Starting with polynomial:
P : t^4 - 6*t^2 + 3
Extension levels are: 4 7 40
-------------------------------------------------
Trying to find an order 7 Kronrod extension for:
P1 : t^4 - 6*t^2 + 3
Solvable: 1
-------------------------------------------------
Trying to find an order 40 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^51 - 3748164888323388940709113950537899378550142119016790958881879731182326663/3452008873983076440048795220156472902343624860042187456485826643003883*t^49 + 1867662153975143239126747310170663868771959483410615928428973830356217307794/3452008873983076440048795220156472902343624860042187456485826643003883*t^47 - 9645462820748284348900674079458197937323782590077354798784549939181641230476050/58684150857712299480829518742660039339841622620717186760259052931066011*t^45 + 2001898753310817709785174891898277834109774058030504081096443704561401313265247080/58684150857712299480829518742660039339841622620717186760259052931066011*t^43 - 301495864114617411182657266311208860481891245837484999425344668122234940927438190520/58684150857712299480829518742660039339841622620717186760259052931066011*t^41 + 34162742987448473160452058187537035434975171414210965748253950046688939464560673180870/58684150857712299480829518742660039339841622620717186760259052931066011*t^39 - 2980479209522555112568047762601975998496108535260758513214115053577363160228980307999510/58684150857712299480829518742660039339841622620717186760259052931066011*t^37 + 203243452789597030050691503467992173492402919793280337313954326655047117550344679666447475/58684150857712299480829518742660039339841622620717186760259052931066011*t^35 - 643377652424242744387387388894962754374675047510034930117480234163207897415365701165043775/3452008873983076440048795220156472902343624860042187456485826643003883*t^33 + 27476852496223251337374246741554493458490364589651131042332274942030700952491573994309494100/3452008873983076440048795220156472902343624860042187456485826643003883*t^31 - 933531118257148004783066320539544069549443692392686586932870741415613564624324868613343318100/3452008873983076440048795220156472902343624860042187456485826643003883*t^29 + 25224924891544866827046731464469060076270604108124767856977137480111782132819024041339399069800/3452008873983076440048795220156472902343624860042187456485826643003883*t^27 - 540572664481302913661416719888677065320999199904967076593093965964937508243355635919048838521000/3452008873983076440048795220156472902343624860042187456485826643003883*t^25 + 9138695881946375161619832694992698888634200976142435902867608043803253288338973769910923142347500/3452008873983076440048795220156472902343624860042187456485826643003883*t^23 - 120894310319415214830795787647419658936386667341183294392092337107478965468164646101075288430727500/3452008873983076440048795220156472902343624860042187456485826643003883*t^21 + 1237392422315911944795478818802237878077863954693368415553287163649238015212293309914139623933430625/3452008873983076440048795220156472902343624860042187456485826643003883*t^19 - 9650442168240241803482983592206329602512174607906942710175573206008188295616586249862587671322923125/3452008873983076440048795220156472902343624860042187456485826643003883*t^17 + 56186933988559690605858199031965034289830650681546059114007122991848499684817384020794127988923256250/3452008873983076440048795220156472902343624860042187456485826643003883*t^15 - 