Home

Elliptic curves by rank › curves from searches

Rank 11 record curves

The ten smallest curves of rank 11 in this collection, in each of four categories. The same curve often heads more than one list. Every curve here has proven rank 11, is given by its global minimal model, and carries 11 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 [0,0,1,-16359067,26274178986] 18031737725935636520843 51.246420000 61.506851000 3.04119414 51.24642037
2 [0,0,0,-127265047,551353318330] 18597723997878981667496 51.277326000 67.598950000 3.43732514 54.74306206
3 [0,0,1,-36274357,72401493396] 19274162508860687844797 51.313052000 63.833468000 3.30985267 55.02662449
4 [1,0,1,-97954983,366459823862] 22725958206446865097358 51.477795000 66.813659000 3.45890632 56.02108954
5 [0,1,1,-2741906470,54779656199380] 27044058554572306801893 51.651754000 76.809361000 4.26000184 65.30046500
6 [0,0,1,-445365697,5911837823466] 27639151875789191304731 51.673520000 72.339101000 4.06403361 64.41524675
7 [1,-1,1,-3391172427,83084924109915] 31433524696380244247818 51.802162000 77.624894000 4.43338469 68.35634864
8 [0,0,0,-70554967,233196593050] 31693061874096605785624 51.810385000 65.873441000 3.38925998 55.27612064
9 [0,0,0,-192733272,1043780189985] 38454953936075825696172 52.003774000 68.870886000 3.61683064 57.78451800
10 [0,0,1,-138457897,681816984246] 43407145415402227336531 52.124911000 68.019200000 3.63003394 58.69439244
generators for these ten

1. [0,0,1,-16359067,26274178986] N = 18031737725935636520843

E = ellinit([0,0,1,-16359067,26274178986]);
generators:
  [951,107595]
  [2023,38196]
  [2047,36939]
  [2194,30716]
  [2204,30413]
  [2238,29528]
  [2553,33903]
  [2837,51935]
  [3656,123825]
  [3866,144260]
  [5753,350096]

2. [0,0,0,-127265047,551353318330] N = 18597723997878981667496

E = ellinit([0,0,0,-127265047,551353318330]);
generators:
  [4637,246838]
  [4997,200462]
  [5569,123814]
  [5949,69226]
  [6157,34438]
  [6191,27332]
  [6799,19276]
  [6873,36446]
  [6953,51154]
  [7151,83428]
  [11077,707662]

3. [0,0,1,-36274357,72401493396] N = 19274162508860687844797

E = ellinit([0,0,1,-36274357,72401493396]);
generators:
  [-6828,41840]
  [-6418,202109]
  [-9559/4,3050929/8]
  [-5295/4,2749251/8]
  [1741,120519]
  [1987,90384]
  [4652,65784]
  [10645,944727]
  [12729,1293488]
  [12791,1304280]
  [16337,1959615]

4. [1,0,1,-97954983,366459823862] N = 22725958206446865097358

E = ellinit([1,0,1,-97954983,366459823862]);
generators:
  [-11339,144648]
  [-10587,471016]
  [-9506,667053]
  [-1516,715933]
  [3795,220318]
  [4641,106390]
  [4979,44198]
  [6765,112408]
  [7226,186008]
  [9529,541422]
  [20198,2564396]

5. [0,1,1,-2741906470,54779656199380] N = 27044058554572306801893

E = ellinit([0,1,1,-2741906470,54779656199380]);
generators:
  [-59161,3151534]
  [-53356,7013020]
  [-3046,7943755]
  [23657,1776136]
  [26753,757552]
  [32774,348124]
  [32816,376228]
  [33074,523675]
  [34622,1162483]
  [48812,6102796]
  [101315,28582594]

6. [0,0,1,-445365697,5911837823466] N = 27639151875789191304731

E = ellinit([0,0,1,-445365697,5911837823466]);
generators:
  [-2820,2673077]
  [653,2370926]
  [19226,2110904]
  [-21946,2261868]
  [11999,1515078]
  [7473,1732326]
  [-25923,191694]
  [206617,93458505]
  [-2233071/100,2172952467/1000]
  [1064869/25,1001731597/125]
  [30103,4447926]

7. [1,-1,1,-3391172427,83084924109915] N = 31433524696380244247818

E = ellinit([1,-1,1,-3391172427,83084924109915]);
generators:
  [-18805,11850250]
  [-2453,9560970]
  [-1557,9400842]
  [1147,8898682]
  [35307,2696202]
  [39723,3305162]
  [57835,8937994]
  [63435,11068970]
  [861419,797297930]
  [118867/9,171860494/27]
  [1049501/25,476706286/125]

