Starting with polynomial:
P : t^2 - 1
Extension levels are: 2 3 34
-------------------------------------------------
Trying to find an order 3 Kronrod extension for:
P1 : t^2 - 1
Solvable: 1
-------------------------------------------------
Trying to find an order 34 Kronrod extension for:
P2 : t^5 - 7*t^3 + 6*t
Solvable: 1
-------------------------------------------------
Ending with final polynomial:
P : t^39 - 42915193612261847582544878238169649/62958174504496216169015452340832*t^37 + 39340257054363348447395763418558876661/188874523513488648507046357022496*t^35 - 3570921378093460968383983741100202952285/94437261756744324253523178511248*t^33 + 35865924641690410601790239642531362277665/7869771813062027021126931542604*t^31 - 6076811986370637275092424936467957709945335/15739543626124054042253863085208*t^29 + 373747031723820140570255406700654936170877045/15739543626124054042253863085208*t^27 - 1416773013178170637001175069798738818763897105/1311628635510337836854488590434*t^25 + 96251371789290534307622641963877676883464458625/2623257271020675673708977180868*t^23 - 425870316780935994041361777291287158950428526375/456218655829682725862430814064*t^21 + 8093559647569396962767066281927055438251951613625/456218655829682725862430814064*t^19 - 56994739075220990836206258099426335082563031762375/228109327914841362931215407032*t^17 + 293367708855621824760224984260177343607382755523875/114054663957420681465607703516*t^15 - 4324014302290064541449472337155420056446552834966875/228109327914841362931215407032*t^13 + 22130846018478748548288483515771569061577613877525625/228109327914841362931215407032*t^11 - 9406293128516292234509094852066879951052844064800625/28513665989355170366401925879*t^9 + 79475809402824538918916129246686620338666559061708125/114054663957420681465607703516*t^7 - 748561094051731693139081998104963914846207735664148125/912437311659365451724861628128*t^5 + 403370034899718385409712554880660963805900293390496875/912437311659365451724861628128*t^3 - 31487846715248591251022549917788619806813814078284375/456218655829682725862430814064*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:   39 out of 39
Indefinite weights: 0 out of 39
Negative weights:   0 out of 39
Extension rule has valid weights: 1
-------------------------------------------------
The nodes are:
| (-3.7765709621516633696 - 1.1636124897139266663e-773j)  +/-  (2.31e-247, 2.31e-247j)
| (-8.4435082819243437975 - 2.3743294447714674653e-771j)  +/-  (3.95e-246, 3.95e-246j)
| (-9.9943243872160398947 - 1.0666201250010724153e-777j)  +/-  (3.51e-247, 3.51e-247j)
| (-4.810267736730986311 + 1.2454129077875573443e-781j)  +/-  (1.68e-246, 1.68e-246j)
| (-10.97772375540709576 - 3.1774796235713140107e-785j)  +/-  (3.75e-248, 3.75e-248j)
| (-9.1764600284686348601 - 2.4620573446863020983e-802j)  +/-  (1.68e-246, 1.68e-246j)
| (-7.764805946718047287 + 1.7120817549977091501e-831j)  +/-  (6.83e-246, 6.83e-246j)
| (-1 + 4.9118373642585990378e-853j)  +/-  (1.96e-252, 1.96e-252j)
| (-5.926581503578052863 + 3.243904112218807442e-851j)  +/-  (5.97e-246, 5.97e-246j)
