Magma V2.15-3 Wed May 6 2009 15:06:57 on host-57-89 [Seed = 4122879581] Type ? for help. Type -D to quit. Loading "JSearch.m" s= -2 it appears that the Torsion subgroup is isomorphic to Abelian Group of order 1 computing the 2-Selmer group might take a very long time 2-Selmer rank is 1 two torsion subgroup is Abelian Group of order 1 Hence rank is at most 1 searching for enough torsion points and for 1 independent points on J this could take forever! found 1 independent points on J and enough torsion basis for torsion is [] torsion subgroup isomorphic to Abelian Group of order 1 basis for a free subgroup of finite index is [ (x - t^2 - t - 1, -t^2 - 2*t - 1, 1) ] it appears that a complete list of rational points on C is [ (1 : 0 : 0), (t^2 + t + 1 : t^2 + 2*t + 1 : 1), (t^2 + t + 1 : -t^2 - 2*t - 1 : 1) ] want to prove this using Chabauty and the Mordell--Weil sieve working with a subgroup A of the Mordell--Weil group isomorphic to Abelian Group isomorphic to Z Defined on 1 generator (free) the basis for this subgroup is [ (x - t^2 - t - 1, -t^2 - 2*t - 1, 1) ] using (t^2 + t + 1 : t^2 + 2*t + 1 : 1) as base point for Abel-Jacobi map the image of the known rational points on C in A is [ A.1, 0, 2*A.1 ] applying the Mordell--Weil sieve verified saturation at 2 verified saturation at 3 verified saturation at 5 verified saturation at 7 verified saturation at 11 verified saturation at 13 verified saturation at 17 verified saturation at 19 verified saturation at 23 verified saturation at 29 verified saturation at 31 verified saturation at 37 verified saturation at 41 verified saturation at 43 verified saturation at 47 verified saturation at 53 verified saturation at 59 verified saturation at 61 verified saturation at 67 verified saturation at 71 verified saturation at 73 getting sieving information 121 14080 5 17 290 9 23 530 24 29 900 4 31 1375 3 43 1850 44 43 1850 44 43 1850 19 47 2210 21 59 3600 6 83 6890 43 89 8100 3 101 10496 5 109 12100 5 109 12100 6 109 12100 3 149 22500 3 157 24650 158 157 24650 77 157 24650 77 173 29930 174 179 32400 3 229 52900 3 229 52900 5 229 52900 4 239 57600 3 269 72900 3 307 94250 43 307 94250 308 307 94250 308 359 129600 4 389 152100 3 419 176400 4 439 193600 3 439 193600 3 439 193600 4 449 202500 4 479 230400 4 499 250000 5 499 250000 5 499 250000 3 509 260100 3 521 266321 21 557 310250 285 569 324900 3 599 360000 4 659 435600 3 719 518400 3 739 547600 5 739 547600 6 739 547600 5 809 656100 4 821 660176 10 839 705600 3 919 846400 5 919 846400 6 919 846400 3 929 864900 3 1019 1040400 3 1049 1102500 3 1051 1073875 59 1109 1232100 3 sorting sieving information Mordell--Weil sieve used 20 places for the Mordell--Weil sieve after the Mordell--Weil sieve W= [ 0, A.1, 2*A.1, 4989601*A.1, -4989599*A.1 ] and L= Abelian Group isomorphic to Z Defined on 1 generator in supergroup A: L.1 = 12474000*A.1 (free) the index of L in A is 12474000 using prime p= 109 for Chabauty there are 3 places above 109 the structures of J(k_v) above these places are Abelian Group isomorphic to Z/110 + Z/110 Defined on 2 generators Relations: 110*$.1 = 0 110*$.2 = 0 Abelian Group isomorphic to Z/110 + Z/110 Defined on 2 generators Relations: 110*$.1 = 0 110*$.2 = 0 Abelian Group isomorphic to Z/110 + Z/110 Defined on 2 generators Relations: 110*B.1 = 0 110*B.2 = 0 the integrals of the basis with place P are [* [ 2067796853351091496762518165408781736934623562427510591670538100203735334\ 58586553965951517394510*109 + O(109^48), 27488388988786322246775393459906607573077641966396309406751701514923681\ 9463911400990180912109291*109 + O(109^48) ] *] the integrals of the basis with place P are [* [ 195001853068928101531440126663673056768547405048305223050451028321063*109 + O(109^35), 30640550608561741854141051317857016779709857208631851360103714\ 8860318*109 + O(109^35) ] *] the integrals of the basis with place P are [* [ 3562053490989579687196796942893200921553842970802481712505516079179242519\ 7783417248088962529216*109 + O(109^48), -17763502098626213271807156718331980601169719103337241791261240858876241229\ 7404865816724025412407*109 + O(109^48) ] *] the integration matrix is [421893*109 + O(109^4) 356166*109 + O(109^4) -509350*109 + O(109^4) 516480*109 + O(109^4) 336436*109 + O(109^4) -99988*109 + O(109^4)] the matrix E is [1 + O(109^3) -60060 + O(109^3) -132423 + O(109^3) -479130 + O(109^3) 191884 + O(109^3) -577324 + O(109^3)] [O(109^50) 1 + O(109^3) O(109^50) O(109^50) O(109^50) O(109^50)] [O(109^50) O(109^50) 1 + O(109^3) O(109^50) O(109^50) O(109^50)] [O(109^50) O(109^50) O(109^50) 1 + O(109^3) O(109^50) O(109^50)] [O(109^50) O(109^50) O(109^50) O(109^50) 1 + O(109^3) O(109^50)] [O(109^50) O(109^50) O(109^50) O(109^50) O(109^50) 1 + O(109^3)] point (1 : 0 : 0) the leading terms matrix