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Rank 13 record curves

The ten smallest curves of rank 13 in this collection, in each of four categories. The same curve often heads more than one list. Every curve here has proven rank 13, is given by its global minimal model, and carries 13 independent generators, listed below each table.

Conductors are given as exact integers. All heights are natural logarithms. The naive height is log max(abs(c4)^3, c6^2); the Faltings height is -(1/2) log abs(Im(conj(w1) w2)) on the minimal model, the normalization used by the Elliptic Curve Rank Leaderboard.

Smallest conductor

# a-invariants conductor N log N naive h Faltings h log abs(disc)
1 [0,-1,1,-1876573260,32705933898332] 13048475524065056885495472921 64.738468820 75.760262570 4.24523710 65.83708111
2 [1,-1,0,-7636625866,271218848904208] 27751906366700157951398581202 65.493102047 79.991025802 4.60744226 70.26378667
3 [0,0,0,-6529826212,195995662948900] 38252643743425234347938185256 65.814010186 79.412541531 4.53993712 69.27974609
4 [0,0,1,-13219745917,236237043476946] 56327033117154259172161550881 66.200972093 81.528535388 4.85154672 73.89582017
5 [0,0,0,-2831015167,93401186864770] 72391879327606181200083239224 66.451891640 77.858995874 4.52309449 69.91762754
6 [0,0,1,-38590905927,2925614575527106] 79782239205970094033438692041 66.549098000 84.747707000 4.87221340 72.04215987
7 [0,0,0,-20005572727,932741657310970] 80533656253906958708883894472 66.558472698 82.771433157 4.88964576 73.99450051
8 [0,1,1,-4070928502,78946908330992] 85536680945963752284246181031 66.618742812 77.995013864 4.51071083 69.56318179
9 [1,1,0,-413296473437,102731720232812529] 87602268226431423767833962510 66.642604000 91.864944000 5.48698410 79.70042926
10 [0,1,0,-74484832620,7829021449464784] 89661775071019961275098888904 66.665842000 86.716380000 4.98123268 72.52947322
generators for these ten

1. [0,-1,1,-1876573260,32705933898332] N = 13048475524065056885495472921

E = ellinit([0,-1,1,-1876573260,32705933898332]);
generators:
  [-50174,740983]
  [-34581,7499665]
  [-7386,6794359]
  [258,5676424]
  [10493,3764339]
  [15191,2775653]
  [15843,2636614]
  [22793,1331960]
  [25138,1190455]
  [26925,1303036]
  [28326,1508983]
  [31670,2244663]
  [41511,5131861]

2. [1,-1,0,-7636625866,271218848904208] N = 27751906366700157951398581202

E = ellinit([1,-1,0,-7636625866,271218848904208]);
generators:
  [-100864,3965436]
  [-70857,21402944]
  [-59446,22725784]
  [-48898,22996540]
  [-44873,22903433]
  [-9662,18554814]
  [557,16338798]
  [63498,6474328]
  [169238,61770214]
  [769523/4,614873529/8]
  [1105366/25,556301646/125]
  [1756288/49,2257341500/343]
  [781732416/14641,6969544831916/1771561]

3. [0,0,0,-6529826212,195995662948900] N = 38252643743425234347938185256

E = ellinit([0,0,0,-6529826212,195995662948900]);
generators:
  [-61858,19058422]
  [-38214,19741382]
  [-23176,18299794]
  [-18394,17603462]
  [-6694,15472762]
  [-104,14024078]
  [2022,13520378]
  [26042,6603578]
  [37328,2064554]
  [53784,614386]
  [54654,1539254]
  [55250,1968970]
  [70414,9237226]

4. [0,0,1,-13219745917,236237043476946] N = 56327033117154259172161550881

E = ellinit([0,0,1,-13219745917,236237043476946]);
generators:
  [-121053,7914354]
  [-118362,11947916]
  [-102866,22530683]
  [-12555,20005797]
  [-3338,16742979]
  [-101,15413378]
  [10364,10017026]
  [15681,5726610]
  [133799,29372453]
  [152311,41906234]
  [173046,55950524]
  [302968,155049641]
  [960493/9,172688158/27]

