Starting with polynomial:
P : t^4 - 6*t^2 + 3
Extension levels are: 4 7 42
-------------------------------------------------
Trying to find an order 7 Kronrod extension for:
P1 : t^4 - 6*t^2 + 3
Solvable: 1
-------------------------------------------------
Trying to find an order 42 Kronrod extension for:
P2 : t^11 - 41*t^9 + 528*t^7 - 2520*t^5 + 4095*t^3 - 1575*t
Solvable: 1
-------------------------------------------------
Ending with final polynomial:
P : t^53 - 15767333081375341737723581901043029147232858209489011672160283897003807298951/12172946534731404259994921631531913551723325534407284336727305941037734402*t^51 + 345792102701007820316307325774342704052706444046638954170893106972790542378858785/450399021785061957619812100366680801413763044773069520458910319818396172874*t^49 - 62374116066591860369371366533819933204872997053474302072525524589587910306385613657/225199510892530978809906050183340400706881522386534760229455159909198086437*t^47 + 15363094522379800363025724339352992881157304878289759464920368630977069822354959033235/225199510892530978809906050183340400706881522386534760229455159909198086437*t^45 - 2746539313221555841428329734499334148323949964868491524148216810669467630417413288825060/225199510892530978809906050183340400706881522386534760229455159909198086437*t^43 + 369838502195314343538092352487399461076816484582542098546852365127256043327031019037131470/225199510892530978809906050183340400706881522386534760229455159909198086437*t^41 - 38419260470448634277199514212340951805874203529857887322265027440125980776863115530599421875/225199510892530978809906050183340400706881522386534760229455159909198086437*t^39 + 4449762295430423971000791673712187865718096049942738396994014167385340032808240088962146930/320340698282405375263024253461366146097982250905454850966508051079940379*t^37 - 573929144964966403140732628991280119305534435707039647716409588853992675636997577320947449125/640681396564810750526048506922732292195964501810909701933016102159880758*t^35 + 29507184643949478593601088125910514219904822933918704413658527166418048622568738486289559177275/640681396564810750526048506922732292195964501810909701933016102159880758*t^33 - 606788295300064404767433429942068727738198279104274539863224201972653612166885983749240669068150/320340698282405375263024253461366146097982250905454850966508051079940379*t^31 + 19981019133143070139544415855555694914213816725302519397655957663683774629032121909479979180274250/320340698282405375263024253461366146097982250905454850966508051079940379*t^29 - 526019505941375089854350338659962138086041404059324268241475278762282884412605234739950466084660300/320340698282405375263024253461366146097982250905454850966508051079940379*t^27 + 11030028478135654339952922146587589609759312370109926185978777434145412777504057536320329414518886000/320340698282405375263024253461366146097982250905454850966508051079940379*t^25 - 183110211388146793955055342168573900921673063562361421509070774123991365836802173846846299604247586250/320340698282405375263024253461366146097982250905454850966508051079940379*t^23 + 2385863160057886703553802456629312463917108649839639423064829682212715313829651923772578683209402169375/320340698282405375263024253461366146097982250905454850966508051079940379*t^21 - 48227230054961722719503346358058833304230386585118466834553445495294945566136990076466750361234951284375/640681396564810750526048506922732292195964501810909701933016102159880758*t^19 + 372209852058176776776775450786136361523226364255206909916742824722562035559939855340148901447464700493125/640681396564810750526048506922732292195964501810909701933016102159880758*t^17 - 