is [O(109^5) -613049 + O(109^3) O(109^5) O(109^5) O(109^5) O(109^5)] [O(109^5) O(109^5) O(109^5) -147618 + O(109^3) O(109^5) O(109^5)] [O(109^5) O(109^5) O(109^5) O(109^5) O(109^5) 113150 + O(109^3)] E*Npt= [O(109^5) -613049 + O(109^3) O(109^5) O(109^5) O(109^5) O(109^5)] [O(109^5) O(109^5) O(109^5) -147618 + O(109^3) O(109^5) O(109^5)] [O(109^5) O(109^5) O(109^5) O(109^5) O(109^5) 113150 + O(109^3)] redSubMat= [ 76 0 0] [ 0 0 0] [ 0 77 0] [ 0 0 0] [ 0 0 8] point (t^2 + t + 1 : t^2 + 2*t + 1 : 1) the leading terms matrix is [-351586 + O(109^3) -293139 + O(109^3) O(109^5) O(109^5) O(109^5) O(109^5)] [O(109^5) O(109^5) -164777 + O(109^3) -342204 + O(109^3) O(109^5) O(109^5)] [O(109^5) O(109^5) O(109^5) O(109^5) 516362 + O(109^3) 635344 + O(109^3)] E*Npt= [-351586 + O(109^3) 514176 + O(109^3) 485299 + O(109^3) 617918 + O(109^3) -487298 + O(109^3) 75491 + O(109^3)] [O(109^5) O(109^5) -164777 + O(109^3) -342204 + O(109^3) O(109^5) O(109^5)] [O(109^5) O(109^5) O(109^5) O(109^5) 516362 + O(109^3) 635344 + O(109^3)] redSubMat= [ 23 0 0] [ 31 31 0] [106 56 0] [ 41 0 29] [ 63 0 92] point (t^2 + t + 1 : -t^2 - 2*t - 1 : 1) the leading terms matrix is [351586 + O(109^3) 293139 + O(109^3) O(109^5) O(109^5) O(109^5) O(109^5)] [O(109^5) O(109^5) 164777 + O(109^3) 342204 + O(109^3) O(109^5) O(109^5)] [O(109^5) O(109^5) O(109^5) O(109^5) -516362 + O(109^3) -635344 + O(109^3)] E*Npt= [351586 + O(109^3) -514176 + O(109^3) -485299 + O(109^3) -617918 + O(109^3) 487298 + O(109^3) -75491 + O(109^3)] [O(109^5) O(109^5) 164777 + O(109^3) 342204 + O(109^3) O(109^5) O(109^5)] [O(109^5) O(109^5) O(109^5) O(109^5) -516362 + O(109^3) -635344 + O(109^3)] redSubMat= [ 86 0 0] [ 78 78 0] [ 3 53 0] [ 68 0 80] [ 46 0 17] having applied Chabauty to the points [ (1 : 0 : 0), (t^2 + t + 1 : t^2 + 2*t + 1 : 1), (t^2 + t + 1 : -t^2 - 2*t - 1 : 1) ] Chabauty is respectively successful at [ true, true, true ] after applying Chabauty with prime 109 W has 0 elements succeeded in proving that the only K-rational points are [ (1 : 0 : 0), (t^2 + t + 1 : t^2 + 2*t + 1 : 1), (t^2 + t + 1 : -t^2 - 2*t - 1 : 1) ] u/v= -1 this solution corresponds to $K$-rational point (t^2 + t + 1 : -t^2 - 2*t - 1 : 1) ############################################################ s= -1 it appears that the Torsion subgroup is isomorphic to Abelian Group of order 1 computing the 2-Selmer group might take a very long time 2-Selmer rank is 3 two torsion subgroup is Abelian Group of order 1 Hence rank is at most 3 searching for enough torsion points and for 3 independent points on J this could take forever! found 3 independent points on J and enough torsion basis for torsion is [] torsion subgroup isomorphic to Abelian Group of order 1 basis for a free subgroup of finite index is [ (x + t^2 + t + 1, 11*t^2 + 13*t + 17, 1), (x^2 + 1/2*(t^2 + t + 2)*x + 1/2*(t^2 + t + 2), (2*t^2 + 2*t + 3)*x + 2*t^2 + 3*t + 4, 2), (x^2 + 1/3*(4*t^2 + 5*t + 4)*x + 1/3*(4*t^2 + 5*t + 7), (-4*t^2 - 6*t - 10)*x - 9*t^2 - 11*t - 13, 2) ] it appears that a complete list of rational points on C is [ (1 : 0 : 0), (-t^2 - t - 1 : -11*t^2 - 13*t - 17 : 1), (1/3*(-t^2 - 2*t - 1) : 1/3*(-t^2 + t - 1) : 1), (1/3*(-t^2 - 2*t - 1) : 1/3*(t^2 - t + 1) : 1), (-t^2 - t - 1 : 11*t^2 + 13*t + 17 : 1) ] want to prove this using Chabauty and the Mordell--Weil sieve working with a subgroup A of the Mordell--Weil group isomorphic to Abelian Group isomorphic to Z + Z + Z Defined on 3 generators (free) the basis for this subgroup is [ (x + t^2 + t + 1, 11*t^2 + 13*t + 17, 1), (x^2 + 1/2*(t^2 + t + 2)*x + 1/2*(t^2 + t + 2), (2*t^2 + 2*t + 3)*x + 2*t^2 + 3*t + 4, 2), (x^2 + 1/3*(4*t^2 + 5*t + 4)*x + 1/3*(4*t^2 + 5*t + 7), (-4*t^2 - 6*t - 10)*x - 9*t^2 - 11*t - 13, 2) ] using (-t^2 - t - 1 : -11*t^2 - 13*t - 17 : 1) as base point for Abel-Jacobi map the image of the known rational points on C in A is [ A.1, 0, A.3, 2*A.1 - A.3, 2*A.1 ] applying the Mordell--Weil sieve verified saturation at 2 verified saturation at 3 verified saturation at 5 verified saturation at 7 verified saturation at 11 verified saturation at 13 verified saturation at 17 verified saturation at 19 verified saturation at 23 verified saturation at 29 verified saturation at 31 verified saturation at 37 verified saturation at 41 verified saturation at 43 verified saturation at 47 verified saturation at 53 verified saturation at 59 verified saturation at 61 verified saturation at 67 verified saturation at 71 verified saturation at 73 getting sieving information 17 290 18 23 530 24 29 900 30 31 880 14 31 1375 43 41 1331 31 43 1850 44 43 1850 44 43 1850 44 47 2210 48 59 3600 60 83 6890 84 89 8100 42 109 12100 110 109 12100 110 109 12100 110 149 22500 150 157 24650 158 157 24650 158 157 24650 158 173 29930 81 179 32400 51 229 52900 107 229 52900 