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Rank 10 record curves

The ten smallest curves of rank 10 in this collection, in each of four categories. The same curve often heads more than one list. Every curve here has proven rank 10, is given by its global minimal model, and carries 10 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,-16312387,25970162646] 10189285026863130793 43.767868000 61.483574000 3.02693637 50.96155417
2 [1,-1,1,-44785887,113653147335] 20762602312568441306 44.479685000 64.465814000 3.25494134 53.49059842
3 [0,-1,0,-9581420,11213539236] 20882718586294952952 44.485454000 59.839612000 2.87786230 49.04980182
4 [1,-1,0,-10194109,12647638369] 22006161865320788846 44.537854000 60.044215000 2.86548777 48.59829718
5 [0,0,1,-21078967,35688990786] 22630148490190627609 44.565815000 62.204962000 3.11465192 52.24967868
6 [1,-1,0,-1536664,648294124] 25440555737235843986 44.682876000 54.348977000 2.51050521 45.37602344
7 [0,0,0,-27718327,55094082970] 43647618141851669768 45.222680000 63.026416000 3.14631419 52.29933420
8 [1,-1,0,-3401281,2458689193] 45699589163661074878 45.268621000 56.768242000 2.62088534 45.96176816
9 [0,0,1,-2917747,2367256296] 48891627275709988459 45.336138000 56.693140000 2.72808302 48.16935118
10 [1,-1,0,-2152789,1315971169] 54491266824353774014 45.444572000 55.518009000 2.58544250 46.13771930
generators for these ten

1. [0,0,1,-16312387,25970162646] N = 10189285026863130793

E = ellinit([0,0,1,-16312387,25970162646]);
generators:
  [3601/4,876883/8]
  [1959,39143]
  [1970,38472]
  [2003,36503]
  [2113,30596]
  [8705/4,223021/8]
  [2322,24744]
  [2564,31641]
  [3048,67578]
  [3653,122996]

2. [1,-1,1,-44785887,113653147335] N = 20762602312568441306

E = ellinit([1,-1,1,-44785887,113653147335]);
generators:
  [-3437,478134]
  [-1873,437930]
  [-1613,427062]
  [-393,362402]
  [3395,25270]
  [4591,66922]
  [4947,112182]
  [5265,151580]
  [5459,175670]
  [6131,260566]

3. [0,-1,0,-9581420,11213539236] N = 20882718586294952952

E = ellinit([0,-1,0,-9581420,11213539236]);
generators:
  [-3566,4652]
  [-3556,17502]
  [-3555,18294]
  [-3400,66894]
  [-3287,84752]
  [-1760,150406]
  [-1454,148556]
  [-1159,144084]
  [253,93838]
  [6230,439644]

4. [1,-1,0,-10194109,12647638369] N = 22006161865320788846

E = ellinit([1,-1,0,-10194109,12647638369]);
generators:
  [-2204,157325]
  [-580,135807]
  [-116,117653]
  [1334,37033]
  [1392,33263]
  [1566,22069]
  [1943,12180]
  [2784,75023]
  [3190,110577]
  [4060,193343]

5. [0,0,1,-21078967,35688990786] N = 22630148490190627609

E = ellinit([0,0,1,-21078967,35688990786]);
generators:
  [-15143/4,1979599/8]
  [-1172,242453]
  [17,187964]
  [2137,20060]
  [2190,5432]
  [3939,117368]
  [4076,132245]
  [4896,223265]
  [49996,11133365]
  [-17231/9,7093745/27]

6. [1,-1,0,-1536664,648294124] N = 25440555737235843986

E = ellinit([1,-1,0,-1536664,648294124]);
generators:
  [-1412,1890]
  [-5561/4,83995/8]
  [-1353,16463]
  [-1350,16832]
  [-1329,19154]
  [-1259,24840]
  [-938,36018]
  [-889,36655]
  [-865,36898]
  [-650,37368]

7. [0,0,0,-27718327,55094082970] N = 43647618141851669768

E = ellinit([0,0,0,-27718327,55094082970]);
generators:
  [-6001,72964]
  [-5981,83204]
  [-5903,114118]
  [-5779,149258]
  [-5339,225598]
  [-5298,230732]
  [-4817,277204]
  [-4127,314954]
  [-3411,331594]
  [24117,3662482]

8. [1,-1,0,-3401281,2458689193] N = 45699589163661074878

E = ellinit([1,-1,0,-3401281,2458689193]);
generators:
  [-1898,46493]
  [644,22809]
  [901,10728]
  [1069,6059]
  [1166,8193]
  [1219,10472]
  [1224,10709]
  [1352,17493]
  [1417,21263]
  [2133,68968]