8. [0,0,0,-70554967,233196593050] N = 31693061874096605785624

E = ellinit([0,0,0,-70554967,233196593050]);
generators:
  [-9723,4682]
  [-9719,29558]
  [-9525,202660]
  [-9275,299510]
  [-8707,432926]
  [-8623,447682]
  [-8147,516974]
  [-8031,530894]
  [-7733,562468]
  [-7118,612182]
  [22439,3154084]

9. [0,0,0,-192733272,1043780189985] N = 38454953936075825696172

E = ellinit([0,0,0,-192733272,1043780189985]);
generators:
  [5214,424989]
  [5802,347643]
  [6330,278235]
  [6972,197343]
  [7584,135279]
  [7944,118449]
  [8244,123201]
  [8406,132813]
  [8664,155889]
  [9156,216063]
  [16632,1561743]

10. [0,0,1,-138457897,681816984246] N = 43407145415402227336531

E = ellinit([0,0,1,-138457897,681816984246]);
generators:
  [-7277,1141938]
  [-2163,985485]
  [-2033,977189]
  [-2695/4,7041801/8]
  [-239,845514]
  [33/4,6600237/8]
  [7920,286397]
  [8316,324809]
  [8501,345149]
  [9967,540314]
  [10331,595005]

Smallest naive height

# a-invariants conductor N log N naive h Faltings h log abs(disc)
1 [0,0,1,-12510607,22378666386] 91029154289442938382923 52.865467000 61.185893000 3.11336358 52.86546678
2 [0,0,1,-15834277,-1218408864] 253441148857651353024037 53.889419000 61.346666000 3.18030906 53.88941859
3 [1,-1,1,-1280502,24610224630] 261505301079507297228953 53.920742000 61.375975000 3.17240343 53.92074151
4 [0,0,1,-16359067,26274178986] 18031737725935636520843 51.246420000 61.506851000 3.04119414 51.24642037
5 [0,-1,1,-18219470,19461310622] 74517748861467542239239 52.665324000 61.767607000 3.18290081 53.76393658
6 [0,0,1,-11098717,35181000996] 447189586584948275626043 54.457270000 62.090690000 3.22499080 54.45726959
7 [1,-1,0,-16935151,35586657241] 118055824521158268588478 53.125445000 62.113381000 3.19202984 53.81859174
8 [1,0,1,-9494793,36701246672] 263569643529999464027834 53.928605000 62.175342000 3.23580643 54.62175177
9 [0,1,1,-4369850,39234978510] 659722595274730880503243 54.846106000 62.308889000 3.25023111 54.84610639
10 [0,-1,1,-22545490,6475058592] 238453475709085744155159 53.828461000 62.406740000 3.26728816 54.92707346
generators for these ten

1. [0,0,1,-12510607,22378666386] N = 91029154289442938382923

E = ellinit([0,0,1,-12510607,22378666386]);
generators:
  [972,105530]
  [929,107508]
  [-2287,197556]
  [2153,-73644]
  [199,-141057]
  [3246,126375]
  [3339,133536]
  [-2814,-187885]
  [-3893,-109920]
  [2817,97419]
  [6692,-488206]

2. [0,0,1,-15834277,-1218408864] N = 253441148857651353024037

E = ellinit([0,0,1,-15834277,-1218408864]);
generators:
  [-3938,8190]
  [-3932,15836]
  [-3661,87651]
  [-3269,124940]
  [-2371,151643]
  [-2354,151694]
  [-2167,151388]
  [-1722,144713]
  [-1094,121634]
  [4065,39827]
  [4571,148019]

3. [1,-1,1,-1280502,24610224630] N = 261505301079507297228953

E = ellinit([1,-1,1,-1280502,24610224630]);
generators:
  [-2442,115982]
  [-1534,152302]
  [5400,-421195]
  [188,156034]
  [2761,203842]
  [961,155296]
  [-136,157490]
  [-1264,156221]
  [-3055,-136]
  [4516,-335294]
  [-12193/4,-95953/8]

4. [0,0,1,-16359067,26274178986] N = 18031737725935636520843

E = ellinit([0,0,1,-16359067,26274178986]);
generators:
  [951,107595]
  [2023,38196]
  [2047,36939]
  [2194,30716]
  [2204,30413]
  [2238,29528]
  [2553,33903]
  [2837,51935]
  [3656,123825]
  [3866,144260]
  [5753,350096]