118 229 52900 118 239 57600 240 269 72900 270 307 94250 308 307 94250 308 307 94250 308 359 129600 184 389 152100 146 419 176400 218 439 193600 440 439 193600 440 439 193600 440 449 202500 109 479 230400 480 499 250000 500 499 250000 500 499 250000 252 509 260100 510 557 310250 558 569 324900 114 599 360000 184 659 435600 132 719 518400 368 739 547600 740 739 547600 740 739 547600 740 809 656100 810 839 705600 168 919 846400 920 919 846400 484 919 846400 920 929 864900 308 1019 1040400 172 1021 999680 52 1049 1102500 1050 1051 1073875 1023 1109 1232100 1110 sorting sieving information Mordell--Weil sieve used 21 places for the Mordell--Weil sieve after the Mordell--Weil sieve W= [ 0, A.3, A.1, 2*A.1 - A.3, 2*A.1, A.1 + 12474000*A.3, 1663201*A.1 + 4158000*A.2 + 33264000*A.3, 1663201*A.1 + 4158000*A.2 + 20790000*A.3, -1663199*A.1 - 4158000*A.2 - 20790000*A.3, -1663199*A.1 - 4158000*A.2 - 33264000*A.3 ] and L= Abelian Group isomorphic to Z + Z + Z Defined on 3 generators in supergroup A: L.1 = 8316000*A.1 + 8316000*A.2 + 16632000*A.3 L.2 = 12474000*A.2 L.3 = 24948000*A.3 (free) the index of L in A is 2587950443232000000000 using prime p= 109 for Chabauty there are 3 places above 109 the structures of J(k_v) above these places are Abelian Group isomorphic to Z/110 + Z/110 Defined on 2 generators Relations: 110*$.1 = 0 110*$.2 = 0 Abelian Group isomorphic to Z/110 + Z/110 Defined on 2 generators Relations: 110*$.1 = 0 110*$.2 = 0 Abelian Group isomorphic to Z/110 + Z/110 Defined on 2 generators Relations: 110*B.1 = 0 110*B.2 = 0 the integrals of the basis with place P are [* [ -283266247851841596327578176006242979839357761404525884807869308157034001\ 27581318346316278097591*109 + O(109^48), -21624201956131038478376123127462068391646950488134390861910990453411598621\ 1209315911184386789034*109 + O(109^48) ], [ -156850626347007012081783948554230433896131587755357063484521213123168128\ 409903062130314088132370*109 + O(109^48), -27356462242991638249741021376746678863560780294474207168190125409059773480\ 2476515265311368708561*109 + O(109^48) ], [ 6831417257732776927481127159592154839034761355194180812671670088614589401\ 2819682794046827686156*109 + O(109^48), 25693623946437278271997713605828108560810435231717908639355301412905531\ 5679947453027050308611058*109 + O(109^48) ] *] the integrals of the basis with place P are [* [ -57256982297879052956259051149897006056656073149909694894110846992837*109 + O(109^35), 90638659977391167986753665613616722289588231288051372575630534\ 1801468*109 + O(109^35) ], [ 9912929045127206804929420392833276577800161825873962441126967968998385549\ 3278074094492665147816*109 + O(109^48), -18866695511040663283618605495978202076883203170292149463395868182568841555\ 0066769614660222736095*109 + O(109^48) ], [ -258561627713982305922665557584802383008703790988619254416909689470242*10\ 9 + O(109^35), 707680658452669674406026036297302248438807526951330370611039\ 142641665*109 + O(109^35) ] *] the integrals of the basis with place P are [* [ 1732560518553305082866272345569177308030663660293337785067966603524282455\ 18627458060904176484376*109 + O(109^48), 26877345373704989872767853874575684106679888930032080104168998968481696\ 4577606998896749305090618*109 + O(109^48) ], [ 717100143444022907556669987651332739575669791998298363106153153662938*109 + O(109^35), 20901806917795556546021289199644656514524409963447385868523837\ 4081760*109 + O(109^35) ], [ -264154486261599946736191037654380656000626588435425029776213769598899532\ 0546929270139493550786280841*109 + O(109^50), -33763250427611615780649827333929688710571062529950973254359355609843750233\ 52128741589472919046850949*109 + O(109^50) ] *] the integration matrix is [332723*109 + O(109^4) -248580*109 + O(109^4) -217732*109 + O(109^4) -311224*109 + O(109^4) -202955*109 + O(109^4) -412436*109 + O(109^4)] [-553163*109 + O(109^4) 102260*109 + O(109^4) -425750*109 + O(109^4) 182842*109 + O(109^4) 225545*109 + O(109^4) -570795*109 + O(109^4)] [522401*109 + O(109^4) -567371*109 + O(109^4) -272292*109 + O(109^4) -262833*109 + O(109^4) 266556*109 + O(109^4) -585069*109 + O(109^4)] the matrix E is [1 + O(109^3) -14922 + O(109^3) 280747 + O(109^3) 262874 + O(109^3) -571033 + O(109^3) 179190 + O(109^3)] [O(109^50) 1 + O(109^3) -508836 + O(109^3) 511949 + O(109^3) 101519 + O(109^3) -87496 + O(109^3)] [O(109^50) O(109^50) 1 + O(109^3) -160877 + O(109^3) -551126 + O(109^3) -183796 + O(109^3)] [O(109^50) O(109^50) O(109^50) 1 + O(109^3) O(109^50) O(109^50)] [O(109^50) O(109^50) O(109^50) O(109^50) 1 + O(109^3) O(109^50)] [O(109^50) O(109^50) O(109^50) O(109^50) O(109^50) 1 + O(109^3)] point (1 : 0 : 0) the leading terms matrix is [O(109^5) 109314 + O(109^3) O(109^5) O(109^5) O(109^5) O(109^5)] [O(109^5) O(109^5) O(109^5) 178186 + O(109^3) O(109^5) O(109^5)] [O(109^5) O(109^5) O(109^5) O(109^5) O(109^5) 360015 + O(109^3)] E*Npt= [O(109^5) 109314 + O(109^3) -107925 + O(109^3) -190220 + O(109^3) 344465 + O(109^3) 546450 + O(109^3)] [O(109^5) O(109^5) O(109^5) 178186 + O(109^3) O(109^5) O(109^5)] [O(109^5) O(109^5) O(109^5) O(109^5) O(109^5) 360015 + O(109^3)] redSubMat= [ 94 80 0] [ 25 0 0] [ 33 0 97] point (-t^2 - t - 1 : -11*t^2 - 13*t - 17 : 1) the leading terms matrix is [247498 + O(109^3) 611889 + O(109^3) O(109^5) O(109^5) O(109^5) O(109^5)] [O(109^5) O(109^5) 281354 + O(109^3) 207853 + O(109^3) O(109^5) O(109^5)] [O(109^5) O(109^5) O(109^5) O(109^5) 507171 + O(109^3) -42724 + O(109^3)] E*Npt= [247498 + O(109^3) -425588 + O(109^3) 556016 + O(109^3) -70857 + O(109^3) 577771 + O(109^3) -442569 + O(109^3)] [O(109^5) O(109^5) 281354 + O(109^3) -621026 + O(109^3) 87740 + O(109^3) 63215 + O(109^3)] [O(109^5) O(109^5) O(109^5) O(109^5) 507171 + O(109^3) -42724 + O(109^3)] redSubMat= [102 56 0] [ 71 104 103] [ 80 104 4] point (1/3*(-t^2 - 2*t - 1) : 1/3*(-t^2 + t - 1) : 1) the leading terms matrix is [-415612 + O(109^3) -639145 + O(109^3) O(109^5) O(109^5) O(109^5) O(109^5)] [O(109^5) O(109^5) -268417 + O(109^3) -225913 + O(109^3) O(109^5) O(109^5)] [O(109^5) O(109^5) O(109^5) O(109^5) -611003 + O(109^3) -429968 + O(109^3)] E*Npt= [-415612 + O(109^3) 524267 + O(109^3) -356814 + O(109^3) 391377 + O(109^3) 629388 + O(109^3) 407065 + O(109^3)] [O(109^5) O(109^5) -268417 + O(109^3) 448820 + O(109^3) 424872 + O(109^3) -158823 + O(109^3)] [O(109^5) O(109^5) O(109^5) O(109^5) -611003 + O(109^3) -429968 + O(109^3)] redSubMat= [ 67 67 0] [ 22 99 51] [ 59 99 37] point (1/3*(-t^2 - 2*t - 1) : 1/3*(t^2 - t + 1) : 1) the leading terms matrix is [415612 + O(109^3) 639145 + O(109^3) O(109^5) O(109^5) O(109^5) O(109^5)] [O(109^5) O(109^5) 268417 + O(109^3) 225913 + O(109^3) O(109^5) O(109^5)] [O(109^5) O(109^5) O(109^5) O(109^5) 611003 + O(109^3) 429968 + O(109^3)] E*Npt= [415612 + O(109^3) -524267 + O(109^3) 356814 + O(109^3) -391377 + O(109^3) -629388 + O(109^3) -407065 + O(109^3)] [O(109^5) O(109^5) 268417 + O(109^3) -448820 + O(109^3) -424872 + O(109^3) 158823 + O(109^3)] [O(109^5) O(109^5) O(109^5) O(109^5) 611003 + O(109^3) 429968 + O(109^3)] redSubMat= [ 42 42 0] [ 87 10 58] [ 50 10 72] point (-t^2 - t - 1 : 11*t^2 + 13*t + 17 : 1) the leading terms matrix is [-247498 + O(109^3) -611889 + O(109^3) O(109^5) O(109^5) O(109^5) O(109^5)] [O(109^5) O(109^5) -281354 + O(109^3) -207853 + O(109^3) O(109^5) O(109^5)] [O(109^5) O(109^5) O(109^5) O(109^5) -507171 + O(109^3) 42724 + O(109^3)] E*Npt= [-247498 + O(109^3) 425588 + O(109^3) -556016 + O(109^3) 70857 + O(109^3) -577771 + O(109^3) 442569 + O(109^3)] [O(109^5) O(109^5) -281354 + O(109^3) 621026 + O(109^3) -87740 + O(109^3) -63215 + O(109^3)] [O(109^5) O(109^5) O(109^5) O(109^5) -507171 + O(109^3) 42724 + O(109^3)] redSubMat= [ 7 53 0] [ 38 5 6] [ 29 5 105] having applied Chabauty to the points [ (1 : 0 : 0), (-t^2 - t - 1 : -11*t^2 - 13*t - 17 : 1), (1/3*(-t^2 - 2*t - 1) : 1/3*(-t^2 + t - 1) : 1), (1/3*(-t^2 - 2*t - 1) : 1/3*(t^2 - t + 1) : 1), (-t^2 - t - 1 : 11*t^2 + 13*t + 17 : 1) ] Chabauty is respectively successful at [ true, true, true, true, true ] after applying Chabauty with prime 109 W has 0 elements succeeded in proving that the only K-rational points are [ (1 : 0 : 0), (-t^2 - t - 1 : -11*t^2 - 13*t - 17 : 1), (1/3*(-t^2 - 2*t - 1) : 1/3*(-t^2 + t - 1) : 1), (1/3*(-t^2 - 2*t - 1) : 1/3*(t^2 - t + 1) : 1), (-t^2 - t - 1 : 11*t^2 + 13*t + 17 : 1) ] u/v= 2 this solution corresponds to $K$-rational point (-t^2 - t - 1 : -11*t^2 - 13*t - 17 : 1) ############################################################ s= 0 it appears that the Torsion subgroup is isomorphic to Abelian Group of order 1 computing the 2-Selmer group might take a very long time 2-Selmer rank is 2 two torsion subgroup is Abelian Group of order 1 Hence rank is at most 2 searching for enough torsion points and for 2 independent points on J this could take forever! found 2 independent points on J and enough torsion basis for torsion is [] torsion subgroup isomorphic to Abelian Group of order 1 basis for a free subgroup of finite index is [ (x - 1, -3, 1), (x + 1/3*(-t^2 - 2*t - 1), 1/3*(10*t^2 + 8*t + 13), 1) ] it appears that a complete list of rational points on C is [ (1 : 0 : 0), (1 : -3 : 1), (1/3*(t^2 + 2*t + 1) : 1/3*(-10*t^2 - 8*t - 13) : 1), (1 : 3 : 1), (1/3*(t^2 + 2*t + 1) : 1/3*(10*t^2 + 8*t + 13) : 1) ] want to prove this using Chabauty and the Mordell--Weil sieve working with a subgroup A of the Mordell--Weil group isomorphic to Abelian Group isomorphic to Z + Z Defined on 2 generators (free) the basis for this subgroup is [ (x - 1, -3, 1), (x + 1/3*(-t^2 - 2*t - 1), 1/3*(10*t^2 + 8*t + 13), 1) ] using (1 : -3 : 1) as base point for Abel-Jacobi map the image of the known rational points on C in A is [ -A.1, 0, -A.1 - A.2, -2*A.1, -A.1 + A.2 ] applying the Mordell--Weil sieve verified saturation at 2 verified saturation at 3 verified saturation at 5 verified saturation at 7 verified saturation at 11 verified saturation at 13 verified saturation at 17 verified saturation at 19 verified saturation at 23 verified saturation at 29 verified saturation at 31 verified saturation at 37 verified saturation at 41 verified saturation at 43 verified saturation at 47 verified saturation at 53 verified saturation at 59 verified saturation at 61 verified saturation at 67 verified saturation at 71 verified saturation at 73 getting sieving information 17 290 9 23 530 24 29 900 5 31 1375 11 31 1375 9 31 1375 9 43 1850 44 43 1850 44 43 1850 19 47 2210 48 59 3600 20 83 6890 84 89 8100 20 109 12100 110 109 12100 8 109 12100 53 149 22500 9 157 24650 77 157 24650 36 157 24650 158 173 29930 174 179 32400 22 229 52900 119 229 52900 23 229 52900 119 239 57600 32 269 72900 25 307 94250 308 307 94250 153 307 94250 308 359 129600 16 389 152100 84 419 176400 25 439 193600 204 439 193600 122 439 193600 204 449 202500 26 479 230400 34 499 250000 252 499 250000 500 499 250000 500 509 260100 106 557 310250 558 569 324900 28 599 360000 32 659 435600 132 719 518400 142 739 547600 378 739 547600 740 739 547600 740 809 656100 21 839 705600 48 919 846400 127 919 846400 226 919 846400 226 929 864900 93 1019 1040400 29 1049 1102500 99 1051 1073875 49 1051 1073875 49 1051 1073875 51 1109 1232100 114 sorting sieving information Mordell--Weil sieve used 21 places for the Mordell--Weil sieve after the Mordell--Weil sieve W= [ 0, -A.1 - A.2, -A.1, -2*A.1, -A.1 + A.2, -1663201*A.1 - 49896000*A.2, -2494801*A.1 - 62370000*A.2, 4157999*A.1 + 112266000*A.2, 2494799*A.1 + 62370000*A.2, 1663199*A.1 + 49896000*A.2 ] and L= Abelian Group isomorphic to Z + Z Defined on 2 generators in supergroup A: L.1 = 8316000*A.1 L.2 = 24948000*A.2 (free) the index of L in A is 207467568000000 using prime p= 31 for Chabauty there are 3 places above 31 the structures of J(k_v) above these places are Abelian Group isomorphic to Z/5 + Z/5 + Z/55 Defined on 3 generators Relations: 5*B.1 = 0 5*B.2 = 0 55*B.3 = 0 Abelian Group isomorphic to Z/5 + Z/5 + Z/55 Defined on 3 generators Relations: 5*B.1 = 0 5*B.2 = 0 55*B.3 = 0 Abelian Group isomorphic to Z/5 + Z/5 + Z/55 Defined on 3 generators Relations: 5*B.1 = 0 5*B.2 = 0 55*B.3 = 0 the integrals of the basis with place P are [* [ 2405654011747568533828295803078765478783187756396600849382753165031719725\ *31 + O(31^50), 3853300341522897064233893509433748563061280678860236032\ 550077634512280361*31 + O(31^50) ], [ -3054895218610312924372328553843810497869120720859753460026082595*31 + O(31^44), 3771255693438480061550443381496543465419553680015337103690346293*\ 31 + O(31^44) ] *] the integrals of the basis with place P are [* [ 2405654011747568533828295803078765478783187756396600849382753165031719725\ *31 + O(31^50), 3853300341522897064233893509433748563061280678860236032\ 550077634512280361*31 + O(31^50) ], [ -521782589599702905194605421691879264761170241160253722633690523727188003\ *31 + O(31^50), 26039520526323705782692347948527010281893105444504698171786\ 95133627670703*31 + O(31^50) ] *] the integrals of the basis with place P are [* [ 2405654011747568533828295803078765478783187756396600849382753165031719725\ *31 + O(31^50), 3853300341522897064233893509433748563061280678860236032\ 550077634512280361*31 + O(31^50) ], [ 1055302281993208308193219755798410440463514663853488548107629021*31 + O(31^44), -6524734484258354859388792784654189373646327424912293577605617036\ *31 + O(31^44) ] *] the integration matrix is [10894*31 + O(31^4) -3341*31 + O(31^4) 10894*31 + O(31^4) -3341*31 + O(31^4) 10894*31 + O(31^4) -3341*31 + O(31^4)] [-14085*31 + O(31^4) 6288*31 + O(31^4) 2618*31 + O(31^4) -3128*31 + O(31^4) 3464*31 + O(31^4) -9842*31 + O(31^4)] the matrix E is [1 + O(31^3) 12905 + O(31^3) -237 + O(31^3) -12261 + O(31^3) 11742 + O(31^3) 1694 + O(31^3)] [O(31^50) 1 + O(31^3) 6997 + O(31^3) -7987 + O(31^3) 3399 + O(31^3) -3519 + O(31^3)] [O(31^50) O(31^50) 1 + O(31^3) O(31^50) O(31^50) O(31^50)] [O(31^50) O(31^50) O(31^50) 1 + O(31^3) O(31^50) O(31^50)] [O(31^50) O(31^50) O(31^50) O(31^50) 1 + O(31^3) O(31^50)] [O(31^50) O(31^50) O(31^50) O(31^50) O(31^50) 1 + O(31^3)] point (1 : 0 : 0) the leading terms matrix is [O(31^5) 4965 + O(31^3) O(31^5) O(31^5) O(31^5) O(31^5)] [O(31^5) O(31^5) O(31^5) 4965 + O(31^3) O(31^5) O(31^5)] [O(31^5) O(31^5) O(31^5) O(31^5) O(31^5) 4965 + O(31^3)] E*Npt= [O(31^5) 4965 + O(31^3) 3799 + O(31^3) -3634 + O(31^3) 14329 + O(31^3) -14309 + O(31^3)] [O(31^5) O(31^5) O(31^5) 4965 + O(31^3) O(31^5) O(31^5)] [O(31^5) O(31^5) O(31^5) O(31^5) O(31^5) 4965 + O(31^3)] redSubMat= [17 0 0] [24 5 0] [ 7 0 0] [13 0 5] point (1 : -3 : 1) the leading terms matrix is [9930 + O(31^3) 9930 + O(31^3) O(31^5) O(31^5) O(31^5) O(31^5)] [O(31^5) O(31^5) 9930 + O(31^3) 9930 + O(31^3) O(31^5) O(31^5)] [O(31^5) O(31^5) O(31^5) O(31^5) 9930 + O(31^3) 9930 + O(31^3)] E*Npt= [9930 + O(31^3) -4302 + O(31^3) 7677 + O(31^3) -3181 + O(31^3) -5047 + O(31^3) -9322 + O(31^3)] [O(31^5) O(31^5) 9930 + O(31^3) 9930 + O(31^3) O(31^5) O(31^5)] [O(31^5) O(31^5) O(31^5) O(31^5) 9930 + O(31^3) 9930 + O(31^3)] redSubMat= [20 10 0] [12 10 0] [ 6 0 10] [ 9 0 10] point (1/3*(t^2 + 2*t + 1) : 1/3*(-10*t^2 - 8*t - 13) : 1) the leading terms matrix is [13307 + O(31^3) -14014 + O(31^3) O(31^5) O(31^5) O(31^5) O(31^5)] [O(31^5) O(31^5) 7623 + O(31^3) 494 + O(31^3) O(31^5) O(31^5)] [O(31^5) O(31^5) O(31^5) O(31^5) -705 + O(31^3) -13811 + O(31^3)] E*Npt= [13307 + O(31^3) -2503 + O(31^3) -9690 + O(31^3) 13211 + O(31^3) -778 + O(31^3) 1432 + O(31^3)] [O(31^5) O(31^5) 7623 + O(31^3) 494 + O(31^3) O(31^5) O(31^5)] [O(31^5) O(31^5) O(31^5) O(31^5) -705 + O(31^3) -13811 + O(31^3)] redSubMat= [13 28 0] [ 5 29 0] [28 0 8] [ 6 0 15] point (1 : 3 : 1) the leading terms matrix is [-9930 + O(31^3) -9930 + O(31^3) O(31^5) O(31^5) O(31^5) O(31^5)] [O(31^5) O(31^5) -9930 + O(31^3) -9930 + O(31^3) O(31^5) O(31^5)] [O(31^5) O(31^5) O(31^5) O(31^5) -9930 + O(31^3) -9930 + O(31^3)] E*Npt= [-9930 + O(31^3) 4302 + O(31^3) -7677 + O(31^3) 3181 + O(31^3) 5047 + O(31^3) 9322 + O(31^3)] [O(31^5) O(31^5) -9930 + O(31^3) -9930 + O(31^3) O(31^5) O(31^5)] [O(31^5) O(31^5) O(31^5) O(31^5) -9930 + O(31^3) -9930 + O(31^3)] redSubMat= [11 21 0] [19 21 0] [25 0 21] [22 0 21] point (1/3*(t^2 + 2*t + 1) : 1/3*(10*t^2 + 8*t + 13) : 1) the leading terms matrix is [-13307 + O(31^3) 14014 + O(31^3) O(31^5) O(31^5) O(31^5) O(31^5)] [O(31^5) O(31^5) -7623 + O(31^3) -494 + O(31^3) O(31^5) O(31^5)] [O(31^5) O(31^5) O(31^5) O(31^5) 705 + O(31^3) 13811 + O(31^3)] E*Npt= [-13307 + O(31^3) 2503 + O(31^3) 9690 + O(31^3) -13211 + O(31^3) 778 + O(31^3) -1432 + O(31^3)] [O(31^5) O(31^5) -7623 + O(31^3) -494 + O(31^3) O(31^5) O(31^5)] [O(31^5) O(31^5) O(31^5) O(31^5) 705 + O(31^3) 13811 + O(31^3)] redSubMat= [18 3 0] [26 2 0] [ 3 0 23] [25 0 16] having applied Chabauty to the points [ (1 : 0 : 0), (1 : -3 : 1), (1/3*(t^2 + 2*t + 1) : 1/3*(-10*t^2 - 8*t - 13) : 1), (1 : 3 : 1), (1/3*(t^2 + 2*t + 1) : 1/3*(10*t^2 + 8*t + 13) : 1) ] Chabauty is respectively successful at [ true, true, true, true, true ] after applying Chabauty with prime 31 W has 0 elements succeeded in proving that the only K-rational points are [ (1 : 0 : 0), (1 : -3 : 1), (1/3*(t^2 + 2*t + 1) : 1/3*(-10*t^2 - 8*t - 13) : 1), (1 : 3 : 1), (1/3*(t^2 + 2*t + 1) : 1/3*(10*t^2 + 8*t + 13) : 1) ] u/v= 0 this solution corresponds to $K$-rational point (1 : -3 : 1) ############################################################ s= 1 it appears that the Torsion subgroup is isomorphic to Abelian Group of order 1 computing the 2-Selmer group might take a very long time 2-Selmer rank is 3 two torsion subgroup is Abelian Group of order 1 Hence rank is at most 3 searching for enough torsion points and for 3 independent points on J this could take forever! found 3 independent points on J and enough torsion basis for torsion is [] torsion subgroup isomorphic to Abelian Group of order 1 basis for a free subgroup of finite index is [ (x + t^2 + t + 1, -40*t^2 - 53*t - 67, 1), (x + 1, 3*t + 3, 1), (x^2 + 1/3*(t^2 - t - 2)*x + 1/3*(-2*t^2 + 2*t + 1), (2*t - 2)*x - t + 1, 2) ] it appears that a complete list of rational points on C is [ (1 : 0 : 0), (-t^2 - t - 1 : 40*t^2 + 53*t + 67 : 1), (-1 : 3*t + 3 : 1), (-1 : -3*t - 3 : 1), (-t^2 - t - 1 : -40*t^2 - 53*t - 67 : 1) ] want to prove this using Chabauty and the Mordell--Weil sieve working with a subgroup A of the Mordell--Weil group isomorphic to Abelian Group isomorphic to Z + Z + Z Defined on 3 generators (free) the basis for this subgroup is [ (x + t^2 + t + 1, -40*t^2 - 53*t - 67, 1), (x + 1, 3*t + 3, 1), (x^2 + 1/3*(t^2 - t - 2)*x + 1/3*(-2*t^2 + 2*t + 1), (2*t - 2)*x - t + 1, 2) ] using (-t^2 - t - 1 : 40*t^2 + 53*t + 67 : 1) as base point for Abel-Jacobi map the image of the known rational points on C in A is [ A.1, 0, A.1 + A.2, A.1 - A.2, 2*A.1 ] applying the Mordell--Weil sieve verified saturation at 2 verified saturation at 3 verified saturation at 5 verified saturation at 7 verified saturation at 11 verified saturation at 13 verified saturation at 17 verified saturation at 19 verified saturation at 23 verified saturation at 29 verified saturation at 31 verified saturation at 37 verified saturation at 41 verified saturation at 43 verified saturation at 47 verified saturation at 53 