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Rank 12 record curves

The ten smallest curves of rank 12 in this collection, in each of four categories. The same curve often heads more than one list. Every curve here has proven rank 12, is given by its global minimal model, and carries 12 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 [1,-1,0,-227292004,882331831684] 12212741842431104280769438 57.764522000 69.338841000 3.81126542 61.29088258
2 [0,0,1,-2558717707,49801921968156] 21585663145423055401656667 58.334071583 76.601919255 4.11608590 61.76805879
3 [1,1,0,-546508862,5450614684680] 23420861912249380220491254 58.415669000 72.176728000 3.98618530 63.04064221
4 [0,0,0,-932733487,11052354147250] 26145292874820119408329144 58.525711000 73.590475000 3.98775654 61.99144730
5 [0,0,0,-1410298617,20439456397265] 26585567392338965114027388 58.542411000 74.820122000 4.04536021 62.12592966
6 [1,-1,0,-266451964,1712164041364] 27805670188643664300931114 58.587282000 69.860626000 3.72231917 59.28042938
7 [0,-1,0,-3750893540,87321690999876] 28049021634623498384808552 58.595996000 77.749383000 4.35384604 66.59433138
8 [1,-1,1,-40458649392,3123308218829715] 31637890823944880544503354 58.716398000 84.884242000 4.88725738 72.27060584
9 [0,-1,0,-1006329305,12582779623101] 33024990362154035341284168 58.759306790 73.849792816 4.05726274 63.32365498
10 [1,-1,0,-1865252986,30958962950848] 33915523108918463487650678 58.785915050 75.653590624 4.09422317 62.42350121
generators for these ten

1. [1,-1,0,-227292004,882331831684] N = 12212741842431104280769438

E = ellinit([1,-1,0,-227292004,882331831684]);
generators:
  [-7017,1463542]
  [-5072,1382630]
  [-1638,1118960]
  [-210,964498]
  [1082,798000]
  [4176,75194]
  [23131,2816954]
  [23658,2945542]
  [35286,6048314]
  [37241,6619599]
  [49855,10627043]
  [54496,12233934]

2. [0,0,1,-2558717707,49801921968156] N = 21585663145423055401656667

E = ellinit([0,0,1,-2558717707,49801921968156]);
generators:
  [-1212,7273328]
  [22286,1961385]
  [22999,1766178]
  [24797,1265249]
  [27711,420189]
  [28703,79778]
  [29633,24753]
  [29912,169275]
  [30997,521249]
  [34807,1705949]
  [39429,3195681]
  [66616,13227684]

3. [1,1,0,-546508862,5450614684680] N = 23420861912249380220491254

E = ellinit([1,1,0,-546508862,5450614684680]);
generators:
  [-27174,499770]
  [-26732,992192]
  [-21326,2732244]
  [-17866,3093122]
  [-4060,2759300]
  [-926,2440932]
  [77,2325587]
  [7693,1300640]
  [13694,724544]
  [15054,789552]
  [16465,948995]
  [73466,18984890]

4. [0,0,0,-932733487,11052354147250] N = 26145292874820119408329144

E = ellinit([0,0,0,-932733487,11052354147250]);
generators:
  [-12777,4569904]
  [11209,1416214]
  [12293,1201646]
  [17037,326378]
  [17321,305078]
  [20561,752782]
  [21463,959296]
  [23925,1559330]
  [24425,1685770]
  [32037,3748622]
  [45450,7908590]
  [87207,24350072]

5. [0,0,0,-1410298617,20439456397265] N = 26585567392338965114027388

E = ellinit([0,0,0,-1410298617,20439456397265]);
generators:
  [-7121,5488269]
  [10120,2683965]
  [17449,1069491]
  [18157,904767]
  [19433,609779]
  [19888,507729]
  [20872,310467]
  [21328,249729]
  [22297,281283]
  [24391,742683]
  [24427,751767]
  [26821,1381287]