5. [0,-1,1,-18219470,19461310622] N = 74517748861467542239239

E = ellinit([0,-1,1,-18219470,19461310622]);
generators:
  [-4651,59740]
  [-3508,200482]
  [-2482,222229]
  [-1933,217837]
  [-895,187216]
  [3611,27253]
  [4253,137419]
  [4367,152182]
  [4586,179821]
  [5276,264898]
  [5368,276245]

6. [0,0,1,-11098717,35181000996] N = 447189586584948275626043

E = ellinit([0,0,1,-11098717,35181000996]);
generators:
  [3046,172149]
  [-2453,218279]
  [-801,-208704]
  [-655,-205353]
  [1570,-147058]
  [446,-174126]
  [879,-161569]
  [-2049,-222081]
  [402,175454]
  [-1312,-217909]
  [-271,-195369]

7. [1,-1,0,-16935151,35586657241] N = 118055824521158268588478

E = ellinit([1,-1,0,-16935151,35586657241]);
generators:
  [904,144515]
  [386,-170801]
  [-710,-217021]
  [1699,-109090]
  [3161,115174]
  [-1724,-243385]
  [2724,96935]
  [18451/4,1867933/8]
  [-198,197411]
  [2820,-102649]
  [2344,-94795]

8. [1,0,1,-9494793,36701246672] N = 263569643529999464027834

E = ellinit([1,0,1,-9494793,36701246672]);
generators:
  [-451,202442]
  [-4115,80136]
  [1211,163648]
  [3497,-216844]
  [-4219,42928]
  [-1103,-213534]
  [4363/4,-1334457/8]
  [1333,-163190]
  [4459,285912]
  [-2981,-194768]
  [-3010,-193462]

9. [0,1,1,-4369850,39234978510] N = 659722595274730880503243

E = ellinit([0,1,1,-4369850,39234978510]);
generators:
  [-3423,118733]
  [-3116,150354]
  [1020,189314]
  [-3824,6453]
  [202,-195859]
  [3944,288730]
  [-1196,-206766]
  [10821/4,-1738395/8]
  [6023,481087]
  [2547,-211269]
  [2360,-205115]

10. [0,-1,1,-22545490,6475058592] N = 238453475709085744155159

E = ellinit([0,-1,1,-22545490,6475058592]);
generators:
  [-4431,139130]
  [-3177,214535]
  [-3143,215122]
  [-2796,218280]
  [-2150,212143]
  [-1739,201046]
  [-1340,185143]
  [-167,101170]
  [240,32831]
  [5107,156547]
  [9541,812281]

Smallest Faltings height

# a-invariants conductor N log N naive h Faltings h log abs(disc)
1 [0,0,1,-16359067,26274178986] 18031737725935636520843 51.246420000 61.506851000 3.04119414 51.24642037
2 [0,0,1,-12510607,22378666386] 91029154289442938382923 52.865467000 61.185893000 3.11336358 52.86546678
3 [1,-1,0,-22075864,43047713164] 55895061045198499766014 52.377763000 62.494039000 3.16460412 53.07091016
4 [1,-1,1,-1280502,24610224630] 261505301079507297228953 53.920742000 61.375975000 3.17240343 53.92074151
5 [0,0,1,-15834277,-1218408864] 253441148857651353024037 53.889419000 61.346666000 3.18030906 53.88941859
6 [0,-1,1,-18219470,19461310622] 74517748861467542239239 52.665324000 61.767607000 3.18290081 53.76393658
7 [1,-1,0,-16935151,35586657241] 118055824521158268588478 53.125445000 62.113381000 3.19202984 53.81859174
8 [0,1,1,-27593420,52869643590] 136664518278987512577317 53.271816104 63.012866758 3.19263167 53.27181610
9 [0,0,1,-11098717,35181000996] 447189586584948275626043 54.457270000 62.090690000 3.22499080 54.45726959
10 [1,0,1,-9494793,36701246672] 263569643529999464027834 53.928605000 62.175342000 3.23580643 54.62175177
generators for these ten

1. [0,0,1,-16359067,26274178986] N = 18031737725935636520843

E = ellinit([0,0,1,-16359067,26274178986]);
generators:
  [951,107595]
  [2023,38196]
  [2047,36939]
  [2194,30716]
  [2204,30413]
  [2238,29528]
  [2553,33903]
  [2837,51935]
  [3656,123825]
  [3866,144260]
  [5753,350096]

2. [0,0,1,-12510607,22378666386] N = 91029154289442938382923

E = ellinit([0,0,1,-12510607,22378666386]);
generators:
  [972,105530]
  [929,107508]
  [-2287,197556]
  [2153,-73644]
  [199,-141057]
  [3246,126375]
  [3339,133536]
  [-2814,-187885]
  [-3893,-109920]
  [2817,97419]
  [6692,-488206]

