Starting with polynomial:
P : t^4 - 6*t^2 + 3
Extension levels are: 4 53
-------------------------------------------------
Trying to find an order 53 Kronrod extension for:
P1 : t^4 - 6*t^2 + 3
Solvable: 1
-------------------------------------------------
Ending with final polynomial:
P : t^57 - 3738560480187344376501980021633463109684731183/2452070369646233148101113457952700592074375*t^55 + 129779700457856755078652986017815704963670965022352/120151448112665424256954559439682329011644375*t^53 - 56804670188253191865034240762097096918352198954683001/120151448112665424256954559439682329011644375*t^51 + 689204639176661967004010848285006854123489954569297433/4806057924506616970278182377587293160465775*t^49 - 3143472473091549111804499431927092550543538849654425594/98082814785849325924044538318108023682975*t^47 + 537314010255767831877366743657006751271935426217757479008/98082814785849325924044538318108023682975*t^45 - 71975733958726497867024071559312521588012495242188283612878/98082814785849325924044538318108023682975*t^43 + 7681196484177677448722115021066351333179038919597442061684119/98082814785849325924044538318108023682975*t^41 - 132081100422742762228970186490994270666206014855685257194833633/19616562957169865184808907663621604736595*t^39 + 9216556685759130054901788158463876875789074463644007630563100736/19616562957169865184808907663621604736595*t^37 - 524305775818059519489424649759267118515056145279269577097941063343/19616562957169865184808907663621604736595*t^35 + 24366565754024542399946839304530554594169843944638856386298259261893/19616562957169865184808907663621604736595*t^33 - 925126476391350951403085218500113919020636100685150734282854713115244/19616562957169865184808907663621604736595*t^31 + 5727991429284773325273093778163660165081576292140941271477944509037056/3923312591433973036961781532724320947319*t^29 - 144040090378495622980236680333468179182181930902883590381021374500389012/3923312591433973036961781532724320947319*t^27 + 2925258296499224360798384470215148186776443200636796944688276230106444685/3923312591433973036961781532724320947319*t^25 - 47598962689173461933335731958732894101587545362888563745629576150575766985/3923312591433973036961781532724320947319*t^23 + 614127301309839199388645479294230609108496384703486728400972040755245158480/3923312591433973036961781532724320947319*t^21 - 6199101455615213327568631954207019595541967780606181462066659171582072237775/3923312591433973036961781532724320947319*t^19 + 48125184607085459638407602219682641533826405637366801795466718718970412967675/3923312591433973036961781532724320947319*t^17 - 281090184683657654815369414117342970332141363351822054937446669094258974921050/3923312591433973036961781532724320947319*t^15 + 1200436701811121540498812868207729947913621028729091393772688681889550119840800/3923312591433973036961781532724320947319*t^13 - 3608739317798414804141109411862142880934308919662114175020338805485477681691150/3923312591433973036961781532724320947319*t^11 + 7247947264735037025073328547528886088855061820657379766651998153267808462274125/3923312591433973036961781532724320947319*t^9 - 9019065417719865757945641029483907669254906708187912205195715531218103545108875/3923312591433973036961781532724320947319*t^7 + 6182806930116120375496226971835082262725981629937903416368087893170328576367200/3923312591433973036961781532724320947319*t^5 - 1897945078608167609349683393488942288554765156353235921551811985119140078546325/3923312591433973036961781532724320947319*t^3 + 163077940747771505123925641765563019736507960416332684540919985583616370623475/3923312591433973036961781532724320947319*t
-------------------------------------------------