6. [1,-1,0,-266451964,1712164041364] N = 27805670188643664300931114

E = ellinit([1,-1,0,-266451964,1712164041364]);
generators:
  [-6310,1775782]
  [-2193,1513030]
  [-824,1390074]
  [-670,1375248]
  [-634,1371754]
  [951,1207675]
  [1178,1182592]
  [1752,1117478]
  [1784,1113778]
  [10261,236495]
  [10761,296119]
  [255159/25,28883167/125]

7. [0,-1,0,-3750893540,87321690999876] N = 28049021634623498384808552

E = ellinit([0,-1,0,-3750893540,87321690999876]);
generators:
  [28256,1973646]
  [29000,1712874]
  [30178,1268768]
  [15484,5740642]
  [30705,1047924]
  [39509,893358]
  [39602,-941160]
  [-31822,13208232]
  [32033,194352]
  [51320,5476026]
  [40268,1254642]
  [-11982,11425608]

8. [1,-1,1,-40458649392,3123308218829715] N = 31637890823944880544503354

E = ellinit([1,-1,1,-40458649392,3123308218829715]);
generators:
  [-75555,75858929]
  [61685,29334569]
  [84621,17439057]
  [100601,8391181]
  [105301,5476617]
  [121375,783009]
  [122033,1763429]
  [123293,2975729]
  [128989,7056945]
  [185577,44697177]
  [232765,79363249]
  [5047277/49,2376896647/343]

9. [0,-1,0,-1006329305,12582779623101] N = 33024990362154035341284168

E = ellinit([0,-1,0,-1006329305,12582779623101]);
generators:
  [-14869,4925274]
  [-6319,4323126]
  [4061,2926266]
  [8145,2219574]
  [14841,957306]
  [16099,744398]
  [17971,549154]
  [19253,586698]
  [21915,1026474]
  [23555,1395474]
  [25163,1786758]
  [29637,2964630]

10. [1,-1,0,-1865252986,30958962950848] N = 33915523108918463487650678

E = ellinit([1,-1,0,-1865252986,30958962950848]);
generators:
  [-49312,1764144]
  [1518,5303108]
  [18632,1625748]
  [21216,956416]
  [21824,792712]
  [22986,466688]
  [24126,5428]
  [25737,8650]
  [25842,105880]
  [27794,751992]
  [1666/9,149387528/27]
  [1548233/49,642375915/343]

Smallest naive height

# a-invariants conductor N log N naive h Faltings h log abs(disc)
1 [1,-1,0,-79922404,945333881884] 176685196983131610721395454 60.436412000 68.672711000 3.77771563 61.12955901
2 [0,0,1,-10424167,1013182725216] 443392455047858519129412443 61.356498000 68.811381000 3.79198184 61.35649751
3 [0,0,1,-129671269,1124584406100] 406802088686884491992318651 61.270369000 69.020015000 3.79657669 61.27036903
4 [0,0,1,-190719817,1221345067746] 200423269507421539440359243 60.562474000 69.185093000 3.76381548 60.56247371
5 [0,1,1,-216826410,990382874670] 228611240971968172580506037 60.694065000 69.197426000 3.77357392 60.69406516
6 [0,0,1,-220280389,937214745522] 304624442109032000008541749 60.981122000 69.244838000 3.79103618 60.98112191
7 [1,-1,0,-227292004,882331831684] 12212741842431104280769438 57.764522000 69.338841000 3.81126542 61.29088258
8 [0,0,1,-234270673,1016774489478] 75252011735942348487679505 59.582885000 69.429566000 3.80793929 61.19232278
9 [0,0,1,-239325247,1007007955806] 439218641837775372384213397 61.347039566 69.493605185 3.81847006 61.34703957
10 [1,-1,0,-241075144,935628365884] 259277654321226224156197706 60.819941744 69.515460766 3.82863083 61.51308892
generators for these ten

1. [1,-1,0,-79922404,945333881884] N = 176685196983131610721395454

E = ellinit([1,-1,0,-79922404,945333881884]);
generators:
  [-9628,911598]
  [-8582,1003842]
  [-8020,1038618]
  [-3478,1088580]
  [-2261,1056818]
  [-1990,1048132]
  [-1445,1029218]
  [-618,997550]
  [170,965188]
  [251,961794]
  [6414,831388]
  [7490,871962]

