Starting with polynomial:
P : t^4 - 6*t^2 + 3
Extension levels are: 4 9 30
-------------------------------------------------
Trying to find an order 9 Kronrod extension for:
P1 : t^4 - 6*t^2 + 3
Solvable: 1
-------------------------------------------------
Trying to find an order 30 Kronrod extension for:
P2 : t^13 - 66*t^11 + 7569/5*t^9 - 74484/5*t^7 + 314307/5*t^5 - 98658*t^3 + 37611*t
Solvable: 1
-------------------------------------------------
Ending with final polynomial:
P : t^43 - 2644215807758474485306420476511785400537336355360319067919/3742279853373169184537541659615992497050465474333728299*t^41 + 468100809892931972792013969426838646024793431517144349909484/2079044362985093991409745366453329165028036374629849055*t^39 - 28117983742049848429713528128995468284852465318656752065367784/655041374639139202772933471622281791721162145431322305*t^37 + 2881567745338180246832140574342392198630693082820668758248031190441/525998223835228779826665577712692278752093202781351810915*t^35 - 4745301864268152753535985620756219000174308978400980345188863342863/9563604069731432360484828685685314159128967323297305653*t^33 + 3471259403731020490546814244459225978787572670277430692874349719468696/105199644767045755965333115542538455750418640556270362183*t^31 - 172996001888939835475703986153784430629544652515686530529669593036146888/105199644767045755965333115542538455750418640556270362183*t^29 + 6537861675465381441725912405873706255351776660883941472454826491835963118/105199644767045755965333115542538455750418640556270362183*t^27 - 188414136438212301278177905418030433473006431793094204833128203867672723174/105199644767045755965333115542538455750418640556270362183*t^25 + 180129874236551204005635493321367683798601292895845876851353730178238366800/4573897598567206781101439806197324163061680024185667921*t^23 - 273650570695034033497934651653311051523862471797235786494101105648414060600/415808872597018798281949073290665833005607274925969811*t^21 + 3446143216916896383806319476214846325353925146435573220636643897129844509850/415808872597018798281949073290665833005607274925969811*t^19 - 32271494482957320142997594759541926232777006926731376712394731225541976291850/415808872597018798281949073290665833005607274925969811*t^17 + 220490602680907608237944047826571522071671035477808932974567406658942204921800/415808872597018798281949073290665833005607274925969811*t^15 - 1070688806835954672753583769984694390619091858871704765148129022052028920283000/415808872597018798281949073290665833005607274925969811*t^13 + 3564862886922773342415208303641124748013721703242879435157549777923234207440375/415808872597018798281949073290665833005607274925969811*t^11 - 7741443024254950983608160035652282151475358493691173252918640435699397210298875/415808872597018798281949073290665833005607274925969811*t^9 + 139778041176480902850660050267511969015861138770141052485830324598244206056500/5696011953383819154547247579324189493227496916794107*t^7 - 7320170395783704957394700509916697318469572089878700557796388089204983283406000/415808872597018798281949073290665833005607274925969811*t^5 + 2385615365261982802356954766028850916696658687318298919930367849522250547348125/415808872597018798281949073290665833005607274925969811*t^3 - 239773608652933945420898189867943007336270399240784270269645330250346076768125/415808872597018798281949073290665833005607274925969811*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: 1
  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:   43 out of 43
Indefinite weights: 0 out of 43
Negative weights:   0 out of 43
Extension rule has valid weights: 1
-------------------------------------------------
The nodes are:
| (7.7832695479259323427 - 2.6788175037098998988e-654j)  +/-  (1.5e-245, 1.5e-245j)