verified saturation at 59 verified saturation at 61 verified saturation at 67 verified saturation at 71 verified saturation at 73 getting sieving information 11 176 5 121 12505 103 17 290 18 23 530 24 29 900 30 31 1375 43 31 880 14 43 1850 44 43 1850 44 43 1850 44 47 2210 48 59 3600 60 83 6890 43 89 8100 13 109 12100 58 109 12100 53 109 12100 58 149 22500 46 157 24650 158 157 24650 158 157 24650 77 173 29930 81 179 32400 180 229 52900 230 229 52900 230 229 52900 20 239 57600 240 269 72900 84 281 68231 241 307 94250 308 307 94250 308 307 94250 308 359 129600 360 389 152100 66 419 176400 218 439 193600 440 439 193600 440 439 193600 440 449 202500 450 479 230400 480 499 250000 252 499 250000 500 499 250000 500 509 260100 510 557 310250 558 569 324900 67 599 360000 100 659 435600 132 719 518400 352 739 547600 740 739 547600 740 739 547600 740 809 656100 810 839 705600 168 919 846400 920 919 846400 484 919 846400 920 929 864900 145 1019 1040400 261 1049 1102500 1050 1051 1073875 1023 1109 1232100 1110 sorting sieving information Mordell--Weil sieve used 22 places for the Mordell--Weil sieve after the Mordell--Weil sieve W= [ 0, A.1 - A.2, A.1, A.1 + A.2, 2*A.1, A.1 + 12474000*A.2 + 87318000*A.3, 277201*A.1 + 5821200*A.2 + 51004800*A.3, 277201*A.1 - 6652800*A.2 - 36313200*A.3, -277199*A.1 + 6652800*A.2 + 36313200*A.3, -277199*A.1 - 5821200*A.2 - 51004800*A.3 ] and L= Abelian Group isomorphic to Z + Z + Z Defined on 3 generators in supergroup A: L.1 = 1386000*A.1 + 16632000*A.2 + 18018000*A.3 L.2 = 24948000*A.2 L.3 = 24948000*A.3 (free) the index of L in A is 862650147744000000000 using prime p= 109 for Chabauty there are 3 places above 109 the structures of J(k_v) above these places are Abelian Group isomorphic to Z/110 + Z/110 Defined on 2 generators Relations: 110*$.1 = 0 110*$.2 = 0 Abelian Group isomorphic to Z/110 + Z/110 Defined on 2 generators Relations: 110*$.1 = 0 110*$.2 = 0 Abelian Group isomorphic to Z/110 + Z/110 Defined on 2 generators Relations: 110*B.1 = 0 110*B.2 = 0 the integrals of the basis with place P are [* [ 520307794098260889907878305562291766566381525865575725684434856447434*109 + O(109^35), 51102542553112817926589710196307336363650458846579489825058221\ 1041954*109 + O(109^35) ], [ 5979989506491715676870233259798434845954872748958515086912770386174742654\ 21638279014129200766620337*109 + O(109^50), -73808993429728400307923370348958267441546629611802762472036013068786778619\ 2573704405280733828220737*109 + O(109^50) ], [ 103904462615259275582540625056156373313844555801618*109 + O(109^26), 227355126512906671038479154203068267937881511925294*109 + O(109^26) ] *] the integrals of the basis with place P are [* [ -202905522892292124873209100852351972535139357609138559508360605823326158\ 18421149624274333741608*109 + O(109^48), -16541075957980130054202025083606206024808099607149116354931172105313715538\ 3083926129544440327432*109 + O(109^48) ], [ 1507761777853558501595283579468132245259356215013861548010510894286310851\ 1034686916070351566*109 + O(109^46), -98704332364357890572623471377259300616854627708023296177738389345514411510\ 16294860971849188*109 + O(109^46) ], [ -168673426222655352199641611670785038761016659989684015176302781332116742\ 182854662122617123698906761*109 + O(109^50), 29920270671891781952492715283506729238415079163890363103334327404948155\ 03449454405443837927645245352*109 + O(109^50) ] *] the integrals of the basis with place P are [* [ 1741969250680323670991590168640969451778161548738633209518552450769198083\ 008498152779217959648881432*109 + O(109^50), 49914097358829145052743544068170198502551524007367702473991041485782586\ 061911549216036516816569847*109 + O(109^50) ], [ -267235109365698864398650994958247013273033587625096018795259282256995*10\ 9 + O(109^35), 116080580140906448659985748718778504604583380624209539560756\ 765410794*109 + O(109^35) ], [ 2734818732111203948637066512432036372252840690601065154243687034586417051\ 266600483792945861212184319*109 + O(109^50), 18655162069554708021579373016141525825542122066364940159772251370754660\ 40696360718099553171855662970*109 + O(109^50) ] *] the integration matrix is [-207948*109 + O(109^4) 165050*109 + O(109^4) -147183*109 + O(109^4) -156049*109 + O(109^4) 158326*109 + O(109^4) -157467*109 + O(109^4)] [-7525*109 + O(109^4) 592698*109 + O(109^4) 73906*109 + O(109^4) 147546*109 + O(109^4) -364764*109 + O(109^4) -28850*109 + O(109^4)] [-24181*109 + O(109^4) -587132*109 + O(109^4) -209299*109 + O(109^4) -464788*109 + O(109^4) 505076*109 + O(109^4) 414577*109 + O(109^4)] the matrix E is [1 + O(109^3) 585438 + O(109^3) -164421 + O(109^3) -107987 + O(109^3) 236693 + O(109^3) -65138 + O(109^3)] [O(109^50) 1 + O(109^3) 246949 + O(109^3) -601156 + O(109^3) 422255 + O(109^3) -341010 + O(109^3)] [O(109^50) O(109^50) 