237599316110131742693749828729981458340494888994263260839926796436216323799769522661436735149756556250/3452008873983076440048795220156472902343624860042187456485826643003883*t^13 + 703070499574945141615546249627729890048730702161076632050583047324635659820693002469622830900063350000/3452008873983076440048795220156472902343624860042187456485826643003883*t^11 - 1382674183315719054388050695773791533962867847426065313660270927979230644844785148711961983089613100000/3452008873983076440048795220156472902343624860042187456485826643003883*t^9 + 1679000703533775438654389550798169606460619949155161151061918041426243802967269835825038315438997268750/3452008873983076440048795220156472902343624860042187456485826643003883*t^7 - 1126401850311081012239995453983557825750585052570062168168090712179077916770482599613086059692053718750/3452008873983076440048795220156472902343624860042187456485826643003883*t^5 + 345717572934617183295563704991174356691094770989938268138052910489820546883088943143102889450063640625/3452008873983076440048795220156472902343624860042187456485826643003883*t^3 - 31800727012789693990202343057044898831806179800030850595352023832416349573784452196817352698884453125/3452008873983076440048795220156472902343624860042187456485826643003883*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:
| (12.493414536808426055 + 1.0853358127337077623e-591j)  +/-  (1.13e-246, 1.13e-246j)
| (-6.6211752527830763906 + 5.2263616971142354654e-598j)  +/-  (1.3e-243, 1.3e-243j)
| (-9.4277276935647708161 - 1.7637291425508216682e-625j)  +/-  (8.93e-244, 8.93e-244j)
| (11.552235096772140366 - 1.4475502375394761045e-646j)  +/-  (1.68e-245, 1.68e-245j)
| (-8.8199606536798438237 - 4.0973896276542879422e-646j)  +/-  (1.56e-243, 1.56e-243j)
| (9.4277276935647708161 - 2.289568620330939487e-658j)  +/-  (8.56e-244, 8.56e-244j)
| (-10.771393978564624202 + 3.0740272213267916502e-657j)  +/-  (9.89e-245, 9.89e-245j)
| (10.771393978564624202 - 3.2352727522225875022e-665j)  +/-  (9.98e-245, 9.98e-245j)
| (8.8199606536798438237 + 3.4236769298923324675e-664j)  +/-  (1.39e-243, 1.39e-243j)
| (6.1099288721057348776 + 1.7660506753607113253e-664j)  +/-  (8.34e-244, 8.34e-244j)
| (0.74196378430272585765 + 1.1711683620459422148e-673j)  +/-  (2.11e-252, 2.11e-252j)
| (3.3747086797406716539 - 1.2537661438963430511e-665j)  +/-  (3.98e-245, 3.98e-245j)
| (-11.552235096772140366 - 1.4395057590970036221e-663j)  +/-  (1.55e-245, 1.55e-245j)
| (-12.493414536808426055 + 1.1576078387589307658e-666j)  +/-  (1.14e-246, 1.14e-246j)
| (-4.6328811754148542753 - 3.86640082616278906e-672j)  +/-  (8.01e-245, 8.01e-245j)
| (10.073113272388543236 + 2.0467812042991414778e-688j)  +/-  (3.62e-244, 3.62e-244j)
| (5.117476997566244678 + 4.587795940013333016e-687j)  +/-  (1.92e-244, 1.92e-244j)
| (7.6837348918550942782 - 4.3963384678197704282e-688j)  +/-  (2.19e-243, 2.19e-243j)
| (-3.3091333397894615165 + 1.3774549437907373761e-692j)  +/-  (2.4e-245, 2.4e-245j)
| (-7.1449641725647205723 + 8.8799249183461941543e-711j)  +/-  (1.85e-243, 1.85e-243j)
| (3.6594867028231916258 + 2.6957792039798310355e-727j)  +/-  (2.3e-245, 2.3e-245j)
| (4.1527208323561910242 - 6.7796255434622677719e-728j)  +/-  (3.58e-245, 3.58e-245j)
| (8.2406459031157235427 - 4.9876826789047167913e-726j)  +/-  (2e-243, 2e-243j)
| (-2.7907071616240613711 - 2.7952194151085545097e-727j)  +/-  (2.59e-247, 2.59e-247j)
| (-8.2406459031157235427 + 1.6618632895181429605e-724j)  +/-  (2.08e-243, 2.08e-243j)