1074263969982152291758628224950542755917583329342234627127495841087224264317580559259049815560619510721875/320340698282405375263024253461366146097982250905454850966508051079940379*t^15 + 4511186473705371948525056930663075034737519006485668680390828153771449799361194357539995267765551303465625/320340698282405375263024253461366146097982250905454850966508051079940379*t^13 - 13275076637172849463472883334828507315562345083127328317657923416531368516568531926978432513028375739275000/320340698282405375263024253461366146097982250905454850966508051079940379*t^11 + 25996033695633918479486874574982081981962139287958172377649767370443395728677214733392152480333497490718750/320340698282405375263024253461366146097982250905454850966508051079940379*t^9 - 31469298621941054344308140001781725247221874412162703326519750530756819627235062541118762304502152432565625/320340698282405375263024253461366146097982250905454850966508051079940379*t^7 + 21068288220735252466816544507863729538874960959399226981218194631352567089607163353929164373208426984562500/320340698282405375263024253461366146097982250905454850966508051079940379*t^5 - 12918673252209893224754169300657155335519160085431261363622093361927833713060258528914584046427006499234375/640681396564810750526048506922732292195964501810909701933016102159880758*t^3 + 1188482173252745613881830180639479086036271425257587123484695822918877507087761617344111972337554040390625/640681396564810750526048506922732292195964501810909701933016102159880758*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
-------------------------------------------------
The nodes are:
| (-14.742555203084679971 - 3.7027156635072062577e-766j)  +/-  (6.23e-248, 6.23e-248j)
| (11.48369674503838461 + 1.1437782971152816549e-765j)  +/-  (8.12e-245, 8.12e-245j)
| (14.742555203084679971 - 2.4956730479535836839e-768j)  +/-  (6.25e-248, 6.25e-248j)
| (9.396503843775963893 + 1.5730542492381327523e-762j)  +/-  (2.14e-243, 2.14e-243j)
| (10.033157169845009656 + 1.2209939398892826952e-763j)  +/-  (1.07e-243, 1.07e-243j)
| (-10.033157169845009656 + 2.5079110802831322974e-765j)  +/-  (1.1e-243, 1.1e-243j)
| (7.6692317790245161834 + 2.0398411222659277989e-774j)  +/-  (3.95e-243, 3.95e-243j)
| (-2.3344142183389772393 + 8.6732248837825173925e-782j)  +/-  (2.03e-248, 2.03e-248j)
| (-8.2216970756860563018 - 1.6543472616498410959e-775j)  +/-  (4.45e-243, 4.45e-243j)
| (1.891112027720486993 + 3.3107299238467793588e-789j)  +/-  (2e-249, 2e-249j)
| (-11.48369674503838461 + 1.7842382537533478825e-782j)  +/-  (7.57e-245, 7.57e-245j)
| (2.7899195375895174427 - 1.9909053107072971105e-787j)  +/-  (2.52e-247, 2.52e-247j)
| (-12.395938929300623527 - 8.3877644440408871884e-786j)  +/-  (8.68e-246, 8.68e-246j)
| (-3.6664211501439944172 - 9.8502494501358001436e-785j)  +/-  (2.26e-245, 2.26e-245j)
| (-6.10459262172341658 + 3.8664103829496043049e-782j)  +/-  (1.25e-243, 1.25e-243j)
| (-10.719752361307239069 - 1.9134731916354302487e-788j)  +/-  (3.42e-244, 3.42e-244j)
| (-8.7955516735233886003 + 6.5797050915942424247e-801j)  +/-  (3.38e-243, 3.38e-243j)
| (-6.6133428944992600391 - 1.2412872126692465848e-823j)  +/-  (2.25e-243, 2.25e-243j)
| (-3.298156079474225201 - 4.4074363911654963912e-838j)  +/-  (2.36e-245, 2.36e-245j)
| (3.6664211501439944172 - 1.353777571704081619e-838j)  +/-  (2.42e-245, 2.42e-245j)
| (3.298156079474225201 + 4.6431730750370844055e-838j)  +/-  (2.07e-245, 2.07e-245j)
| (12.395938929300623527 - 3.4074941020837671917e-839j)  +/-  (8.45e-246, 8.45e-246j)
| (5.6059932285035207981 + 9.0898858352956322453e-837j)  +/-  (6.63e-244, 6.63e-244j)
| (-4.6328811754148542753 + 3.0715790645798868048e-836j)  +/-  (9.38e-245, 9.38e-245j)