2. [0,0,1,-10424167,1013182725216] N = 443392455047858519129412443

E = ellinit([0,0,1,-10424167,1013182725216]);
generators:
  [-10348,113915]
  [-9995,344777]
  [-8831,645398]
  [-7828,784284]
  [-4426,986213]
  [-4353,987965]
  [-3854,998054]
  [-2152,1012743]
  [-1746,1012946]
  [130,1005897]
  [6672,1113840]
  [10496,1435295]

3. [0,0,1,-129671269,1124584406100] N = 406802088686884491992318651

E = ellinit([0,0,1,-129671269,1124584406100]);
generators:
  [-10223,1175504]
  [-2438,1194249]
  [-2309,1188143]
  [3186,862433]
  [4656,788523]
  [7040,748740]
  [13007,1280039]
  [15865,1749440]
  [25708,3844664]
  [36728,6775404]
  [70955,18685680]
  [75088/9,21319145/27]

4. [0,0,1,-190719817,1221345067746] N = 200423269507421539440359243

E = ellinit([0,0,1,-190719817,1221345067746]);
generators:
  [-15980,434037]
  [-1782,1247216]
  [-1752,1245033]
  [-118,1115279]
  [5419,589038]
  [8623,466841]
  [9134,491283]
  [9278,500508]
  [10094,569811]
  [12145,834537]
  [25270,3540987]
  [47686,10028084]

5. [0,1,1,-216826410,990382874670] N = 228611240971968172580506037

E = ellinit([0,1,1,-216826410,990382874670]);
generators:
  [2241,718146]
  [-16287,449117]
  [-1596,-1154286]
  [11400,15354]
  [12579,503460]
  [71777/16,21031257/64]
  [-7063,-1472934]
  [11412,-48142]
  [11846,290378]
  [4095,-413721]
  [4272,376938]
  [12099,-371845]

6. [0,0,1,-220280389,937214745522] N = 304624442109032000008541749

E = ellinit([0,0,1,-220280389,937214745522]);
generators:
  [-65943/4,2375953/8]
  [-7435,1471053]
  [-1281,1103309]
  [-611,1035170]
  [944,854465]
  [2979,554469]
  [4109,318525]
  [4663,106946]
  [12335,311213]
  [12950,506291]
  [16245,1282893]
  [65617/4,10546819/8]

7. [1,-1,0,-227292004,882331831684] N = 12212741842431104280769438

E = ellinit([1,-1,0,-227292004,882331831684]);
generators:
  [-7017,1463542]
  [-5072,1382630]
  [-1638,1118960]
  [-210,964498]
  [1082,798000]
  [4176,75194]
  [23131,2816954]
  [23658,2945542]
  [35286,6048314]
  [37241,6619599]
  [49855,10627043]
  [54496,12233934]

8. [0,0,1,-234270673,1016774489478] N = 75252011735942348487679505

E = ellinit([0,0,1,-234270673,1016774489478]);
generators:
  [-13249,1339757]
  [-2424,1253157]
  [-1524,1170582]
  [86,998312]
  [2041,739682]
  [2641,645357]
  [3639,460925]
  [4711,132962]
  [4816,15117]
  [12456,176837]
  [12472,187029]
  [16291,1234442]

9. [0,0,1,-239325247,1007007955806] N = 439218641837775372384213397

E = ellinit([0,0,1,-239325247,1007007955806]);
generators:
  [-9053,1559379]
  [-2399,1251935]
  [-1850,1201427]
  [1312,833829]
  [2836,592529]
  [3968,346181]
  [4614,31443]
  [4618,16943]
  [12982,296604]
  [14203,687731]
  [15059,904428]
  [23362,2857704]

10. [1,-1,0,-241075144,935628365884] N = 259277654321226224156197706

E = ellinit([1,-1,0,-241075144,935628365884]);
generators:
  [-17180,88338]
  [-16550,634478]
  [-15443,995360]
  [-11425,1488447]
  [-2314,1218154]
  [-1116,1097498]
  [-324,1006990]
  [2876,514390]
  [2940,500798]
  [4138,92280]
  [13970,535288]
  [51142,11036378]