| (-9.1910018238358979707 + 2.5180840116968000582e-663j)  +/-  (2.52e-246, 2.52e-246j)
| (-10.007395373769291038 - 3.1782123276853559352e-670j)  +/-  (5.63e-247, 5.63e-247j)
| (10.989426046679475214 - 2.642803438613649989e-675j)  +/-  (5.14e-248, 5.14e-248j)
| (-10.989426046679475214 - 4.2599434554337728491e-685j)  +/-  (5.98e-248, 5.98e-248j)
| (5.962430283195914526 + 8.5597131131313168007e-704j)  +/-  (2.95e-245, 2.95e-245j)
| (-7.1461777002773528398 + 3.3034995246540150677e-715j)  +/-  (2.38e-245, 2.38e-245j)
| (3.2290040429144049805 + 4.8206257244786642177e-716j)  +/-  (1.49e-247, 1.49e-247j)
| (-5.4228479113280690393 - 3.5578670405762471642e-717j)  +/-  (2.88e-245, 2.88e-245j)
| (5.4228479113280690393 - 5.5168312753071378457e-720j)  +/-  (2.91e-245, 2.91e-245j)
| (1.0889178662485273507 - 3.9881730684170576437e-732j)  +/-  (1.43e-251, 1.43e-251j)
| (4.2283965787258560516 - 1.6194576177961492102e-725j)  +/-  (3.63e-246, 3.63e-246j)
| (-8.4597801014425319993 - 2.4054068273386211627e-730j)  +/-  (7.79e-246, 7.79e-246j)
| (-6.5401984326203223666 + 5.9870932459392805619e-730j)  +/-  (2.49e-245, 2.49e-245j)
| (-4.6926889424259592592 - 2.2020338521523371826e-730j)  +/-  (1.79e-245, 1.79e-245j)
| (9.1910018238358979707 + 4.1792615839791204443e-729j)  +/-  (2.74e-246, 2.74e-246j)
| (10.007395373769291038 - 1.6308536095575124672e-738j)  +/-  (5.62e-247, 5.62e-247j)
| (-5.962430283195914526 - 6.997978257351264797e-742j)  +/-  (2.93e-245, 2.93e-245j)
| (8.4597801014425319993 + 9.4868976699093763972e-747j)  +/-  (7.75e-246, 7.75e-246j)
| (-4.2283965787258560516 - 1.5027757902285039095e-757j)  +/-  (3.63e-246, 3.63e-246j)
| (1.5122552015009576337 + 6.6137301693355531847e-772j)  +/-  (1.03e-250, 1.03e-250j)
| (7.1461777002773528398 - 3.5715654456106333643e-767j)  +/-  (2.13e-245, 2.13e-245j)
| (-3.2290040429144049805 - 9.5175707422490628878e-785j)  +/-  (1.47e-247, 1.47e-247j)
| (-1.9334495166203157698 - 5.6366663078098158739e-791j)  +/-  (1.04e-249, 1.04e-249j)
| (-2.7600174129340520741 - 7.8506154255332089314e-788j)  +/-  (3.38e-248, 3.38e-248j)
| (4.6926889424259592592 - 4.3305938643686125108e-793j)  +/-  (1.62e-245, 1.62e-245j)
| (0.41540312448578587575 - 1.7818193519739176554e-812j)  +/-  (1.41e-253, 1.41e-253j)
| (2.3344142183389772393 + 7.3354532298562198652e-807j)  +/-  (6.23e-249, 6.23e-249j)
| (6.5401984326203223666 - 2.7942881667529219726e-817j)  +/-  (2.42e-245, 2.42e-245j)
| (-3.7244524467757354907 + 4.3081021754439092785e-832j)  +/-  (8.01e-247, 8.01e-247j)
| (-2.3344142183389772393 - 1.4260974965243023776e-838j)  +/-  (6.12e-249, 6.12e-249j)
| (3.7244524467757354907 - 1.3028483683077648035e-837j)  +/-  (7.14e-247, 7.14e-247j)
| (1.9334495166203157698 + 3.9379408077969816702e-851j)  +/-  (1.14e-249, 1.14e-249j)
| (5.004198327719700136 - 1.706059570746623168e-860j)  +/-  (2.95e-245, 2.95e-245j)
| (-5.004198327719700136 + 4.7412054353247452476e-869j)  +/-  (2.98e-245, 2.98e-245j)
| (-0.74196378430272585765 - 1.7152562085234429792e-882j)  +/-  (1.52e-252, 1.52e-252j)
| (-7.7832695479259323427 - 5.1870133988985884912e-875j)  +/-  (1.51e-245, 1.51e-245j)
| (0.74196378430272585765 - 3.7249620299354292268e-889j)  +/-  (1.52e-252, 1.52e-252j)
| (-1.5122552015009576337 - 3.4812331007849441502e-888j)  +/-  (1.08e-250, 1.08e-250j)
| (-1.0889178662485273507 + 1.9357687795782252503e-890j)  +/-  (1.18e-251, 1.18e-251j)
| (-0.41540312448578587575 + 3.7734873372014698648e-891j)  +/-  (1.41e-253, 1.41e-253j)
| (-7.0337832569969166338e-891 - 1.5257005419365011696e-890j)  +/-  (7.22e-889, 7.22e-889j)
| (2.7600174129340520741 + 3.0195825357991539374e-885j)  +/-  (3.2e-248, 3.2e-248j)
-------------------------------------------------
The weights are:
