J[5]=db4a-192/49b2a3+128/49c2b-64/7cda-64/147cb4+4/63d2ba+2db3a-4db4+60/4\
   9db2a-8db3+8/7dba-656/147db2-32/7db
==> J[6]=db6+333576da3-189/4b8+15876b4a2-5186640/2401b2a3-2860371662240/17294\
   403c2b-779017667308341275/6537284334cd2+1815960/49cdb2-1455089310448/2470\
   629cda+8395442084/7203cb4-2304/49cb2a-826877376/2401ca2-77901889382621354\
   /3268642167d2ba-43215/7db5+199982018/49db3a-4664160/7dba2-126b7+72/7b5a+3\
   1752b3a2+1728/2401ba3-5770564112896/17294403c2-1012272/49cdb+5598114168/2\
   401cb3-4608/49cba+604630/21609d2b2-129552/49db4+864216/2401db2a-117b6+144\
   /7b4a+9072b2a2-20160cd+4663152/7cb2-3307509504/2401ca-7757580/16807d2b-17\
   2796/49db3+38880570713960/17294403dba-1296/7b5+288/49b3a+18144ba2+9326304\
   /7cb+185320d2-19529120/7203db2-1188/7b4+576/49b2a+2752/2401db-413460864/3\
   43b3-3312/49b2-1653754752/2401b
==> J[7]=cb3a-5/8da3-245/1024b8-7/32b6a+2/343b2a3-472622182763/533655864c2b-4\
   290617508534270775/6455101330944cd2+5/112cdb2-472618682105/152473104cda-1\
   08013/49392cb4+25/7cb2a-966391/1372ca2-429060350297371105/3227550665472d2\
   ba-3/256db5-15431/2016db3a+5/4dba2-245/256b7-25/32b5a+5/1372ba3-472620756\
   569/266827932c2+5/84cdb-108035/24696cb3+16/7cba+1512605/3556224d2b2-47/26\
   88db4+53/24696db2a-91/256b6-5/8b4a-5/84cd-5/4cb2-966979/343ca-3232325/138\
   2976d2b+1/192db3-477292060535/1067311728dba+65/32b5-1/14b3a-5/2cb-25/72d2\
   -667/98784db2+85/64b4-2/7b2a+109/12348db-1932803/784b3+3/14ba+93/112b2-96\
   6979/686b
==> J[8]=cb5-4/7db2a2+96/7b2a3-64/7c2b-4/441cdb2+32cda+116/21cb4+1152/7ca2-8/\
   7dba2-8/441cdb+32/7cb3+14db4-16/7db2a+16/7cb2+4608/7ca+28db3+16/7cb+328/2\
   1db2+16db+576b3+2304/7b
==> J[9]=cdb3+18da3+2cdb2+63cb4+441/2db3a-36dba2+126cb3+36cb2+126dba+72cb+10d\
   2
==> J[10]=ba4-2/3c2a+1/9cb3a+14/3cda-1/6dba2+49/12db3+5/9dba+7/3db
==> J[11]=b3a3+2744/5c2d+382/15c2b2-1127/15cdb3+7357/45cdba+129659/45cb5+2352\
   /5cb3a-535/1134d2b2a+906269/90db4a-8232/5db2a2-1029/10b6a+236/5c2b+2675/5\
   67cd2+161/5cdb2+5474/45cda+28812/5cb4+1680cb2a+129631/945d2ba-98/15db5-25\
   49/90db3a-735/2b5a-152/15c2-224/15cdb+8232/5cb3+5376/5cba+2156/45db4+2595\
   88/45db2a+2058/5b6-294b4a-28cd+16464/5cb2-4032/5ca+1069/9db3-898/45dba+14\
   70b5-168/5b3a+1232/45db2+1176b4-672/5b2a+3136/45db+672/5b3+504/5ba+2688/5\
   b2-2016/5b
==> J[12]=b5a2+182/3c2d+32/9c2b2-71/9cdb3+28204/1323cdba+931/2cb5+64cb3a-116/\
   7cba2+17/1134d2b4+17225981/1058841d2b2a-7/18db6+175759/108db4a-117310/441\
   db2a2-14b6a+4b4a2+2354384/453789c2b-172259810/1058841cd2+194/63cdb2+79360\
   /7203cda+1225cb4+1600/7cb2a-128/7ca2-600502807/57177414d2ba-11/6db5+19380\