| (10.97772375540709576 + 1.0494156705508465954e-865j)  +/-  (3.61e-248, 3.61e-248j)
| (9.1764600284686348601 + 4.1694079051577984106e-863j)  +/-  (1.53e-246, 1.53e-246j)
| (2.8830420156555552017 + 4.7737720342487893994e-866j)  +/-  (2.03e-248, 2.03e-248j)
| (6.5138676610279647721 + 1.0638580662985663365e-863j)  +/-  (7.84e-246, 7.84e-246j)
| (9.9943243872160398947 - 2.4368503097211532238e-865j)  +/-  (3.97e-247, 3.97e-247j)
| (4.810267736730986311 + 1.6097367090695287349e-864j)  +/-  (1.66e-246, 1.66e-246j)
| (7.764805946718047287 - 6.0831516262127064359e-864j)  +/-  (6.77e-246, 6.77e-246j)
| (7.1246723154052708987 + 5.9290758861403970685e-865j)  +/-  (8.07e-246, 8.07e-246j)
| (-6.5138676610279647721 - 1.0373258272060876727e-867j)  +/-  (7.66e-246, 7.66e-246j)
| (5.3592448315564107421 - 1.6870346771338407329e-876j)  +/-  (3.1e-246, 3.1e-246j)
| (3.7765709621516633696 - 1.7759585492394302838e-878j)  +/-  (2.43e-247, 2.43e-247j)
| (-5.3592448315564107421 + 1.050661923735809015e-875j)  +/-  (3.54e-246, 3.54e-246j)
| (3.3106568969158138507 - 8.4825991482494424515e-884j)  +/-  (8.1e-248, 8.1e-248j)
| (-2.8830420156555552017 + 4.0712284326677074981e-884j)  +/-  (2.21e-248, 2.21e-248j)
| (-4.2807185344466849722 + 4.001593096999584198e-883j)  +/-  (6.92e-247, 6.92e-247j)
| (-7.1246723154052708987 + 5.5419359335296933773e-894j)  +/-  (9.32e-246, 9.32e-246j)
| (4.2807185344466849722 - 2.8842904952154958453e-895j)  +/-  (7.21e-247, 7.21e-247j)
| (1.9831222306167850178 + 9.7502734567141751231e-901j)  +/-  (4.04e-250, 4.04e-250j)
| (8.4435082819243437975 + 7.4716462761370204837e-895j)  +/-  (4.1e-246, 4.1e-246j)
| (1.4959716277147448352 - 7.8802270216719836251e-902j)  +/-  (3.61e-251, 3.61e-251j)
| (-1.9831222306167850178 - 6.9133457379175351146e-902j)  +/-  (4.15e-250, 4.15e-250j)
| (3.6874489941109678494e-912 - 1.1468409135404227972e-911j)  +/-  (4.7e-910, 4.7e-910j)
| (-0.50058938498617141876 + 1.9793903040745968819e-903j)  +/-  (1.22e-253, 1.22e-253j)
| (1 + 2.1306812037288279419e-903j)  +/-  (2.07e-252, 2.07e-252j)
| (5.926581503578052863 - 8.0542816330060380933e-899j)  +/-  (5.86e-246, 5.86e-246j)
| (-3.3106568969158138507 - 1.5614975894412037876e-901j)  +/-  (7.57e-248, 7.57e-248j)
| (-1.4959716277147448352 + 7.874411325194221475e-905j)  +/-  (3.29e-251, 3.29e-251j)
| (2.4494897427831780982 - 2.4972847828269799435e-905j)  +/-  (3.86e-249, 3.86e-249j)
| (0.50058938498617141876 - 7.9507882937586836755e-909j)  +/-  (1.22e-253, 1.22e-253j)
| (-2.4494897427831780982 + 3.8431088823542872077e-904j)  +/-  (3.74e-249, 3.74e-249j)
-------------------------------------------------
The weights are:
| (0.00015554384474828760935 - 4.8913914682088616146e-777j)  +/-  (1.45e-66, 4.66e-190j)
| (9.2568057470925125536e-17 + 3.4521086181890916497e-785j)  +/-  (3.97e-80, 1.28e-203j)
| (7.1564955686511235933e-23 + 1.6232976164600676482e-788j)  +/-  (2.67e-83, 8.59e-207j)
| (2.0351473753195190507e-06 + 9.6425806525141835263e-779j)  +/-  (1.65e-72, 5.33e-196j)