| (1.8302170767659181139e-14 + 2.3784032295799348027e-667j)  +/-  (9.69e-86, 1.48e-207j)
| (1.3870509721234899507e-19 + 2.4187364292882631305e-672j)  +/-  (4.36e-89, 6.67e-211j)
| (6.26961427073452621e-23 - 2.7474768187022084243e-674j)  +/-  (8.77e-91, 1.34e-212j)
| (2.6943046156583249361e-27 - 5.3635389621170664262e-676j)  +/-  (4.04e-94, 6.18e-216j)
| (2.6943046156583249361e-27 + 9.1602962459729343154e-677j)  +/-  (7.11e-93, 1.09e-214j)
| (4.2794007036140764859e-09 - 1.2499634244784458909e-664j)  +/-  (7.71e-84, 1.18e-205j)
| (2.0158678142023796635e-12 - 6.6476639518767596064e-668j)  +/-  (1.62e-86, 2.48e-208j)
| (0.0010540154992074051822 - 3.3796450246514412701e-661j)  +/-  (2.43e-75, 3.72e-197j)
| (8.3067754220526910425e-08 + 2.2873125919461708775e-664j)  +/-  (3.93e-83, 6.01e-205j)
| (8.3067754220526910425e-08 + 1.2797086031608401991e-663j)  +/-  (7.01e-83, 1.07e-204j)
| (0.088313657809850212483 + 2.2070144506948967952e-659j)  +/-  (2.21e-62, 3.39e-184j)
| (2.6120921170157798831e-05 - 3.9710042437784137318e-662j)  +/-  (1.29e-79, 1.97e-201j)
| (8.0454075884523301583e-17 - 1.0894091795759819936e-670j)  +/-  (2.6e-89, 3.99e-211j)
| (1.2158528597383463317e-10 + 1.1235904450645779746e-666j)  +/-  (5.35e-86, 8.19e-208j)
| (2.5997493414620316707e-06 + 4.6227521843853048621e-663j)  +/-  (1.24e-82, 1.89e-204j)
| (1.3870509721234899507e-19 - 2.9164841353969918793e-671j)  +/-  (1.41e-93, 2.15e-215j)
| (6.26961427073452621e-23 + 2.1976921855190049857e-673j)  +/-  (3e-95, 4.59e-217j)
| (4.2794007036140764859e-09 - 1.65577781474157469e-665j)  +/-  (1.5e-86, 2.3e-208j)
| (8.0454075884523301583e-17 + 2.6156764734855722664e-669j)  +/-  (2.49e-92, 3.82e-214j)
| (2.6120921170157798831e-05 - 1.1752254431817473928e-662j)  +/-  (3.36e-84, 5.14e-206j)
| (0.054840050723882520947 - 1.0912067350845120268e-659j)  +/-  (8.66e-76, 1.32e-197j)
| (2.0158678142023796635e-12 - 1.5577965508976259395e-666j)  +/-  (1.69e-90, 2.58e-212j)
| (0.0010540154992074051822 - 1.3976950924966257526e-661j)  +/-  (7.28e-83, 1.12e-204j)
| (0.025071219559143311489 + 3.4894153102523261929e-660j)  +/-  (1.09e-79, 1.67e-201j)
| (0.0039733387021454058235 + 5.0162163097525864896e-661j)  +/-  (5.73e-82, 8.77e-204j)
| (2.5997493414620316707e-06 + 1.8660980483630863359e-662j)  +/-  (5.12e-87, 7.84e-209j)
| (0.13892177146966289163 + 3.658103207481673831e-659j)  +/-  (1.41e-77, 2.16e-199j)
| (0.010585178903207895927 - 2.7925945490280996581e-660j)  +/-  (4.62e-83, 7.07e-205j)
| (1.2158528597383463317e-10 + 1.2946734554639356488e-665j)  +/-  (1.98e-90, 3.03e-212j)
| (0.00019509776669949086076 + 3.8405415244339707592e-662j)  +/-  (6.97e-86, 1.07e-207j)
| (0.010585178903207895927 - 1.5039453736153318082e-660j)  +/-  (1.06e-83, 1.62e-205j)
| (0.00019509776669949086076 + 1.0888858273287609251e-661j)  +/-  (2.71e-86, 4.14e-208j)
| (0.025071219559143311489 + 5.7960190377765282931e-660j)  +/-  (1.61e-83, 2.46e-205j)
| (4.4731033821033293551e-07 - 7.9262853321782043228e-663j)  +/-  (1.06e-87, 1.62e-209j)
| (4.4731033821033293551e-07 - 1.7226015083320573972e-663j)  +/-  (1.63e-88, 2.49e-210j)
| (0.090803807855317919196 - 2.9334903334011044548e-659j)  +/-  (3.4e-84, 5.2e-206j)
| (1.8302170767659181139e-14 + 3.1495874009637829872e-669j)  +/-  (2.37e-94, 3.63e-216j)
| (0.090803807855317919196 - 3.5517119138921209269e-659j)  +/-  (4.94e-84, 7.56e-206j)
| (0.054840050723882520947 - 7.3615780444119572645e-660j)  +/-  (2.18e-85, 3.33e-207j)
| (0.088313657809850212483 + 1.6652636164952163581e-659j)  +/-  (4.38e-85, 6.69e-207j)
| (0.13892177146966289163 + 3.2874121059394572212e-659j)  +/-  (2.32e-85, 3.58e-207j)
| (0.17242521251851731121 - 3.1228746053856812806e-659j)  +/-  (1.49e-85, 2.2e-207j)
| (0.0039733387021454058235 + 1.0528519490890458956e-660j)  +/-  (1.12e-87, 1.83e-209j)