   5/189db3a-74096/441dba2-50b5a+32/7b3a2-1745120/453789c2-124/21cdb+854cb3+\
   80cba+34/3969d2b2+1079/189db4+1232881/1323db2a+56b6-40b4a+16/7b2a2-400/63\
   cd+700cb2-1280/7ca+406/27db3+88449644/151263dba+200b5-32/7b3a+16/7ba2+336\
   cb+4484/1323db2+102b4-128/7b2a+12160/1323db-320/7b3+96/7ba+40b2-640/7b
==> J[13]=b7a-13606662/245c2d+20584/15435c2b2-26750/583443cd2b+23004293/2205c\
   db3+865156/5145cdba+758/5cb5+3904/245cb3a+12618cba2+264792689149/8168202d\
   2b2a+4/63db6+8178503/15435db4a-3032/35db2a2-52/35b6a-1008b4a2+8917192/108\
   045c2b-441254711815/1361367cd2+20176/15435cdb2+3119812/5145cda+20430577/3\
   5cb4+26496/343cb2a+9216ca2-147068092022/2268945d2ba+863872/2205db5+700382\
   3691/3430db3a-16335524/49dba2-4b7-772/49b5a-2016b3a2+17258032/108045c2+28\
   92256/2205cdb+40842962/35cb3+12382792/245cba+53656/315db4+520568/1715db2a\
   +208/35b6-736/49b4a-576b2a2+3946912/3087cd+81722308/245cb2+63150592/1715c\
   a+100/441d2b+709600/3087db3+42025695302/36015dba+3088/49b5-432/1715b3a-11\
   52ba2+32671944/49cb+1502048/15435db2+2166931/49b4-1984/245b2a+7552/2205db\
   +55320768/1715b3+8896/1715ba+6190756/245b2+31575296/1715b
==> J[14]=b9-48392128/5145c2d+60496264498816/16343210835c2b2+1716247029309933\
   910/617773369563cd2b+8065888/5145cdb3+30248793311936/2334744405cdba+384/3\
   5cb5+3072/1715cb3a-11648320/16807cba2+346635773341463204/617773369563d2b2\
   a-4/441db6+1380928/36015db4a-1536/245db2a2+4b8-96/245b6a-256/16807b2a3+11\
   9797764967424/16343210835c2b-522968750240/9529569cd2+7936/36015cdb2-30657\
   28/12005cda+74726439712/756315cb4+16384/2401cb2a-2420168/1361367d2b3-5229\
   38599168/47647845d2ba-664/15435db5+12451636912/36015db3a-19361152/343dba2\
   +36/7b7-512/343b5a-110595584/756315c2+256/108045cdb+48404416/245cb3-23247\
   8976/84035cba+10343440/1058841d2b2+304/1715db4+30606364522688/16343210835\
   db2a+1504/245b6-512/343b4a-1024/21609cd+96827264/1715cb2-16384/12005ca-64\
   0/3087d2b+1056/2401db3+49797330496/252105dba+3728/343b5-6656/12005b3a+387\
   22304/343cb+72448/756315db2-5806688/2401b4-8704/12005b2a+4096/15435db+423\
   04/12005b3+2048/12005ba-116239488/84035b2-8192/12005b
==> J[15]=dba3-2/3c2d+1/9cdb3-1/6d2ba+5/9d2b
==> J[16]=db3a2-147/2cdb3-10cdba-288cba2+63b4a2-32c2b-132cda-576ca2-49/2db5+4\
   db3a+126b3a2-64c2-82cdb-1152cba-21/2db4+36b2a2-80cd-2304ca-14db3-16dba+72\
   ba2-6db2-1008b4-2016b3-576b2-1152b
==> J[17]=a5+1/9cb2a2-4/3c2a-7/6cb3+5/9da2-2/3cb
==> J[18]=da4-4/21c3+11/63cdba+8/63c2b-1/63db3a+10/63cdb+32/7cba+5/9d2a+2/63d\
   b2a-b4a+4b4-4/7b2a+16/7b2
See also: modular.


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