| (-2.7899195375895174427 - 3.170898660013735117e-840j)  +/-  (2.63e-247, 2.63e-247j)
| (8.2216970756860563018 - 2.6138261486077883631e-835j)  +/-  (4.24e-243, 4.24e-243j)
| (1.0780655736595501984 + 6.4971114090591320249e-848j)  +/-  (2.04e-251, 2.04e-251j)
| (-7.1341148051874572235 + 1.9701391929065043746e-838j)  +/-  (3.5e-243, 3.5e-243j)
| (4.1543764535318397252 - 2.0434183253941969389e-855j)  +/-  (3.6e-245, 3.6e-245j)
| (-1.891112027720486993 + 5.04069882608630347e-860j)  +/-  (1.91e-249, 1.91e-249j)
| (-4.1543764535318397252 - 7.6865294527919291406e-854j)  +/-  (3.74e-245, 3.74e-245j)
| (1.4655215253859805014 - 2.7562957587060515083e-860j)  +/-  (1.88e-250, 1.88e-250j)
| (-1.4655215253859805014 + 4.4975881295042110983e-860j)  +/-  (1.85e-250, 1.85e-250j)
| (6.6133428944992600391 - 7.2728224171649898821e-854j)  +/-  (2.23e-243, 2.23e-243j)
| (-1.0780655736595501984 - 1.1303518174798809809e-862j)  +/-  (1.88e-251, 1.88e-251j)
| (-3.3747086797406716539 + 2.0148287640749682622e-857j)  +/-  (3.26e-245, 3.26e-245j)
| (6.10459262172341658 + 8.2164062375977665727e-856j)  +/-  (1.34e-243, 1.34e-243j)
| (3.3747086797406716539 + 6.4494760557681155757e-857j)  +/-  (3.36e-245, 3.36e-245j)
| (4.6328811754148542753 - 1.7770170572795542685e-856j)  +/-  (1e-244, 1e-244j)
| (10.719752361307239069 - 3.0062849111315359275e-854j)  +/-  (3.59e-244, 3.59e-244j)
| (8.7955516735233886003 + 3.0008679800000417354e-853j)  +/-  (3.78e-243, 3.78e-243j)
| (7.1341148051874572235 + 1.6414042210187773494e-854j)  +/-  (3.38e-243, 3.38e-243j)
| (0.74196378430272585765 - 8.1367483643208607762e-866j)  +/-  (2.12e-252, 2.12e-252j)
| (2.3344142183389772393 - 2.5527934741999772379e-861j)  +/-  (2.15e-248, 2.15e-248j)
| (-7.6692317790245161834 - 2.7435431067241352373e-855j)  +/-  (4e-243, 4e-243j)
| (-0.39090034640620168249 + 4.6979966463086490531e-882j)  +/-  (1.31e-253, 1.31e-253j)
| (6.3286246456872136531e-897 - 3.1833679650787867788e-897j)  +/-  (3.75e-895, 3.75e-895j)
| (5.1159471599880115149 - 1.1650013409931606736e-872j)  +/-  (2.67e-244, 2.67e-244j)
| (0.39090034640620168249 - 5.2030049350147144518e-881j)  +/-  (1.54e-253, 1.54e-253j)
| (-9.396503843775963893 - 3.6256490175050001465e-872j)  +/-  (2.19e-243, 2.19e-243j)
| (-5.6059932285035207981 - 4.2300141189695463509e-886j)  +/-  (6.96e-244, 6.96e-244j)
| (-0.74196378430272585765 + 1.6786023656548381498e-899j)  +/-  (2.14e-252, 2.14e-252j)
| (-5.1159471599880115149 - 4.8302771182454810231e-895j)  +/-  (2.55e-244, 2.55e-244j)
-------------------------------------------------
The weights are:
| (9.8013693316204496703e-46 + 2.0111368606385858401e-796j)  +/-  (3.64e-78, 1.96e-199j)
| (7.5476753569619103477e-30 - 1.371431123151942645e-787j)  +/-  (7.92e-73, 4.26e-194j)
| (9.8013693316204496703e-46 - 9.0783812686355412109e-796j)  +/-  (9.14e-79, 4.91e-200j)
| (1.653320356183165976e-20 - 1.3212249959145941486e-781j)  +/-  (4.88e-69, 2.62e-190j)
| (3.6352485522501728868e-23 - 1.5411582340396926896e-783j)  +/-  (2.87e-70, 1.54e-191j)
| (3.6352485522501728868e-23 + 4.9987320264798692558e-785j)  +/-  (2.56e-73, 1.38e-194j)
| (3.6636223238077396973e-14 + 4.9467304728359190585e-779j)  +/-  (2.73e-66, 1.47e-187j)
| (0.01175443796477297362 + 3.1566820633023202391e-772j)  +/-  (6.28e-45, 3.38e-166j)
| (4.7052799530441290694e-16 - 4.1397151124118618887e-781j)  +/-  (9.98e-70, 5.36e-191j)
| (0.029117214107713100573 - 9.3303990936149210267e-772j)  +/-  (2.76e-42, 1.48e-163j)
| (7.5476753569619103477e-30 + 1.3719957782138427149e-788j)  +/-  (1.4e-76, 7.55e-198j)
| (0.0037691867256882447994 - 3.5267871226965924051e-772j)  +/-  (6.14e-51, 3.3e-172j)