| (-10.073113272388543236 - 6.688938104900983444e-736j)  +/-  (3.5e-244, 3.5e-244j)
| (2.3344142183389772393 + 2.4179111693071676292e-743j)  +/-  (1.84e-248, 1.84e-248j)
| (2.7907071616240613711 + 4.8371795314476720824e-743j)  +/-  (2.5e-247, 2.5e-247j)
| (-5.117476997566244678 - 5.0786345996061258876e-738j)  +/-  (1.87e-244, 1.87e-244j)
| (-2.3344142183389772393 + 1.4378567767046498479e-743j)  +/-  (1.79e-248, 1.79e-248j)
| (-3.3747086797406716539 - 2.6748669408747183698e-738j)  +/-  (3.92e-245, 3.92e-245j)
| (1.4655215253859805014 - 4.1767345532200939041e-747j)  +/-  (1.67e-250, 1.67e-250j)
| (7.1449641725647205723 - 6.8418048808892010291e-741j)  +/-  (1.78e-243, 1.78e-243j)
| (-1.8909565122839236978 - 4.5307130116666453814e-747j)  +/-  (1.66e-249, 1.66e-249j)
| (-3.6594867028231916258 + 1.8989871294582478959e-740j)  +/-  (2.35e-245, 2.35e-245j)
| (-1.0783095564328941651 + 6.5203997292709021569e-752j)  +/-  (1.9e-251, 1.9e-251j)
| (4.6328811754148542753 + 1.9374151687961311208e-744j)  +/-  (8.54e-245, 8.54e-245j)
| (-4.1527208323561910242 + 5.6874318959801172885e-743j)  +/-  (3.56e-245, 3.56e-245j)
| (6.6211752527830763906 + 2.2251276841086814086e-746j)  +/-  (1.29e-243, 1.29e-243j)
| (-7.6837348918550942782 + 3.6107346304318386176e-746j)  +/-  (2.15e-243, 2.15e-243j)
| (-1.4655215253859805014 + 5.9999446535012169517e-762j)  +/-  (1.68e-250, 1.68e-250j)
| (-6.1099288721057348776 + 1.1590487236851790756e-754j)  +/-  (7.52e-244, 7.52e-244j)
| (1.0783095564328941651 + 2.8731710050150707138e-771j)  +/-  (2.03e-251, 2.03e-251j)
| (0.39067267601683762037 + 1.3201994041295177975e-773j)  +/-  (1.48e-253, 1.48e-253j)
| (5.6092593934732976311 - 1.005304605318826969e-764j)  +/-  (4.33e-244, 4.33e-244j)
| (-0.74196378430272585765 - 3.4299470194160162613e-773j)  +/-  (2.27e-252, 2.27e-252j)
| (-0.39067267601683762037 + 6.893418992193967303e-775j)  +/-  (1.48e-253, 1.48e-253j)
| (3.3091333397894615165 - 2.0079043697352077382e-769j)  +/-  (2.61e-245, 2.61e-245j)
| (-5.7260817623080341456e-787 + 1.1274841210108373049e-787j)  +/-  (2.98e-785, 2.98e-785j)
| (1.8909565122839236978 - 8.2098388222236465234e-778j)  +/-  (1.72e-249, 1.72e-249j)
| (-5.6092593934732976311 + 3.0906197502515346812e-780j)  +/-  (4.19e-244, 4.19e-244j)
-------------------------------------------------
The weights are:
| (5.5372858576481390888e-35 - 7.5116773027918681581e-623j)  +/-  (2.14e-85, 5.11e-207j)
| (6.23456336466155266e-11 + 1.3273930850531071482e-607j)  +/-  (3.91e-72, 9.35e-194j)
| (1.2476095762525836415e-20 + 2.9284968866107751023e-614j)  +/-  (5.95e-80, 1.42e-201j)
| (3.5152327296238765484e-30 + 2.8802052494537754481e-620j)  +/-  (2.9e-84, 6.93e-206j)
| (3.0284636823479302981e-18 - 8.0152211661551311445e-613j)  +/-  (9.58e-79, 2.29e-200j)
| (1.2476095762525836415e-20 - 5.1223656375727000434e-615j)  +/-  (5.18e-81, 1.24e-202j)
| (1.8694926425858515398e-26 + 1.2831899171184407658e-617j)  +/-  (4.32e-84, 1.03e-205j)
| (1.8694926425858515398e-26 - 3.0632749640445396565e-618j)  +/-  (1.99e-83, 4.77e-205j)
| (3.0284636823479302981e-18 + 1.1415575358253693883e-613j)  +/-  (3.62e-80, 8.66e-202j)
| (1.5790462361504481796e-09 - 1.944734841884207419e-608j)  +/-  (5.82e-75, 1.39e-196j)
| (0.099844733636181286197 + 3.5246337266560282318e-602j)  +/-  (2.5e-65, 5.97e-187j)
| (-0.0010165142900650483914 - 4.7910709902492043433e-603j)  +/-  (2.5e-65, 5.97e-187j)