Computing nodes and weights
  current precision for roots: 53
  current precision for roots: 106
 current precision for weights: 53
Linear system for weights solvable: 0
  current precision for roots: 106
  current precision for roots: 212
 current precision for weights: 106
Linear system for weights solvable: 0
  current precision for roots: 212
  current precision for roots: 424
 current precision for weights: 212
Linear system for weights solvable: 0
  current precision for roots: 424
  current precision for roots: 848
 current precision for weights: 424
Linear system for weights solvable: 1
Sufficient bits for target precision reached
Positive weights:   57 out of 57
Indefinite weights: 0 out of 57
Negative weights:   0 out of 57
Extension rule has valid weights: 1
-------------------------------------------------
The nodes are:
| (9.1192720645091910828 - 1.6478867779021752748e-910j)  +/-  (1.8e-241, 1.8e-241j)
| (-0.74196378430272585765 + 1.0732433643355424054e-923j)  +/-  (1.93e-252, 1.93e-252j)
| (12.850803669793543044 + 2.0493597748853143745e-916j)  +/-  (3.39e-244, 3.39e-244j)
| (6.66735912066171512 - 1.9603886659364992628e-913j)  +/-  (4.92e-242, 4.92e-242j)
| (-10.211630043548552749 - 3.2572870661413830331e-912j)  +/-  (7.86e-242, 7.86e-242j)
| (10.211630043548552749 - 8.6366744511878864774e-917j)  +/-  (8.1e-242, 8.1e-242j)
| (-8.1011119281187209104 + 8.3773092245411292865e-928j)  +/-  (1.97e-241, 1.97e-241j)
| (-8.6025710485851192661 - 2.1739539268687238328e-939j)  +/-  (2.11e-241, 2.11e-241j)
| (-12.094987024303670608 + 2.5859159680102844396e-951j)  +/-  (2.78e-243, 2.78e-243j)
| (13.763297283250047764 - 2.3598889740190396564e-959j)  +/-  (2.02e-245, 2.02e-245j)
| (-13.763297283250047764 + 2.8030028649017494605e-957j)  +/-  (1.9e-245, 1.9e-245j)
| (-6.66735912066171512 + 2.1962443202645610322e-958j)  +/-  (4.75e-242, 4.75e-242j)
| (-5.3108702301458024293 - 5.8001648463894107803e-965j)  +/-  (3.45e-243, 3.45e-243j)
| (-2.7467412580028503487 - 8.3829345013491909699e-970j)  +/-  (2.38e-247, 2.38e-247j)
| (-4.8715319107231122256 + 1.4462809626909249313e-965j)  +/-  (9.07e-244, 9.07e-244j)
| (1.5240488928812602109 + 2.1398336074044110338e-972j)  +/-  (3.1e-250, 3.1e-250j)
| (9.6542508508450064551 + 7.438986740098320406e-964j)  +/-  (1.44e-241, 1.44e-241j)
| (2.7467412580028503487 - 5.4663096445236616345e-970j)  +/-  (2.24e-247, 2.24e-247j)
| (-9.1192720645091910828 - 1.1931134489585719413e-963j)  +/-  (1.84e-241, 1.84e-241j)
| (7.1351468571133623415 - 1.287430915093490361e-967j)  +/-  (8.75e-242, 8.75e-242j)
| (5.3108702301458024293 + 2.285257622302629827e-970j)  +/-  (3.2e-243, 3.2e-243j)
| (-10.797278463130324633 + 2.4757440570954501112e-968j)  +/-  (3.59e-242, 3.59e-242j)
| (-11.420107827048906971 - 1.5445504105599901691e-975j)  +/-  (1.25e-242, 1.25e-242j)
| (-2.3344142183389772393 + 4.0467646941057457755e-985j)  +/-  (2.72e-248, 2.72e-248j)
| (-6.2080182602318516879 - 1.1908020785315223642e-978j)  +/-  (2.39e-242, 2.39e-242j)
| (11.420107827048906971 + 1.1247018028166430349e-980j)  +/-  (1.19e-242, 1.19e-242j)
| (-9.6542508508450064551 - 1.4869957750987873481e-983j)  +/-  (1.34e-241, 1.34e-241j)
| (-12.850803669793543044 + 3.6315564776260833824e-986j)  +/-  (3.55e-244, 3.55e-244j)
| (-3.163186298761166201 - 8.4360464501456620887e-989j)  +/-  (1.61e-246, 1.61e-246j)
| (-4.4375293551893118631 - 1.2693977442760573342e-985j)  +/-  (2.39e-244, 2.39e-244j)
| (-5.7561323548821068346 - 3.979502385061033224e-983j)  +/-  (9.34e-243, 9.34e-243j)
| (12.094987024303670608 - 5.4092389589799750998e-985j)  +/-  (2.86e-243, 2.86e-243j)