Smallest Faltings height

# a-invariants conductor N log N naive h Faltings h log abs(disc)
1 [1,-1,0,-266451964,1712164041364] 27805670188643664300931114 58.587282000 69.860626000 3.72231917 59.28042938
2 [0,0,1,-332557057,2359264710636] 50719803766045246971171323 59.188359000 70.501888000 3.74272059 59.18835867
3 [0,0,1,-190719817,1221345067746] 200423269507421539440359243 60.562474000 69.185093000 3.76381548 60.56247371
4 [0,0,1,-291185797,1832954911656] 128721058559431142077373797 60.119690000 70.082019000 3.77072630 60.11968996
5 [0,1,1,-216826410,990382874670] 228611240971968172580506037 60.694065000 69.197426000 3.77357392 60.69406516
6 [1,-1,0,-326831689,2215060677769] 57453726914665426475770586 59.313022000 70.428471000 3.77739060 60.00616929
7 [1,-1,0,-79922404,945333881884] 176685196983131610721395454 60.436412000 68.672711000 3.77771563 61.12955901
8 [0,0,1,-220280389,937214745522] 304624442109032000008541749 60.981122000 69.244838000 3.79103618 60.98112191
9 [0,0,1,-10424167,1013182725216] 443392455047858519129412443 61.356498000 68.811381000 3.79198184 61.35649751
10 [0,0,1,-129671269,1124584406100] 406802088686884491992318651 61.270369000 69.020015000 3.79657669 61.27036903
generators for these ten

1. [1,-1,0,-266451964,1712164041364] N = 27805670188643664300931114

E = ellinit([1,-1,0,-266451964,1712164041364]);
generators:
  [-6310,1775782]
  [-2193,1513030]
  [-824,1390074]
  [-670,1375248]
  [-634,1371754]
  [951,1207675]
  [1178,1182592]
  [1752,1117478]
  [1784,1113778]
  [10261,236495]
  [10761,296119]
  [255159/25,28883167/125]

2. [0,0,1,-332557057,2359264710636] N = 50719803766045246971171323

E = ellinit([0,0,1,-332557057,2359264710636]);
generators:
  [-1751/4,12661137/8]
  [292,1504055]
  [4562,968030]
  [8759,344076]
  [9738,210393]
  [11752,272244]
  [11833,284556]
  [12246,351050]
  [12373,372468]
  [12815,449532]
  [15648,993468]
  [27696,3793875]

3. [0,0,1,-190719817,1221345067746] N = 200423269507421539440359243

E = ellinit([0,0,1,-190719817,1221345067746]);
generators:
  [-15980,434037]
  [-1782,1247216]
  [-1752,1245033]
  [-118,1115279]
  [5419,589038]
  [8623,466841]
  [9134,491283]
  [9278,500508]
  [10094,569811]
  [12145,834537]
  [25270,3540987]
  [47686,10028084]

4. [0,0,1,-291185797,1832954911656] N = 128721058559431142077373797

E = ellinit([0,0,1,-291185797,1832954911656]);
generators:
  [-13814,1794245]
  [-2181,1567691]
  [6529,458388]
  [8114,66893]
  [8161,11220]
  [11451,10164]
  [11562,108954]
  [11772,190995]
  [11822,206870]
  [12013,261843]
  [20928,2214743]
  [23057,2716020]

5. [0,1,1,-216826410,990382874670] N = 228611240971968172580506037

E = ellinit([0,1,1,-216826410,990382874670]);
generators:
  [2241,718146]
  [-16287,449117]
  [-1596,-1154286]
  [11400,15354]
  [12579,503460]
  [71777/16,21031257/64]
  [-7063,-1472934]
  [11412,-48142]
  [11846,290378]
  [4095,-413721]
  [4272,376938]
  [12099,-371845]

6. [1,-1,0,-326831689,2215060677769] N = 57453726914665426475770586

E = ellinit([1,-1,0,-326831689,2215060677769]);
generators:
  [8967,-77360]
  [2336,1208925]
  [16899,-1240387]
  [8824,-138957]
  [13556,-531671]
  [-18204,-1451005]
  [13349,473728]
  [15583,-959568]
  [106597/9,1833664/27]
  [7840,-370723]
  [8511,-227536]
  [5822,710925]