3. [1,-1,0,-22075864,43047713164] N = 55895061045198499766014

E = ellinit([1,-1,0,-22075864,43047713164]);
generators:
  [-2166,285158]
  [-2130,284618]
  [-1475,269808]
  [-1324,265150]
  [546,176240]
  [2895,56818]
  [3141,66904]
  [3158,67842]
  [3159,67898]
  [3642,102800]
  [3938,129070]

4. [1,-1,1,-1280502,24610224630] N = 261505301079507297228953

E = ellinit([1,-1,1,-1280502,24610224630]);
generators:
  [-2442,115982]
  [-1534,152302]
  [5400,-421195]
  [188,156034]
  [2761,203842]
  [961,155296]
  [-136,157490]
  [-1264,156221]
  [-3055,-136]
  [4516,-335294]
  [-12193/4,-95953/8]

5. [0,0,1,-15834277,-1218408864] N = 253441148857651353024037

E = ellinit([0,0,1,-15834277,-1218408864]);
generators:
  [-3938,8190]
  [-3932,15836]
  [-3661,87651]
  [-3269,124940]
  [-2371,151643]
  [-2354,151694]
  [-2167,151388]
  [-1722,144713]
  [-1094,121634]
  [4065,39827]
  [4571,148019]

6. [0,-1,1,-18219470,19461310622] N = 74517748861467542239239

E = ellinit([0,-1,1,-18219470,19461310622]);
generators:
  [-4651,59740]
  [-3508,200482]
  [-2482,222229]
  [-1933,217837]
  [-895,187216]
  [3611,27253]
  [4253,137419]
  [4367,152182]
  [4586,179821]
  [5276,264898]
  [5368,276245]

7. [1,-1,0,-16935151,35586657241] N = 118055824521158268588478

E = ellinit([1,-1,0,-16935151,35586657241]);
generators:
  [904,144515]
  [386,-170801]
  [-710,-217021]
  [1699,-109090]
  [3161,115174]
  [-1724,-243385]
  [2724,96935]
  [18451/4,1867933/8]
  [-198,197411]
  [2820,-102649]
  [2344,-94795]

8. [0,1,1,-27593420,52869643590] N = 136664518278987512577317

E = ellinit([0,1,1,-27593420,52869643590]);
generators:
  [-6014,36566]
  [-659/4,1916847/8]
  [1697,104563]
  [2215,51209]
  [2257,45756]
  [3596,12541]
  [3758,47542]
  [3802,54153]
  [4208,106237]
  [4707,165218]
  [5836,301058]

9. [0,0,1,-11098717,35181000996] N = 447189586584948275626043

E = ellinit([0,0,1,-11098717,35181000996]);
generators:
  [3046,172149]
  [-2453,218279]
  [-801,-208704]
  [-655,-205353]
  [1570,-147058]
  [446,-174126]
  [879,-161569]
  [-2049,-222081]
  [402,175454]
  [-1312,-217909]
  [-271,-195369]

10. [1,0,1,-9494793,36701246672] N = 263569643529999464027834

E = ellinit([1,0,1,-9494793,36701246672]);
generators:
  [-451,202442]
  [-4115,80136]
  [1211,163648]
  [3497,-216844]
  [-4219,42928]
  [-1103,-213534]
  [4363/4,-1334457/8]
  [1333,-163190]
  [4459,285912]
  [-2981,-194768]
  [-3010,-193462]

Smallest absolute discriminant

# a-invariants conductor N log N naive h Faltings h log abs(disc)
1 [0,0,1,-16359067,26274178986] 18031737725935636520843 51.246420000 61.506851000 3.04119414 51.24642037
2 [0,0,1,-12510607,22378666386] 91029154289442938382923 52.865467000 61.185893000 3.11336358 52.86546678
3 [1,-1,0,-22075864,43047713164] 55895061045198499766014 52.377763000 62.494039000 3.16460412 53.07091016
4 [1,-1,0,-93228004,346879627684] 57609056314579788665554 52.407967000 66.667498000 3.33282754 53.10111392
5 [0,1,1,-27593420,52869643590] 136664518278987512577317 53.271816104 63.012866758 3.19263167 53.27181610
6 [0,0,1,-43111327,110577482226] 154155590299449574815563 53.392249000 64.381110000 3.24704296 53.39224937
7 [1,-1,0,-103672594,405866860144] 80075902593366056993306 52.737262000 66.983848000 3.35968631 53.43040910
8 [1,-1,0,-84320329,297391514569] 83637942651649673596346 52.780784000 66.364004000 3.33308771 53.47393141
9 [1,-1,0,-108131509,432366460969] 84065693672563262871566 52.785886000 67.110179000 3.36735836 53.47903269
10 [0,1,1,-118684050,497211843240] 177544977951211262981957 53.533511000 67.389529000 3.38246389 53.53351093
generators for these ten