5. [0,0,0,-2831015167,93401186864770] N = 72391879327606181200083239224

E = ellinit([0,0,0,-2831015167,93401186864770]);
generators:
  [-58759,7541638]
  [-44543,11451038]
  [-43847,11542714]
  [-17115,11697890]
  [-9563,10936162]
  [-1367,9862486]
  [1049,9509606]
  [13461,7598158]
  [18534,6877336]
  [26561,6078242]
  [31471,5956168]
  [50281,8841622]
  [55485,10350730]

6. [0,0,1,-38590905927,2925614575527106] N = 79782239205970094033438692041

E = ellinit([0,0,1,-38590905927,2925614575527106]);
generators:
  [109675,3520552]
  [110998,-3107777]
  [78544,19470001]
  [-135368,75293014]
  [97511,-9473641]
  [112078,-2878952]
  [80827,18288535]
  [375748,203655901]
  [122518,-6050624]
  [111277,3038035]
  [95650,-10463693]
  [116191,-3212105]
  [128719,9535351]

7. [0,0,0,-20005572727,932741657310970] N = 80533656253906958708883894472

E = ellinit([0,0,0,-20005572727,932741657310970]);
generators:
  [-109303,42585818]
  [-85877,44915804]
  [-28690,38510860]
  [-3833,31770532]
  [2421,29737558]
  [22137,22376918]
  [46093,10418846]
  [105771,213908]
  [109799,7737364]
  [113774,11374436]
  [487161/4,139199147/8]
  [2650741/25,236291914/125]
  [2897569/9,4506918706/27]

8. [0,1,1,-4070928502,78946908330992] N = 85536680945963752284246181031

E = ellinit([0,1,1,-4070928502,78946908330992]);
generators:
  [-54368,11814095]
  [-34665,13357066]
  [-33000,13317343]
  [-8619,10648663]
  [428,8786616]
  [7420,7010648]
  [9282,6477679]
  [21955,391086]
  [49885,103008]
  [61798,7961256]
  [133637,43835384]
  [167338,63902688]
  [-532955/9,286204672/27]

9. [1,1,0,-413296473437,102731720232812529] N = 87602268226431423767833962510

E = ellinit([1,1,0,-413296473437,102731720232812529]);
generators:
  [356708,26155271]
  [413458,50100521]
  [420818,-57925879]
  [82633,262911221]
  [370178,-21743599]
  [68008,273715521]
  [95458,253229021]
  [382208,-24701479]
  [-604792,-362290729]
  [366008,-22390729]
  [16676192/49,-13312562047/343]
  [304688,71198921]
  [519293,167506121]

10. [0,1,0,-74484832620,7829021449464784] N = 89661775071019961275098888904

E = ellinit([0,1,0,-74484832620,7829021449464784]);
generators:
  [150956,-5005396]
  [169392,8508236]
  [123434,22709258]
  [92876,41380196]
  [150758,5126770]
  [618333/4,23807597/8]
  [162214,3868338]
  [169658,8689690]
  [146787,-7640204]
  [50119,-64975482]
  [158658,2286590]
  [155754,2493878]
  [146986,7512054]