| (6.2080182602318516879 - 1.0263212230593907022e-983j)  +/-  (2.47e-242, 2.47e-242j)
| (4.0083761460932215041 - 5.6436422134418783507e-987j)  +/-  (5.4e-245, 5.4e-245j)
| (2.3344142183389772393 + 2.2260002468104252147e-990j)  +/-  (2.45e-248, 2.45e-248j)
| (-4.0083761460932215041 + 1.8970264010109015488e-986j)  +/-  (5.16e-245, 5.16e-245j)
| (-7.1351468571133623415 + 2.5731593307715714927e-982j)  +/-  (8.84e-242, 8.84e-242j)
| (10.797278463130324633 - 3.6077627348613080353e-986j)  +/-  (3.68e-242, 3.68e-242j)
| (7.6125771730017602846 - 9.7154735107479953124e-995j)  +/-  (1.37e-241, 1.37e-241j)
| (8.1011119281187209104 + 2.0982331085570460307e-1010j)  +/-  (1.87e-241, 1.87e-241j)
| (-1.9265694629169259933 - 2.1537357847120299481e-1030j)  +/-  (2.72e-249, 2.72e-249j)
| (3.163186298761166201 - 1.1912587344007491294e-1027j)  +/-  (1.59e-246, 1.59e-246j)
| (4.8715319107231122256 + 5.2868683234679755357e-1023j)  +/-  (8.9e-244, 8.9e-244j)
| (0.36659399668862391531 + 4.7893120722451858565e-1034j)  +/-  (1.36e-253, 1.36e-253j)
| (4.4375293551893118631 - 5.4500950091495070571e-1024j)  +/-  (2.26e-244, 2.26e-244j)
| (-3.5836865283730310318 + 9.2407551822815373325e-1027j)  +/-  (9.23e-246, 9.23e-246j)
| (1.9265694629169259933 + 6.3146854176418829176e-1030j)  +/-  (2.7e-249, 2.7e-249j)
| (3.5836865283730310318 + 6.1766351962777180129e-1027j)  +/-  (9.52e-246, 9.52e-246j)
| (0.74196378430272585765 + 1.0356015373348997946e-1033j)  +/-  (1.95e-252, 1.95e-252j)
| (5.7561323548821068346 + 3.1970797462198036041e-1022j)  +/-  (9.23e-243, 9.23e-243j)
| (-1.5240488928812602109 + 3.4012030559437363906e-1031j)  +/-  (2.76e-250, 2.76e-250j)
| (-0.36659399668862391531 - 2.8315637450039338896e-1034j)  +/-  (1.18e-253, 1.18e-253j)
| (8.6025710485851192661 + 8.0623918623838207155e-1032j)  +/-  (1.97e-241, 1.97e-241j)
| (1.1284088220207113168 - 8.9166506282154880482e-1042j)  +/-  (2.17e-251, 2.17e-251j)
| (-2.2571078243370812739e-1065 - 3.4404992269237883942e-1065j)  +/-  (2.35e-1063, 2.35e-1063j)
| (-7.6125771730017602846 + 2.9408763402708530881e-1041j)  +/-  (1.44e-241, 1.44e-241j)
| (-1.1284088220207113168 + 1.8083353304600066042e-1052j)  +/-  (2.17e-251, 2.17e-251j)
-------------------------------------------------
The weights are:
| (1.8326317123556569466e-19 + 2.7886297770279383841e-928j)  +/-  (1.81e-56, 4.83e-177j)
| (0.11543492009590669254 + 3.8338089049454802647e-919j)  +/-  (6.81e-31, 1.82e-151j)
| (4.4766595546569932974e-37 + 4.0042586222790601216e-938j)  +/-  (3.47e-65, 9.27e-186j)
| (4.1101471245989587671e-11 + 2.8312818716471073406e-923j)  +/-  (1.22e-49, 3.26e-170j)
| (5.170087054759359143e-24 - 8.024143199976435812e-932j)  +/-  (1.81e-61, 4.83e-182j)
| (5.170087054759359143e-24 + 2.9256656195095293737e-931j)  +/-  (1.3e-59, 3.47e-180j)
| (1.1073188368278708339e-15 - 1.9498864052497621159e-927j)  +/-  (1.98e-56, 5.28e-177j)
| (1.7278624818979163764e-17 + 2.1432007623994520902e-928j)  +/-  (9.27e-58, 2.48e-178j)
| (4.8429249603369553784e-33 + 1.6519237815546588693e-936j)  +/-  (1.13e-66, 3.02e-187j)
| (3.0895015136782014074e-42 - 8.2278330981481951427e-941j)  +/-  (8.16e-70, 2.18e-190j)
| (3.0895015136782014074e-42 + 3.0485298052478149438e-941j)  +/-  (3.17e-70, 8.47e-191j)
| (4.1101471245989587671e-11 + 5.5798947068392503045e-925j)  +/-  (2.25e-54, 6.01e-175j)
| (1.3237621925036499563e-07 - 4.8384246834797335621e-923j)  +/-  (1.08e-50, 2.88e-171j)
| (0.0038024314271254897604 - 2.2471463168260147128e-920j)  +/-  (8.41e-40, 2.25e-160j)
| (1.2237003144079373502e-06 + 1.7095454615357852548e-922j)  +/-  (2.47e-49, 6.61e-170j)
| (0.049883396640429460269 + 2.5060172435557606636e-919j)  +/-  (6.81e-31, 1.82e-151j)