1. [0,0,1,-16359067,26274178986] N = 18031737725935636520843

E = ellinit([0,0,1,-16359067,26274178986]);
generators:
  [951,107595]
  [2023,38196]
  [2047,36939]
  [2194,30716]
  [2204,30413]
  [2238,29528]
  [2553,33903]
  [2837,51935]
  [3656,123825]
  [3866,144260]
  [5753,350096]

2. [0,0,1,-12510607,22378666386] N = 91029154289442938382923

E = ellinit([0,0,1,-12510607,22378666386]);
generators:
  [972,105530]
  [929,107508]
  [-2287,197556]
  [2153,-73644]
  [199,-141057]
  [3246,126375]
  [3339,133536]
  [-2814,-187885]
  [-3893,-109920]
  [2817,97419]
  [6692,-488206]

3. [1,-1,0,-22075864,43047713164] N = 55895061045198499766014

E = ellinit([1,-1,0,-22075864,43047713164]);
generators:
  [-2166,285158]
  [-2130,284618]
  [-1475,269808]
  [-1324,265150]
  [546,176240]
  [2895,56818]
  [3141,66904]
  [3158,67842]
  [3159,67898]
  [3642,102800]
  [3938,129070]

4. [1,-1,0,-93228004,346879627684] N = 57609056314579788665554

E = ellinit([1,-1,0,-93228004,346879627684]);
generators:
  [96,581270]
  [2605,347543]
  [5264,41758]
  [5488,19842]
  [5552,17058]
  [5683,21275]
  [6114,70448]
  [6352,101490]
  [6482,118800]
  [6495,120542]
  [6739,153563]

5. [0,1,1,-27593420,52869643590] N = 136664518278987512577317

E = ellinit([0,1,1,-27593420,52869643590]);
generators:
  [-6014,36566]
  [-659/4,1916847/8]
  [1697,104563]
  [2215,51209]
  [2257,45756]
  [3596,12541]
  [3758,47542]
  [3802,54153]
  [4208,106237]
  [4707,165218]
  [5836,301058]

6. [0,0,1,-43111327,110577482226] N = 154155590299449574815563

E = ellinit([0,0,1,-43111327,110577482226]);
generators:
  [-7590,23387]
  [-1009,391215]
  [2389,145668]
  [2698,117908]
  [3337,62244]
  [3558,47216]
  [3835,40592]
  [4327,71055]
  [4436,81410]
  [4849,124676]
  [5626,214724]

7. [1,-1,0,-103672594,405866860144] N = 80075902593366056993306

E = ellinit([1,-1,0,-103672594,405866860144]);
generators:
  [5281,72410]
  [5578,30560]
  [5592,28360]
  [5717,-1827]
  [6114,20008]
  [6146,25590]
  [6190,32702]
  [24819/4,279985/8]
  [6490,76502]
  [6702,106430]
  [6705,106853]

8. [1,-1,0,-84320329,297391514569] N = 83637942651649673596346

E = ellinit([1,-1,0,-84320329,297391514569]);
generators:
  [-1588,654465]
  [-1306,637273]
  [4156,134769]
  [4260,122297]
  [4887,42448]
  [5086,6429]
  [22019/4,6705/8]
  [6069,92683]
  [6272,120285]
  [6488,149715]
  [6711,180304]

9. [1,-1,0,-108131509,432366460969] N = 84065693672563262871566

E = ellinit([1,-1,0,-108131509,432366460969]);
generators:
  [-11692,319227]
  [4293,215255]
  [5260,92793]
  [22899/4,221317/8]
  [5792,15795]
  [5826,7789]
  [6440,52083]
  [6596,74625]
  [6812,105403]
  [7273,171300]
  [7283,172740]

10. [0,1,1,-118684050,497211843240] N = 177544977951211262981957

E = ellinit([0,1,1,-118684050,497211843240]);
generators:
  [4145,276584]
  [4738,203152]
  [4837,190603]
  [5339,125594]
  [5646,84456]
  [6129,8279]
  [6437,971]
  [6576,34056]
  [6699,53109]
  [7227,130373]
  [8435,310365]