Smallest naive height

# a-invariants conductor N log N naive h Faltings h log abs(disc)
1 [0,0,1,-1129201567,23953648602246] 155721986935837682842530035243 67.217870000 75.137429000 4.29745552 67.21786979
2 [0,0,1,-1141015057,26061092520366] 198333672870666343333189775723 67.459748000 75.306075000 4.31537831 67.45974834
3 [0,-1,1,-1876573260,32705933898332] 13048475524065056885495472921 64.738468820 75.760262570 4.24523710 65.83708111
4 [0,0,1,-2217077947,28425800438316] 348398420573690733009119729797 68.023144222 76.171970810 4.37487054 68.02314422
5 [0,0,1,-2147907577,47888102069226] 356492038662688596396428055563 68.046109419 76.522921935 4.38307354 68.04610942
6 [1,-1,0,-2049670219,48898999201369] 240918419474403489984588164194 67.654256000 76.564681000 4.40046584 68.34740306
7 [1,-1,0,-2599487089,39173361400609] 230646920098311714063851989286 67.610685568 76.649343000 4.40343634 68.30383275
8 [0,0,1,-1726817617,57975709345266] 1122482466080847283271138480843 69.193096000 76.905236000 4.45559955 69.19309551
9 [1,-1,0,-2834795014,46808291571064] 255733741895151889311142284086 67.713934000 76.909309000 4.41629412 68.40708152
10 [1,0,1,-2977992463,43200225886902] 442007947048813769354369373646 68.261125373 77.057148753 4.45128179 68.95427255
generators for these ten

1. [0,0,1,-1129201567,23953648602246] N = 155721986935837682842530035243

E = ellinit([0,0,1,-1129201567,23953648602246]);
generators:
  [-41321,246731]
  [-31532,5311158]
  [-18067,6201416]
  [-2305,5152107]
  [86,4884315]
  [847,4795604]
  [3304,4500981]
  [19999,3060971]
  [21377,3095720]
  [28464,3856626]
  [30009,4134216]
  [38281,6068360]
  [50563,9804491]

2. [0,0,1,-1141015057,26061092520366] N = 198333672870666343333189775723

E = ellinit([0,0,1,-1141015057,26061092520366]);
generators:
  [-30096,5756826]
  [-28332,5970441]
  [-23223,6327285]
  [-22420,6353982]
  [-10314,6060720]
  [-5762,5695991]
  [-1867,5308941]
  [-873,5201589]
  [1709,4910813]
  [10639,3889221]
  [25205,3648882]
  [28849,4141733]
  [39589,6552621]

3. [0,-1,1,-1876573260,32705933898332] N = 13048475524065056885495472921

E = ellinit([0,-1,1,-1876573260,32705933898332]);
generators:
  [-50174,740983]
  [-34581,7499665]
  [-7386,6794359]
  [258,5676424]
  [10493,3764339]
  [15191,2775653]
  [15843,2636614]
  [22793,1331960]
  [25138,1190455]
  [26925,1303036]
  [28326,1508983]
  [31670,2244663]
  [41511,5131861]

4. [0,0,1,-2217077947,28425800438316] N = 348398420573690733009119729797

E = ellinit([0,0,1,-2217077947,28425800438316]);
generators:
  [-1992,5730123]
  [607,5203869]
  [10905,2354862]
  [38511,399330]
  [43592,3822980]
  [47123,5347026]
  [76053,17312001]
  [102842,29801394]
  [448418,298666298]
  [-13012501/289,29872122205/4913]
  [-34298519/1296,386329606955/46656]
  [1493376/25,1305873751/125]
  [3154275529/3364,176932931836535/195112]

5. [0,0,1,-2147907577,47888102069226] N = 356492038662688596396428055563

E = ellinit([0,0,1,-2147907577,47888102069226]);
generators:
  [-219071/4,8936925/8]
  [-45112,7278558]
  [-38508,8573069]
  [-30707,9213558]
  [-9063,8161505]
  [-668,7023005]
  [606,6825444]
  [1724,6647573]
  [19378,3680016]
  [22558,3303716]
  [33568,3689433]
  [58477,11056694]
  [60591,11840190]

6. [1,-1,0,-2049670219,48898999201369] N = 240918419474403489984588164194

E = ellinit([1,-1,0,-2049670219,48898999201369]);
generators:
  [-47765,6174018]
  [-41117,7999362]
  [-40722,8072435]
  [-508,7067085]
  [33789,4251433]
  [40830,5748203]
  [77964,19013293]
  [82167,20820898]
  [531863/4,368438139/8]
  [208745,93260298]
  [381728,234096037]
  [12475639,44058514338]
  [-868995/16,7471163/64]