1 + O(109^3) -313455 + O(109^3) 248551*109 + O(109^4) 23132 + O(109^3)] [O(109^50) O(109^50) O(109^50) 1 + O(109^3) O(109^50) O(109^50)] [O(109^50) O(109^50) O(109^50) O(109^50) 1 + O(109^3) O(109^50)] [O(109^50) O(109^50) O(109^50) O(109^50) O(109^50) 1 + O(109^3)] point (1 : 0 : 0) the leading terms matrix is [O(109^5) -362408 + O(109^3) O(109^5) O(109^5) O(109^5) O(109^5)] [O(109^5) O(109^5) O(109^5) 260574 + O(109^3) O(109^5) O(109^5)] [O(109^5) O(109^5) O(109^5) O(109^5) O(109^5) -545681 + O(109^3)] E*Npt= [O(109^5) -362408 + O(109^3) 570940 + O(109^3) -280051 + O(109^3) -193226 + O(109^3) 134610 + O(109^3)] [O(109^5) O(109^5) O(109^5) 260574 + O(109^3) O(109^5) O(109^5)] [O(109^5) O(109^5) O(109^5) O(109^5) O(109^5) -545681 + O(109^3)] redSubMat= [ 79 64 0] [ 31 0 0] [104 0 82] point (-t^2 - t - 1 : 40*t^2 + 53*t + 67 : 1) the leading terms matrix is [-108466 + O(109^3) -9142 + O(109^3) O(109^5) O(109^5) O(109^5) O(109^5)] [O(109^5) O(109^5) -248536 + O(109^3) -236044 + O(109^3) O(109^5) O(109^5)] [O(109^5) O(109^5) O(109^5) O(109^5) -119036 + O(109^3) 465290 + O(109^3)] E*Npt= [-108466 + O(109^3) 324736 + O(109^3) -128384 + O(109^3) 340742 + O(109^3) -261803 + O(109^3) -41299 + O(109^3)] [O(109^5) O(109^5) -248536 + O(109^3) -443717 + O(109^3) 306993*109 + O(109^4) -501021 + O(109^3)] [O(109^5) O(109^5) O(109^5) O(109^5) -119036 + O(109^3) 465290 + O(109^3)] redSubMat= [ 8 22 0] [ 15 0 101] [ 12 52 78] point (-1 : 3*t + 3 : 1) the leading terms matrix is [-19482 + O(109^3) 19482 + O(109^3) O(109^5) O(109^5) O(109^5) O(109^5)] [O(109^5) O(109^5) 490819 + O(109^3) -490819 + O(109^3) O(109^5) O(109^5)] [O(109^5) O(109^5) O(109^5) O(109^5) 392016 + O(109^3) -392016 + O(109^3)] E*Npt= [-19482 + O(109^3) -163231 + O(109^3) -624141 + O(109^3) -98307 + O(109^3) -602084 + O(109^3) -167954 + O(109^3)] [O(109^5) O(109^5) 490819 + O(109^3) 579765 + O(109^3) 526440*109 + O(109^4) 105865 + O(109^3)] [O(109^5) O(109^5) O(109^5) O(109^5) 392016 + O(109^3) -392016 + O(109^3)] redSubMat= [ 11 103 0] [ 32 0 52] [ 15 26 57] point (-1 : -3*t - 3 : 1) the leading terms matrix is [19482 + O(109^3) -19482 + O(109^3) O(109^5) O(109^5) O(109^5) O(109^5)] [O(109^5) O(109^5) -490819 + O(109^3) 490819 + O(109^3) O(109^5) O(109^5)] [O(109^5) O(109^5) O(109^5) O(109^5) -392016 + O(109^3) 392016 + O(109^3)] E*Npt= [19482 + O(109^3) 163231 + O(109^3) 624141 + O(109^3) 98307 + O(109^3) 602084 + O(109^3) 167954 + O(109^3)] [O(109^5) O(109^5) -490819 + O(109^3) -579765 + O(109^3) -526440*109 + O(109^4) -105865 + O(109^3)] [O(109^5) O(109^5) O(109^5) O(109^5) -392016 + O(109^3) 392016 + O(109^3)] redSubMat= [ 98 6 0] [ 77 0 57] [ 94 83 52] point (-t^2 - t - 1 : -40*t^2 - 53*t - 67 : 1) the leading terms matrix is [108466 + O(109^3) 9142 + O(109^3) O(109^5) O(109^5) O(109^5) O(109^5)] [O(109^5) O(109^5) 248536 + O(109^3) 236044 + O(109^3) O(109^5) O(109^5)] [O(109^5) O(109^5) O(109^5) O(109^5) 119036 + O(109^3) -465290 + O(109^3)] E*Npt= [108466 + O(109^3) -324736 + O(109^3) 128384 + O(109^3) -340742 + O(109^3) 261803 + O(109^3) 41299 + O(109^3)] [O(109^5) O(109^5) 248536 + O(109^3) 443717 + O(109^3) -306993*109 + O(109^4) 501021 + O(109^3)] [O(109^5) O(109^5) O(109^5) O(109^5) 119036 + O(109^3) -465290 + O(109^3)] redSubMat= [101 87 0] [ 94 0 8] [ 97 57 31] having applied Chabauty to the points [ (1 : 0 : 0), (-t^2 - t - 1 : 40*t^2 + 53*t + 67 : 1), (-1 : 3*t + 3 : 1), (-1 : -3*t - 3 : 1), (-t^2 - t - 1 : -40*t^2 - 53*t - 67 : 1) ] Chabauty is respectively successful at [ true, true, true, true, true ] after applying Chabauty with prime 109 W has 0 elements succeeded in proving that the only K-rational points are [ (1 : 0 : 0), (-t^2 - t - 1 : 40*t^2 + 53*t + 67 : 1), (-1 : 3*t + 3 : 1), (-1 : -3*t - 3 : 1), (-t^2 - t - 1 : -40*t^2 - 53*t - 67 : 1) ] u/v= 5/4 this solution corresponds to $K$-rational point (-t^2 - t - 1 : 40*t^2 + 53*t + 67 : 1) u/v= 1 this solution corresponds to $K$-rational point (-1 : 3*t + 3 : 1) ############################################################ s= 2 it appears that the Torsion subgroup is isomorphic to Abelian Group of order 1 computing the 2-Selmer group might take a very long time 2-Selmer rank is 0 two torsion subgroup is Abelian Group of order 1 Hence rank is at most 0 searching for enough torsion points and for 0 independent points on J this could take forever! found 0 independent points on J and enough torsion basis for torsion is [] torsion subgroup isomorphic to Abelian Group of order 1 basis for a free subgroup of finite index is [] Mordell--Weil rank is 0 and we know the torsion succeeded in showing that the only $K$-rational points are [ (1 : 0 : 0) ] ############################################################ Total time: 8740.729 seconds, Total memory usage: 175.33MB