7. [1,-1,0,-79922404,945333881884] N = 176685196983131610721395454

E = ellinit([1,-1,0,-79922404,945333881884]);
generators:
  [-9628,911598]
  [-8582,1003842]
  [-8020,1038618]
  [-3478,1088580]
  [-2261,1056818]
  [-1990,1048132]
  [-1445,1029218]
  [-618,997550]
  [170,965188]
  [251,961794]
  [6414,831388]
  [7490,871962]

8. [0,0,1,-220280389,937214745522] N = 304624442109032000008541749

E = ellinit([0,0,1,-220280389,937214745522]);
generators:
  [-65943/4,2375953/8]
  [-7435,1471053]
  [-1281,1103309]
  [-611,1035170]
  [944,854465]
  [2979,554469]
  [4109,318525]
  [4663,106946]
  [12335,311213]
  [12950,506291]
  [16245,1282893]
  [65617/4,10546819/8]

9. [0,0,1,-10424167,1013182725216] N = 443392455047858519129412443

E = ellinit([0,0,1,-10424167,1013182725216]);
generators:
  [-10348,113915]
  [-9995,344777]
  [-8831,645398]
  [-7828,784284]
  [-4426,986213]
  [-4353,987965]
  [-3854,998054]
  [-2152,1012743]
  [-1746,1012946]
  [130,1005897]
  [6672,1113840]
  [10496,1435295]

10. [0,0,1,-129671269,1124584406100] N = 406802088686884491992318651

E = ellinit([0,0,1,-129671269,1124584406100]);
generators:
  [-10223,1175504]
  [-2438,1194249]
  [-2309,1188143]
  [3186,862433]
  [4656,788523]
  [7040,748740]
  [13007,1280039]
  [15865,1749440]
  [25708,3844664]
  [36728,6775404]
  [70955,18685680]
  [75088/9,21319145/27]

Smallest absolute discriminant

# a-invariants conductor N log N naive h Faltings h log abs(disc)
1 [0,0,1,-332557057,2359264710636] 50719803766045246971171323 59.188359000 70.501888000 3.74272059 59.18835867
2 [1,-1,0,-266451964,1712164041364] 27805670188643664300931114 58.587282000 69.860626000 3.72231917 59.28042938
3 [1,-1,0,-326831689,2215060677769] 57453726914665426475770586 59.313022000 70.428471000 3.77739060 60.00616929
4 [0,0,1,-291185797,1832954911656] 128721058559431142077373797 60.119690000 70.082019000 3.77072630 60.11968996
5 [1,-1,0,-490124959,4213132611529] 66233056438848840448971334 59.455221912 71.661542521 3.83069541 60.14836909
6 [0,0,1,-708531019,7237725024300] 134244761086677411732182149 60.161707000 72.749716000 3.87706134 60.16170694
7 [0,0,1,-190719817,1221345067746] 200423269507421539440359243 60.562474000 69.185093000 3.76381548 60.56247371
8 [1,-1,0,-348460744,2408671455724] 100901682201334257815641106 59.876189000 70.620711000 3.81124926 60.56933601
9 [0,1,1,-332553930,2225142738210] 214624425812282864316406997 60.630932000 70.480541000 3.80951426 60.63093188
10 [0,1,1,-1026395222,12636415148832] 217769389285474480258939109 60.645479000 73.861559000 3.94541618 60.64547889
generators for these ten

1. [0,0,1,-332557057,2359264710636] N = 50719803766045246971171323

E = ellinit([0,0,1,-332557057,2359264710636]);
generators:
  [-1751/4,12661137/8]
  [292,1504055]
  [4562,968030]
  [8759,344076]
  [9738,210393]
  [11752,272244]
  [11833,284556]
  [12246,351050]
  [12373,372468]
  [12815,449532]
  [15648,993468]
  [27696,3793875]