7. [1,-1,0,-2599487089,39173361400609] N = 230646920098311714063851989286

E = ellinit([1,-1,0,-2599487089,39173361400609]);
generators:
  [-37735,9158448]
  [-1806,6623753]
  [-177,6295600]
  [-37,6266560]
  [557,6141840]
  [2292,5763173]
  [56492,8492685]
  [124692,40603973]
  [-631661/25,1178988577/125]
  [-413012/9,212912381/27]
  [15745684/961,29247194873/29791]
  [959762629/17424,18208243461517/2299968]
  [16845344559906/1024960225,30680473388418927859/32814101603375]

8. [0,0,1,-1726817617,57975709345266] N = 1122482466080847283271138480843

E = ellinit([0,0,1,-1726817617,57975709345266]);
generators:
  [401,7568574]
  [52318,10527843]
  [-29122,9141443]
  [-33509,8843865]
  [893,7512281]
  [30523,5805593]
  [-47872,5561681]
  [-226151/9,249660170/27]
  [-207703/4,22105873/8]
  [-104487/4,73868697/8]
  [35904,6500753]
  [27267,5582409]
  [44289/4,-50731203/8]

9. [1,-1,0,-2834795014,46808291571064] N = 255733741895151889311142284086

E = ellinit([1,-1,0,-2834795014,46808291571064]);
generators:
  [-50050,7981942]
  [-2388,7319948]
  [-1316,7109568]
  [2684,6261168]
  [13418,3337942]
  [15173,2692210]
  [41660,985488]
  [41710,1042902]
  [42013,1344842]
  [44988,3191092]
  [46886,4095270]
  [99585,27374638]
  [-3174287/121,13533550930/1331]

10. [1,0,1,-2977992463,43200225886902] N = 442007947048813769354369373646

E = ellinit([1,0,1,-2977992463,43200225886902]);
generators:
  [-60711,507738]
  [-60033,2401166]
  [-21051,9837092]
  [-1035,6803556]
  [1288,6273645]
  [8191,4395570]
  [44911,181270]
  [45642,1513496]
  [214359/4,48784185/8]
  [55081,6775498]
  [99865,27185356]
  [103759,29124678]
  [110785,32701068]

Smallest Faltings height

# a-invariants conductor N log N naive h Faltings h log abs(disc)
1 [0,-1,1,-1876573260,32705933898332] 13048475524065056885495472921 64.738468820 75.760262570 4.24523710 65.83708111
2 [0,0,1,-1129201567,23953648602246] 155721986935837682842530035243 67.217870000 75.137429000 4.29745552 67.21786979
3 [0,0,1,-1141015057,26061092520366] 198333672870666343333189775723 67.459748000 75.306075000 4.31537831 67.45974834
4 [0,1,1,-3915257350,93169964920370] 91004323775371737344808170357 66.680704530 77.878043729 4.36269022 66.68070453
5 [0,0,1,-2217077947,28425800438316] 348398420573690733009119729797 68.023144222 76.171970810 4.37487054 68.02314422
6 [0,0,1,-2147907577,47888102069226] 356492038662688596396428055563 68.046109419 76.522921935 4.38307354 68.04610942
7 [1,-1,0,-2049670219,48898999201369] 240918419474403489984588164194 67.654256000 76.564681000 4.40046584 68.34740306
8 [1,-1,0,-2599487089,39173361400609] 230646920098311714063851989286 67.610685568 76.649343000 4.40343634 68.30383275
9 [1,-1,0,-2834795014,46808291571064] 255733741895151889311142284086 67.713934000 76.909309000 4.41629412 68.40708152
10 [0,0,1,-3491340307,85062365501016] 402099048666566052137511629723 68.166496000 77.671957000 4.42527041 68.16649596
generators for these ten