| (1.2545634343126444147e-21 - 4.1509133209203434141e-930j)  +/-  (6.58e-64, 1.76e-184j)
| (0.0038024314271254897604 - 5.4896636163743367347e-920j)  +/-  (1.85e-42, 4.94e-163j)
| (1.8326317123556569466e-19 - 1.9442958079895509266e-929j)  +/-  (1.99e-61, 5.32e-182j)
| (1.6602562684423833192e-12 + 3.2012073551226062144e-924j)  +/-  (3.97e-59, 1.06e-179j)
| (1.3237621925036499563e-07 - 4.470913748852411965e-922j)  +/-  (9.25e-55, 2.47e-175j)
| (1.1627943026525208923e-26 + 3.3427983081330502604e-933j)  +/-  (5.22e-66, 1.4e-186j)
| (1.2325298607190500084e-29 - 9.5489957439363552263e-935j)  +/-  (1.37e-67, 3.66e-188j)
| (0.0107285734248013407 + 4.6216847673603369856e-920j)  +/-  (2.93e-44, 7.85e-165j)
| (7.7735383434797329816e-10 - 2.7842332873125943637e-924j)  +/-  (3.2e-57, 8.56e-178j)
| (1.2325298607190500084e-29 + 3.2102182141697750143e-934j)  +/-  (1.13e-71, 3.02e-192j)
| (1.2545634343126444147e-21 + 1.4178629441048908931e-930j)  +/-  (7.83e-64, 2.09e-184j)
| (4.4766595546569932974e-37 - 1.3725108941466264153e-938j)  +/-  (1.37e-71, 3.67e-192j)
| (0.001121625463896485243 + 1.009302673788320531e-920j)  +/-  (7.52e-51, 2.01e-171j)
| (9.1181104944258696506e-06 - 5.4538205480366697716e-922j)  +/-  (1.25e-54, 3.34e-175j)
| (1.1425333978325444654e-08 + 1.2291750531969789346e-923j)  +/-  (5.12e-57, 1.37e-177j)
| (4.8429249603369553784e-33 - 5.1820752112413867869e-936j)  +/-  (1.01e-74, 2.71e-195j)
| (7.7735383434797329816e-10 - 8.3265681988246089774e-923j)  +/-  (5.5e-64, 1.47e-184j)
| (5.52432852290791807e-05 + 6.5224855400172792814e-921j)  +/-  (8.02e-60, 2.14e-180j)
| (0.0107285734248013407 + 9.7396746520487056148e-920j)  +/-  (1.15e-53, 3.08e-174j)
| (5.52432852290791807e-05 + 1.5792725347513261362e-921j)  +/-  (5.97e-56, 1.6e-176j)
| (1.6602562684423833192e-12 - 9.8048520000629227008e-926j)  +/-  (7.82e-62, 2.09e-182j)
| (1.1627943026525208923e-26 - 1.1934065832535129435e-932j)  +/-  (1.46e-72, 3.92e-193j)
| (5.0191150037018741073e-14 - 2.586075025321263051e-925j)  +/-  (1.16e-67, 3.1e-188j)
| (1.1073188368278708339e-15 + 2.4557674053765297171e-926j)  +/-  (9.61e-69, 2.57e-189j)
| (0.025280605372138695183 - 8.8103709891712675635e-920j)  +/-  (4.87e-56, 1.3e-176j)
| (0.001121625463896485243 + 2.890151187981256818e-920j)  +/-  (8.5e-61, 2.27e-181j)
| (1.2237003144079373502e-06 + 1.1417003788173878422e-921j)  +/-  (2.05e-64, 5.49e-185j)
| (0.13802012658169727467 - 5.7262661573242491886e-919j)  +/-  (4.02e-58, 1.07e-178j)
| (9.1181104944258696506e-06 - 2.8072502145689563244e-921j)  +/-  (7.92e-64, 2.12e-184j)
| (0.00027416842478606973034 - 4.1710526004105717711e-921j)  +/-  (1.48e-61, 3.96e-182j)
| (0.025280605372138695183 - 1.615634590883347506e-919j)  +/-  (4.73e-60, 1.27e-180j)
| (0.00027416842478606973034 - 1.4204672016833293247e-920j)  +/-  (9.48e-63, 2.54e-183j)
| (0.11543492009590669254 + 4.8144576309818321057e-919j)  +/-  (4.87e-60, 1.28e-180j)
| (1.1425333978325444654e-08 + 1.7685473026154377251e-922j)  +/-  (3.77e-66, 1.01e-186j)
| (0.049883396640429460269 + 1.5594512801344781997e-919j)  +/-  (8.06e-62, 2.08e-182j)
| (0.13802012658169727467 - 5.1186474225380741183e-919j)  +/-  (5.87e-61, 1.49e-181j)
| (1.7278624818979163764e-17 - 2.4572037880510088281e-927j)  +/-  (3.95e-71, 1.06e-191j)
| (0.082625337333766145046 - 3.621633895617885824e-919j)  +/-  (3.19e-61, 8.06e-182j)
| (0.14552617103538865478 + 5.8801870130444652543e-919j)  +/-  (4.09e-61, 1.05e-181j)
| (5.0191150037018741073e-14 + 1.4943858435988444297e-926j)  +/-  (8.15e-70, 2.54e-190j)
| (0.082625337333766145046 - 2.5563575220001003699e-919j)  +/-  (7.99e-62, 1.88e-182j)