9. [0,0,1,-2917747,2367256296] N = 48891627275709988459

E = ellinit([0,0,1,-2917747,2367256296]);
generators:
  [-2022,161]
  [-2005,12537]
  [-798,64710]
  [86,46010]
  [324,38156]
  [545,30642]
  [988,21188]
  [1446,34229]
  [1820,55547]
  [2347,91910]

10. [1,-1,0,-2152789,1315971169] N = 54491266824353774014

E = ellinit([1,-1,0,-2152789,1315971169]);
generators:
  [599,15228]
  [812,9733]
  [822,9653]
  [985,11772]
  [987,11833]
  [1140,17933]
  [1312,26705]
  [1431,33404]
  [1790,55633]
  [2360,95647]

Smallest naive height

# a-invariants conductor N log N naive h Faltings h log abs(disc)
1 [1,-1,0,-1536664,648294124] 25440555737235843986 44.682876000 54.348977000 2.51050521 45.37602344
2 [0,0,1,-1628809,524691642] 157630523181786606709 46.506786000 54.523682000 2.57847146 46.50678551
3 [0,0,1,-1344889,804658212] 124026809008982622011 46.267029000 54.535002000 2.56193761 46.26702942
4 [0,0,1,-1788817,843180666] 59202439687694448757 45.527494000 54.804798000 2.53237753 45.52749443
5 [1,0,0,-1792372,168435171] 356245105339268177309 47.322151000 54.810754000 2.63374770 47.32215067
6 [0,1,1,-1863480,494369610] 308300047454008759397 47.177605000 54.927472000 2.62780471 47.17760516
7 [0,0,1,-1900297,554782206] 306219504787781099797 47.170834000 54.986165000 2.62881967 47.17083385
8 [0,0,1,-1915627,625734516] 280749843930745706437 47.083996000 55.010270000 2.62430554 47.08399571
9 [0,1,1,-1856500,1072474760] 87950374485438204043 45.923304000 55.110769000 2.56137179 45.92330440
10 [1,0,0,-1605567,1172516536] 329157169581291693751 47.243067000 55.288210000 2.63665863 47.24306703
generators for these ten

1. [1,-1,0,-1536664,648294124] N = 25440555737235843986

E = ellinit([1,-1,0,-1536664,648294124]);
generators:
  [-1412,1890]
  [-5561/4,83995/8]
  [-1353,16463]
  [-1350,16832]
  [-1329,19154]
  [-1259,24840]
  [-938,36018]
  [-889,36655]
  [-865,36898]
  [-650,37368]

2. [0,0,1,-1628809,524691642] N = 157630523181786606709

E = ellinit([0,0,1,-1628809,524691642]);
generators:
  [-1171,28745]
  [-822,36168]
  [-681,36305]
  [-560,35513]
  [-427,33798]
  [-152,27726]
  [245,11846]
  [345,1953]
  [1089,6510]
  [1107,8841]

3. [0,0,1,-1344889,804658212] N = 124026809008982622011

E = ellinit([0,0,1,-1344889,804658212]);
generators:
  [-1387,1323]
  [-1383,4403]
  [-1370,8706]
  [-1136,29435]
  [-956,34880]
  [-757,37268]
  [-685,37476]
  [4128,256116]
  [8474,773249]
  [-12359/9,212500/27]

4. [0,0,1,-1788817,843180666] N = 59202439687694448757

E = ellinit([0,0,1,-1788817,843180666]);
generators:
  [-1529,1929]
  [-1395,24977]
  [-911,41433]
  [-316,37106]
  [96,25929]
  [1392,32409]
  [11401/4,1099809/8]
  [3148,162506]
  [7623,655881]
  [-12041/9,783298/27]

5. [1,0,0,-1792372,168435171] N = 356245105339268177309

E = ellinit([1,0,0,-1792372,168435171]);
generators:
  [-495,30816]
  [-958,32205]
  [-1246,22249]
  [-1379,4959]
  [62,7555]
  [-799,33424]
  [-194,-22461]
  [-981,31842]
  [-9,-13581]
  [3870,-228201]

6. [0,1,1,-1863480,494369610] N = 308300047454008759397

E = ellinit([0,1,1,-1863480,494369610]);
generators:
  [-14,22813]
  [1206,1586]
  [-1474,6386]
  [129,16004]
  [264,4569]
  [4572,-295914]
  [-1321,25545]
  [270,3314]
  [-310,-32286]
  [4828,-322583]

7. [0,0,1,-1900297,554782206] N = 306219504787781099797

E = ellinit([0,0,1,-1900297,554782206]);
generators:
  [-1067,36981]
  [-1506,983]
  [-61,25893]
  [-1043,-37446]
  [156,16190]
  [296,-4270]
  [-1408,-20955]
  [-1057,-37182]
  [-1347,25893]
  [-1186,-33769]