1. [0,-1,1,-1876573260,32705933898332] N = 13048475524065056885495472921

E = ellinit([0,-1,1,-1876573260,32705933898332]);
generators:
  [-50174,740983]
  [-34581,7499665]
  [-7386,6794359]
  [258,5676424]
  [10493,3764339]
  [15191,2775653]
  [15843,2636614]
  [22793,1331960]
  [25138,1190455]
  [26925,1303036]
  [28326,1508983]
  [31670,2244663]
  [41511,5131861]

2. [0,0,1,-1129201567,23953648602246] N = 155721986935837682842530035243

E = ellinit([0,0,1,-1129201567,23953648602246]);
generators:
  [-41321,246731]
  [-31532,5311158]
  [-18067,6201416]
  [-2305,5152107]
  [86,4884315]
  [847,4795604]
  [3304,4500981]
  [19999,3060971]
  [21377,3095720]
  [28464,3856626]
  [30009,4134216]
  [38281,6068360]
  [50563,9804491]

3. [0,0,1,-1141015057,26061092520366] N = 198333672870666343333189775723

E = ellinit([0,0,1,-1141015057,26061092520366]);
generators:
  [-30096,5756826]
  [-28332,5970441]
  [-23223,6327285]
  [-22420,6353982]
  [-10314,6060720]
  [-5762,5695991]
  [-1867,5308941]
  [-873,5201589]
  [1709,4910813]
  [10639,3889221]
  [25205,3648882]
  [28849,4141733]
  [39589,6552621]

4. [0,1,1,-3915257350,93169964920370] N = 91004323775371737344808170357

E = ellinit([0,1,1,-3915257350,93169964920370]);
generators:
  [-58520,11040254]
  [-799,9813141]
  [3837,8843281]
  [4705,8651754]
  [28301,2243358]
  [29377,1872138]
  [32403,571492]
  [39824,639729]
  [40642,1085841]
  [41215,1347320]
  [65150,10706345]
  [66436,11237958]
  [178900,71543654]

5. [0,0,1,-2217077947,28425800438316] N = 348398420573690733009119729797

E = ellinit([0,0,1,-2217077947,28425800438316]);
generators:
  [-1992,5730123]
  [607,5203869]
  [10905,2354862]
  [38511,399330]
  [43592,3822980]
  [47123,5347026]
  [76053,17312001]
  [102842,29801394]
  [448418,298666298]
  [-13012501/289,29872122205/4913]
  [-34298519/1296,386329606955/46656]
  [1493376/25,1305873751/125]
  [3154275529/3364,176932931836535/195112]

6. [0,0,1,-2147907577,47888102069226] N = 356492038662688596396428055563

E = ellinit([0,0,1,-2147907577,47888102069226]);
generators:
  [-219071/4,8936925/8]
  [-45112,7278558]
  [-38508,8573069]
  [-30707,9213558]
  [-9063,8161505]
  [-668,7023005]
  [606,6825444]
  [1724,6647573]
  [19378,3680016]
  [22558,3303716]
  [33568,3689433]
  [58477,11056694]
  [60591,11840190]

7. [1,-1,0,-2049670219,48898999201369] N = 240918419474403489984588164194

E = ellinit([1,-1,0,-2049670219,48898999201369]);
generators:
  [-47765,6174018]
  [-41117,7999362]
  [-40722,8072435]
  [-508,7067085]
  [33789,4251433]
  [40830,5748203]
  [77964,19013293]
  [82167,20820898]
  [531863/4,368438139/8]
  [208745,93260298]
  [381728,234096037]
  [12475639,44058514338]
  [-868995/16,7471163/64]

8. [1,-1,0,-2599487089,39173361400609] N = 230646920098311714063851989286

E = ellinit([1,-1,0,-2599487089,39173361400609]);
generators:
  [-37735,9158448]
  [-1806,6623753]
  [-177,6295600]
  [-37,6266560]
  [557,6141840]
  [2292,5763173]
  [56492,8492685]
  [124692,40603973]
  [-631661/25,1178988577/125]
  [-413012/9,212912381/27]
  [15745684/961,29247194873/29791]
  [959762629/17424,18208243461517/2299968]
  [16845344559906/1024960225,30680473388418927859/32814101603375]