| (3.5152327296238765484e-30 - 1.0606280806688921052e-619j)  +/-  (2.72e-87, 6.49e-209j)
| (5.5372858576481390888e-35 + 2.4321164248458171152e-622j)  +/-  (1.74e-89, 4.17e-211j)
| (4.1969126094327172161e-06 + 3.1445953445164988536e-605j)  +/-  (4.22e-73, 1.01e-194j)
| (2.4695256574207775565e-23 + 1.5967692655599982372e-616j)  +/-  (2.04e-83, 4.86e-205j)
| (4.0035711787673869026e-07 - 8.8797170211847140297e-607j)  +/-  (1.24e-74, 2.97e-196j)
| (3.3018783504066663661e-14 + 2.5069791193127843108e-611j)  +/-  (3.25e-80, 7.77e-202j)
| (0.0018384625596751064814 + 1.4863776967809293532e-602j)  +/-  (1.18e-70, 2.83e-192j)
| (1.7393294197001432959e-12 + 7.1016723248373841131e-609j)  +/-  (8.3e-81, 1.98e-202j)
| (0.00026873367006038457705 + 4.8876024068849667983e-604j)  +/-  (6.63e-71, 1.58e-192j)
| (3.4467563905259585203e-05 - 3.7340967991592598643e-605j)  +/-  (9.46e-73, 2.26e-194j)
| (4.060898080435662142e-16 - 1.9051867150137726651e-612j)  +/-  (2.42e-81, 5.79e-203j)
| (0.0037726443180705676572 - 3.7420189185912086255e-603j)  +/-  (8.77e-71, 2.09e-192j)
| (4.060898080435662142e-16 + 1.7481198621903001066e-611j)  +/-  (5.69e-84, 1.36e-205j)
| (2.4695256574207775565e-23 - 7.7200170756039423346e-616j)  +/-  (4.2e-87, 1e-208j)
| (0.011764223655187952749 + 2.5932893119334910046e-603j)  +/-  (5.86e-70, 1.4e-191j)
| (0.0037726443180705676572 - 1.52298098734508803e-603j)  +/-  (2.86e-71, 6.82e-193j)
| (4.0035711787673869026e-07 - 6.9315305964712717236e-606j)  +/-  (2.27e-79, 5.41e-201j)
| (0.011764223655187952749 + 5.4175764218164896493e-603j)  +/-  (3.77e-72, 9e-194j)
| (-0.0010165142900650483914 - 1.4751173855051126816e-602j)  +/-  (1.82e-74, 4.33e-196j)
| (0.05605804142976254744 + 1.1333772119717410885e-602j)  +/-  (9.61e-70, 2.29e-191j)
| (1.7393294197001432959e-12 - 2.7024165675745859496e-610j)  +/-  (1.26e-83, 3.01e-205j)
| (0.029124190816978930991 - 9.4168331233432053912e-603j)  +/-  (3.74e-72, 8.93e-194j)
| (0.00026873367006038457705 + 1.696522587922521485e-603j)  +/-  (3.01e-76, 7.2e-198j)
| (0.079928480360799832414 - 3.2299147109649565844e-602j)  +/-  (3.29e-70, 7.82e-192j)
| (4.1969126094327172161e-06 + 5.5559757246761723544e-606j)  +/-  (2.36e-79, 5.6e-201j)
| (3.4467563905259585203e-05 - 1.6297175655094047481e-604j)  +/-  (4.07e-78, 9.73e-200j)
| (6.23456336466155266e-11 + 2.4608301317323165801e-609j)  +/-  (2.54e-83, 6.04e-205j)
| (3.3018783504066663661e-14 - 3.3746382420989295832e-610j)  +/-  (8.84e-86, 2.12e-207j)
| (0.05605804142976254744 + 1.7776867829273321808e-602j)  +/-  (3.8e-75, 8.92e-197j)
| (1.5790462361504481796e-09 - 4.8423957034829749556e-607j)  +/-  (1.93e-83, 4.56e-205j)
| (0.079928480360799832414 - 2.3252443960914153495e-602j)  +/-  (1.48e-75, 3.32e-197j)
| (0.13902277347417826149 - 3.9571984432484845544e-602j)  +/-  (3.34e-75, 7.58e-197j)
| (2.9112428822913068676e-08 + 1.3699981477412944694e-607j)  +/-  (2.48e-82, 5.58e-204j)
| (0.099844733636181286197 + 4.4142261545761007613e-602j)  +/-  (1.25e-75, 2.84e-197j)
| (0.13902277347417826149 - 4.4534387938484973489e-602j)  +/-  (6.76e-76, 1.53e-197j)
| (0.0018384625596751064814 + 4.9576765380164463587e-603j)  +/-  (5.13e-78, 1.15e-199j)
| (0.15871026955988831862 + 4.0767968561508924341e-602j)  +/-  (1.27e-76, 3.1e-198j)
| (0.029124190816978930991 - 5.2330685408882216973e-603j)  +/-  (6.82e-78, 1.45e-199j)
| (2.9112428822913068676e-08 + 1.6557166817046471311e-606j)  +/-  (5.22e-83, 1.35e-204j)