2. [1,-1,0,-266451964,1712164041364] N = 27805670188643664300931114

E = ellinit([1,-1,0,-266451964,1712164041364]);
generators:
  [-6310,1775782]
  [-2193,1513030]
  [-824,1390074]
  [-670,1375248]
  [-634,1371754]
  [951,1207675]
  [1178,1182592]
  [1752,1117478]
  [1784,1113778]
  [10261,236495]
  [10761,296119]
  [255159/25,28883167/125]

3. [1,-1,0,-326831689,2215060677769] N = 57453726914665426475770586

E = ellinit([1,-1,0,-326831689,2215060677769]);
generators:
  [8967,-77360]
  [2336,1208925]
  [16899,-1240387]
  [8824,-138957]
  [13556,-531671]
  [-18204,-1451005]
  [13349,473728]
  [15583,-959568]
  [106597/9,1833664/27]
  [7840,-370723]
  [8511,-227536]
  [5822,710925]

4. [0,0,1,-291185797,1832954911656] N = 128721058559431142077373797

E = ellinit([0,0,1,-291185797,1832954911656]);
generators:
  [-13814,1794245]
  [-2181,1567691]
  [6529,458388]
  [8114,66893]
  [8161,11220]
  [11451,10164]
  [11562,108954]
  [11772,190995]
  [11822,206870]
  [12013,261843]
  [20928,2214743]
  [23057,2716020]

5. [1,-1,0,-490124959,4213132611529] N = 66233056438848840448971334

E = ellinit([1,-1,0,-490124959,4213132611529]);
generators:
  [-3172,2396557]
  [-972,2165807]
  [3801,1548929]
  [4164,1496063]
  [13180,200013]
  [13570,239883]
  [13626,247079]
  [13831,275739]
  [13847,278110]
  [14188,332205]
  [15096,496685]
  [337676/25,29021259/125]

6. [0,0,1,-708531019,7237725024300] N = 134244761086677411732182149

E = ellinit([0,0,1,-708531019,7237725024300]);
generators:
  [-2945,3049394]
  [-2028,2943855]
  [-64,2698716]
  [8569/4,19149439/8]
  [2367,2360909]
  [4033,2108510]
  [64753/4,805237/8]
  [17358,411110]
  [17771,508578]
  [21405,1370705]
  [90313/4,13267015/8]
  [363704/25,11861983/125]

7. [0,0,1,-190719817,1221345067746] N = 200423269507421539440359243

E = ellinit([0,0,1,-190719817,1221345067746]);
generators:
  [-15980,434037]
  [-1782,1247216]
  [-1752,1245033]
  [-118,1115279]
  [5419,589038]
  [8623,466841]
  [9134,491283]
  [9278,500508]
  [10094,569811]
  [12145,834537]
  [25270,3540987]
  [47686,10028084]

8. [1,-1,0,-348460744,2408671455724] N = 100901682201334257815641106

E = ellinit([1,-1,0,-348460744,2408671455724]);
generators:
  [-9412,2208010]
  [-3587,1902435]
  [-1438,1705648]
  [-46,1557168]
  [1203,1410502]
  [3106,1163054]
  [4572,952210]
  [5180,859162]
  [12493,65027]
  [12911,242050]
  [14098,538800]
  [98641,30410660]

9. [0,1,1,-332553930,2225142738210] N = 214624425812282864316406997

E = ellinit([0,1,1,-332553930,2225142738210]);
generators:
  [-1896,1687854]
  [-1024,1601438]
  [1202,1351721]
  [5991,669236]
  [6884,511974]
  [8595,43160]
  [12374,70209]
  [12569,176289]
  [14603,695072]
  [18564,1565094]
  [18726,1601438]
  [46897,9474886]

10. [0,1,1,-1026395222,12636415148832] N = 217769389285474480258939109

E = ellinit([0,1,1,-1026395222,12636415148832]);
generators:
  [-25464,4718256]
  [-16505,5008106]
  [-1484,3762488]
  [876,3426071]
  [1161,3383243]
  [1950,3262263]
  [17886,22856]
  [20620,489768]
  [21296,660991]
  [23083,1115286]
  [41586,6470943]
  [89271919/4489,90240863879/300763]