| (3.0676156230103030003e-27 - 7.4829576990646561731e-791j)  +/-  (2.22e-85, 7.15e-209j)
| (1.5884823282157663943e-19 - 1.0663934290971651913e-786j)  +/-  (5.44e-82, 1.75e-205j)
| (2.120856980873003087e-14 - 7.5491128708040674924e-784j)  +/-  (4.62e-80, 1.49e-203j)
| (0.12055772182343694821 - 1.6161702910408577282e-776j)  +/-  (6.53e-72, 2.1e-195j)
| (5.4304234232227991115e-09 + 1.3665969683329907816e-780j)  +/-  (4.47e-77, 1.44e-200j)
| (3.0676156230103030003e-27 + 3.6421726394916702219e-791j)  +/-  (1.09e-91, 3.5e-215j)
| (1.5884823282157663943e-19 + 4.4068781686746445344e-787j)  +/-  (2.96e-89, 9.54e-213j)
| (0.0026340625065102582753 - 1.9037026255760004405e-777j)  +/-  (1.26e-74, 4.05e-198j)
| (1.4593475485928267437e-10 + 3.7080465610134262827e-782j)  +/-  (2.87e-85, 9.26e-209j)
| (7.1564955686511235933e-23 - 7.2978703919271117261e-789j)  +/-  (1e-90, 3.24e-214j)
| (2.0351473753195190507e-06 - 1.1622246270415910645e-779j)  +/-  (1.86e-82, 5.98e-206j)
| (2.120856980873003087e-14 + 2.6316436527913784013e-784j)  +/-  (6.19e-88, 1.99e-211j)
| (2.3641979800627171022e-12 - 3.5938433346479128625e-783j)  +/-  (8.15e-87, 2.62e-210j)
| (1.4593475485928267437e-10 - 1.3900909111434262377e-781j)  +/-  (1.01e-85, 3.26e-209j)
| (1.2906360546276143602e-07 + 2.0418164762046986992e-780j)  +/-  (2.1e-84, 6.76e-208j)
| (0.00015554384474828760935 - 2.3982223327316308071e-778j)  +/-  (6.19e-82, 1.99e-205j)
| (1.2906360546276143602e-07 - 1.17678150945325071e-779j)  +/-  (1.89e-84, 6.09e-208j)
| (0.00073685883773214424839 + 8.014663856929365062e-778j)  +/-  (3.46e-81, 1.12e-204j)
| (0.0026340625065102582753 - 1.4180563236123789566e-776j)  +/-  (9.98e-82, 3.21e-205j)
| (2.1700829521746146731e-05 - 9.1052588731504020069e-778j)  +/-  (2.52e-83, 8.13e-207j)
| (2.3641979800627171022e-12 + 1.1651145755031729405e-782j)  +/-  (1.99e-87, 6.42e-211j)
| (2.1700829521746146731e-05 + 5.7028948980251373399e-779j)  +/-  (3.36e-84, 1.08e-207j)
| (0.026770461161271274424 - 5.0526475696965072289e-777j)  +/-  (1.54e-81, 4.96e-205j)
| (9.2568057470925125536e-17 - 1.3573589252944430234e-785j)  +/-  (2.51e-91, 8.06e-215j)
| (0.0642231702730168224 + 7.0742900312276537241e-777j)  +/-  (5.47e-81, 1.75e-204j)
| (0.026770461161271274424 - 1.6220759731403514813e-776j)  +/-  (4.41e-83, 1.41e-206j)
| (0.19977261990680679822 - 1.3871359865280861047e-776j)  +/-  (8.95e-81, 2.89e-204j)
| (0.17606613969416888786 + 1.5374891329287869903e-776j)  +/-  (5.65e-81, 1.82e-204j)
| (0.12055772182343694821 - 9.3962833433103833369e-777j)  +/-  (1.2e-81, 3.89e-205j)
| (5.4304234232227991115e-09 - 3.0342354757570693655e-781j)  +/-  (2.28e-87, 7.36e-211j)
| (0.00073685883773214424839 + 1.2183189441882818196e-776j)  +/-  (1.71e-84, 5.48e-208j)
| (0.0642231702730168224 + 1.6350745461965244051e-776j)  +/-  (1.97e-83, 6.72e-207j)
| (0.0089458612864657720744 + 3.3631277930940851105e-777j)  +/-  (6.98e-84, 2.67e-207j)
| (0.17606613969416888786 + 1.1777031476542692167e-776j)  +/-  (3.07e-83, 1.37e-206j)
| (0.0089458612864657720744 + 1.577094160960530979e-776j)  +/-  (4.65e-84, 1.42e-207j)