9. [1,-1,0,-2834795014,46808291571064] N = 255733741895151889311142284086

E = ellinit([1,-1,0,-2834795014,46808291571064]);
generators:
  [-50050,7981942]
  [-2388,7319948]
  [-1316,7109568]
  [2684,6261168]
  [13418,3337942]
  [15173,2692210]
  [41660,985488]
  [41710,1042902]
  [42013,1344842]
  [44988,3191092]
  [46886,4095270]
  [99585,27374638]
  [-3174287/121,13533550930/1331]

10. [0,0,1,-3491340307,85062365501016] N = 402099048666566052137511629723

E = ellinit([0,0,1,-3491340307,85062365501016]);
generators:
  [32014,2470146]
  [101,9203789]
  [11355,6847062]
  [59037,9203789]
  [41873,-3505342]
  [-65278,-5899690]
  [45978,4661966]
  [232009,-108460414]
  [152658,-55764634]
  [-35901,-12811434]
  [32266,-2450050]
  [-48106,11903381]
  [3021,8633805]

Smallest absolute discriminant

# a-invariants conductor N log N naive h Faltings h log abs(disc)
1 [0,-1,1,-1876573260,32705933898332] 13048475524065056885495472921 64.738468820 75.760262570 4.24523710 65.83708111
2 [0,1,1,-3915257350,93169964920370] 91004323775371737344808170357 66.680704530 77.878043729 4.36269022 66.68070453
3 [0,0,1,-1129201567,23953648602246] 155721986935837682842530035243 67.217870000 75.137429000 4.29745552 67.21786979
4 [0,0,1,-1141015057,26061092520366] 198333672870666343333189775723 67.459748000 75.306075000 4.31537831 67.45974834
5 [1,0,1,-6489409453,202730435387172] 132452629998841421797163345894 67.056023000 79.408947000 4.47003817 67.74916976
6 [0,0,1,-6265772149,189142776829182] 288794189909830943337356727829 67.835511800 79.288706030 4.46902532 67.83551180
7 [0,0,1,-6703514287,209526656923296] 313723684682894471293495831717 67.918310124 79.491296272 4.48073556 67.91831012
8 [1,-1,0,-25949190094,1608685635288664] 165782089380280872920512734806 67.280471722 83.551821740 4.67077074 67.97361890
9 [0,0,1,-2217077947,28425800438316] 348398420573690733009119729797 68.023144222 76.171970810 4.37487054 68.02314422
10 [0,0,1,-2147907577,47888102069226] 356492038662688596396428055563 68.046109419 76.522921935 4.38307354 68.04610942
generators for these ten

1. [0,-1,1,-1876573260,32705933898332] N = 13048475524065056885495472921

E = ellinit([0,-1,1,-1876573260,32705933898332]);
generators:
  [-50174,740983]
  [-34581,7499665]
  [-7386,6794359]
  [258,5676424]
  [10493,3764339]
  [15191,2775653]
  [15843,2636614]
  [22793,1331960]
  [25138,1190455]
  [26925,1303036]
  [28326,1508983]
  [31670,2244663]
  [41511,5131861]

2. [0,1,1,-3915257350,93169964920370] N = 91004323775371737344808170357

E = ellinit([0,1,1,-3915257350,93169964920370]);
generators:
  [-58520,11040254]
  [-799,9813141]
  [3837,8843281]
  [4705,8651754]
  [28301,2243358]
  [29377,1872138]
  [32403,571492]
  [39824,639729]
  [40642,1085841]
  [41215,1347320]
  [65150,10706345]
  [66436,11237958]
  [178900,71543654]

3. [0,0,1,-1129201567,23953648602246] N = 155721986935837682842530035243

E = ellinit([0,0,1,-1129201567,23953648602246]);
generators:
  [-41321,246731]
  [-31532,5311158]
  [-18067,6201416]
  [-2305,5152107]
  [86,4884315]
  [847,4795604]
  [3304,4500981]
  [19999,3060971]
  [21377,3095720]
  [28464,3856626]
  [30009,4134216]
  [38281,6068360]
  [50563,9804491]