| (1.791900431102503319e-34 - 5.3397836954566588969e-791j)  +/-  (1.88e-78, 1.01e-199j)
| (0.00025286725993809680904 + 5.8783414993187011631e-773j)  +/-  (2.27e-57, 1.22e-178j)
| (1.6240372016448792901e-09 - 3.1823356234281278313e-777j)  +/-  (2.17e-66, 1.17e-187j)
| (3.1952231088705036431e-26 - 1.1352063921446048924e-786j)  +/-  (1.72e-75, 9.22e-197j)
| (3.7163729897564537688e-18 + 2.7372933566572670822e-782j)  +/-  (2.89e-72, 1.56e-193j)
| (6.5314299606799617006e-11 + 4.2712684230704214245e-778j)  +/-  (4.23e-68, 2.27e-189j)
| (0.0015722299288428408655 + 5.3755731087367357001e-772j)  +/-  (5.46e-58, 2.94e-179j)
| (0.00025286725993809680904 + 1.3402147025245769778e-772j)  +/-  (3.87e-62, 2.08e-183j)
| (0.0015722299288428408655 + 1.1190993176891254385e-771j)  +/-  (2.57e-60, 1.38e-181j)
| (1.791900431102503319e-34 + 3.8776362322242024514e-790j)  +/-  (1.59e-83, 8.56e-205j)
| (2.9537744402830089234e-08 + 8.4133731276630359046e-776j)  +/-  (4.21e-70, 2.26e-191j)
| (4.1838608025781330308e-06 + 7.8373811568254842346e-775j)  +/-  (7.88e-66, 4.24e-187j)
| (0.0037691867256882447994 - 1.9118392168410993274e-772j)  +/-  (3.51e-59, 1.88e-180j)
| (4.7052799530441290694e-16 - 6.2115799184890398228e-780j)  +/-  (6.98e-76, 3.75e-197j)
| (0.079969867443172082165 - 3.3359011079848559753e-771j)  +/-  (4.01e-51, 2.16e-172j)
| (1.8676806947368095799e-12 - 5.0093058691559061614e-779j)  +/-  (1.09e-71, 5.86e-193j)
| (3.4086372599384002436e-05 - 1.2987413771566837554e-773j)  +/-  (2.99e-66, 1.61e-187j)
| (0.029117214107713100573 - 6.2037290568076501003e-772j)  +/-  (1.37e-57, 7.37e-179j)
| (3.4086372599384002436e-05 - 5.0236030686624702336e-774j)  +/-  (8.83e-65, 4.75e-186j)
| (0.056093934061562650282 + 1.7995508726322914387e-771j)  +/-  (1.77e-56, 9.52e-178j)
| (0.056093934061562650282 + 1.3139115588833317895e-771j)  +/-  (8.68e-57, 4.67e-178j)
| (6.5314299606799617006e-11 + 2.4576288053167279982e-777j)  +/-  (1.27e-73, 6.83e-195j)
| (0.079969867443172082165 - 2.6491617156710753906e-771j)  +/-  (1.92e-56, 1.03e-177j)
| (-0.00072622367899343512757 - 5.1238671492205293602e-772j)  +/-  (1.92e-62, 1.03e-183j)
| (1.6240372016448792901e-09 - 1.498828497469562405e-776j)  +/-  (1.58e-72, 8.47e-194j)
| (-0.00072622367899343512757 - 1.0867759140887274095e-771j)  +/-  (6.48e-65, 3.48e-186j)
| (4.1838608025781330308e-06 + 2.3084896882080512681e-774j)  +/-  (1.45e-69, 7.78e-191j)
| (3.1952231088705036431e-26 + 1.7229058231098284841e-785j)  +/-  (1.17e-82, 6.32e-204j)
| (3.7163729897564537688e-18 + 8.2931979033113152104e-781j)  +/-  (1.3e-78, 6.98e-200j)
| (1.8676806947368095799e-12 - 3.661303992629878715e-778j)  +/-  (2.09e-75, 1.12e-196j)
| (0.099706808890770860186 + 4.6339835697143072913e-771j)  +/-  (1.44e-64, 7.72e-186j)
| (0.01175443796477297362 + 5.2438826317781723407e-772j)  +/-  (4.83e-67, 2.6e-188j)
| (3.6636223238077396973e-14 + 5.0047179187513113679e-780j)  +/-  (1.62e-78, 8.69e-200j)
| (0.13903597780269829238 - 4.3663936217886117396e-771j)  +/-  (2.46e-66, 1.32e-187j)
| (0.15882999171158220523 + 4.4161666387509790345e-771j)  +/-  (1.64e-66, 8.82e-188j)
| (4.0217564053342919982e-07 - 4.4477520157730418106e-775j)  +/-  (2.32e-72, 1.25e-193j)
| (0.13903597780269829238 - 4.7454924740862314919e-771j)  +/-  (2.3e-66, 1.23e-187j)
| (1.653320356183165976e-20 - 1.3816040796305451916e-783j)  +/-  (3.66e-82, 1.97e-203j)
| (2.9537744402830089234e-08 + 2.1253342405336788689e-776j)  +/-  (7.69e-76, 4.14e-197j)
| (0.099706808890770860186 + 3.9556613749702108368e-771j)  +/-  (3.91e-70, 2.76e-191j)
| (4.0217564053342919982e-07 - 1.3117004557427573026e-775j)  +/-  (4.89e-75, 2.62e-196j)