4. [0,0,1,-1141015057,26061092520366] N = 198333672870666343333189775723

E = ellinit([0,0,1,-1141015057,26061092520366]);
generators:
  [-30096,5756826]
  [-28332,5970441]
  [-23223,6327285]
  [-22420,6353982]
  [-10314,6060720]
  [-5762,5695991]
  [-1867,5308941]
  [-873,5201589]
  [1709,4910813]
  [10639,3889221]
  [25205,3648882]
  [28849,4141733]
  [39589,6552621]

5. [1,0,1,-6489409453,202730435387172] N = 132452629998841421797163345894

E = ellinit([1,0,1,-6489409453,202730435387172]);
generators:
  [40159,2604588]
  [40170,2601123]
  [1191,13963760]
  [45438,-1317709]
  [46769,1212522]
  [53105,-2832732]
  [8362,12204451]
  [63825,6935466]
  [397012,-245561184]
  [51615,-2325412]
  [55444,3628683]
  [176375/4,-12356847/8]
  [15781,-10218240]

6. [0,0,1,-6265772149,189142776829182] N = 288794189909830943337356727829

E = ellinit([0,0,1,-6265772149,189142776829182]);
generators:
  [-46306,19493439]
  [30635,5093318]
  [34263,3831681]
  [39722,1711323]
  [41743,571761]
  [41922,380876]
  [49353,343721]
  [198473/4,5089953/8]
  [49703,707678]
  [50530,1245228]
  [50699,1338105]
  [54119,2924510]
  [621337/9,249825122/27]

7. [0,0,1,-6703514287,209526656923296] N = 313723684682894471293495831717

E = ellinit([0,0,1,-6703514287,209526656923296]);
generators:
  [-12776,17119736]
  [26713,7036878]
  [40823,1975178]
  [41468,1689228]
  [42512,1173680]
  [50836,349604]
  [53094,1811301]
  [53770,2130542]
  [75783,11693621]
  [82753,14882478]
  [218196,95577444]
  [-1076701/25,2556697218/125]
  [16501489/324,2752651765/5832]

8. [1,-1,0,-25949190094,1608685635288664] N = 165782089380280872920512734806

E = ellinit([1,-1,0,-25949190094,1608685635288664]);
generators:
  [-120149,54759245]
  [-100654,56626110]
  [-1434,40570334]
  [-308,40208088]
  [15758,34686372]
  [132103/4,224397919/8]
  [69578,11798350]
  [103747,5710205]
  [124943,17740860]
  [133006,22521764]
  [1376179/4,1460773185/8]
  [20177011/196,14458679971/2744]
  [1686891884/13225,29316901631272/1520875]

9. [0,0,1,-2217077947,28425800438316] N = 348398420573690733009119729797

E = ellinit([0,0,1,-2217077947,28425800438316]);
generators:
  [-1992,5730123]
  [607,5203869]
  [10905,2354862]
  [38511,399330]
  [43592,3822980]
  [47123,5347026]
  [76053,17312001]
  [102842,29801394]
  [448418,298666298]
  [-13012501/289,29872122205/4913]
  [-34298519/1296,386329606955/46656]
  [1493376/25,1305873751/125]
  [3154275529/3364,176932931836535/195112]

10. [0,0,1,-2147907577,47888102069226] N = 356492038662688596396428055563

E = ellinit([0,0,1,-2147907577,47888102069226]);
generators:
  [-219071/4,8936925/8]
  [-45112,7278558]
  [-38508,8573069]
  [-30707,9213558]
  [-9063,8161505]
  [-668,7023005]
  [606,6825444]
  [1724,6647573]
  [19378,3680016]
  [22558,3303716]
  [33568,3689433]
  [58477,11056694]
  [60591,11840190]