8. [0,0,1,-1915627,625734516] N = 280749843930745706437

E = ellinit([0,0,1,-1915627,625734516]);
generators:
  [207,-15430]
  [1258,-14379]
  [1522,35154]
  [-395,36342]
  [-247,-32922]
  [1364,23463]
  [323,6378]
  [268,11471]
  [326,5990]
  [1349,-22282]

9. [0,1,1,-1856500,1072474760] N = 87950374485438204043

E = ellinit([0,1,1,-1856500,1072474760]);
generators:
  [-1588,4311]
  [-1061,42999]
  [-1979/4,345955/8]
  [-463,42813]
  [26,32003]
  [1733/4,149589/8]
  [1744,56055]
  [22365/4,3254705/8]
  [8098,719132]
  [12047,1314243]

10. [1,0,0,-1605567,1172516536] N = 329157169581291693751

E = ellinit([1,0,0,-1605567,1172516536]);
generators:
  [569,20769]
  [-648,44380]
  [-1480,-16796]
  [-1469,19750]
  [1295,-36221]
  [-1531,7294]
  [1400,-41556]
  [3780,219734]
  [1617,52153]
  [-49,-35346]

Smallest Faltings height

# a-invariants conductor N log N naive h Faltings h log abs(disc)
1 [1,-1,0,-1536664,648294124] 25440555737235843986 44.682876000 54.348977000 2.51050521 45.37602344
2 [0,0,1,-1788817,843180666] 59202439687694448757 45.527494000 54.804798000 2.53237753 45.52749443
3 [0,1,1,-1856500,1072474760] 87950374485438204043 45.923304000 55.110769000 2.56137179 45.92330440
4 [0,0,1,-1344889,804658212] 124026809008982622011 46.267029000 54.535002000 2.56193761 46.26702942
5 [0,0,1,-1628809,524691642] 157630523181786606709 46.506786000 54.523682000 2.57847146 46.50678551
6 [1,-1,0,-2152789,1315971169] 54491266824353774014 45.444572000 55.518009000 2.58544250 46.13771930
7 [0,0,1,-2324077,1272484926] 103897289596322300437 46.089934000 55.590104000 2.58628621 46.08993449
8 [0,0,1,-2438527,1545098346] 103294665688000244363 46.084117000 55.839852000 2.59343344 46.08411741
9 [0,1,1,-2719370,1634287200] 131912458262686619717 46.328670000 56.061336000 2.61375846 46.32867018
10 [1,-1,0,-3401281,2458689193] 45699589163661074878 45.268621000 56.768242000 2.62088534 45.96176816
generators for these ten

1. [1,-1,0,-1536664,648294124] N = 25440555737235843986

E = ellinit([1,-1,0,-1536664,648294124]);
generators:
  [-1412,1890]
  [-5561/4,83995/8]
  [-1353,16463]
  [-1350,16832]
  [-1329,19154]
  [-1259,24840]
  [-938,36018]
  [-889,36655]
  [-865,36898]
  [-650,37368]

2. [0,0,1,-1788817,843180666] N = 59202439687694448757

E = ellinit([0,0,1,-1788817,843180666]);
generators:
  [-1529,1929]
  [-1395,24977]
  [-911,41433]
  [-316,37106]
  [96,25929]
  [1392,32409]
  [11401/4,1099809/8]
  [3148,162506]
  [7623,655881]
  [-12041/9,783298/27]

3. [0,1,1,-1856500,1072474760] N = 87950374485438204043

E = ellinit([0,1,1,-1856500,1072474760]);
generators:
  [-1588,4311]
  [-1061,42999]
  [-1979/4,345955/8]
  [-463,42813]
  [26,32003]
  [1733/4,149589/8]
  [1744,56055]
  [22365/4,3254705/8]
  [8098,719132]
  [12047,1314243]

4. [0,0,1,-1344889,804658212] N = 124026809008982622011

E = ellinit([0,0,1,-1344889,804658212]);
generators:
  [-1387,1323]
  [-1383,4403]
  [-1370,8706]
  [-1136,29435]
  [-956,34880]
  [-757,37268]
  [-685,37476]
  [4128,256116]
  [8474,773249]
  [-12359/9,212500/27]

5. [0,0,1,-1628809,524691642] N = 157630523181786606709

E = ellinit([0,0,1,-1628809,524691642]);
generators:
  [-1171,28745]
  [-822,36168]
  [-681,36305]
  [-560,35513]
  [-427,33798]
  [-152,27726]
  [245,11846]
  [345,1953]
  [1089,6510]
  [1107,8841]

6. [1,-1,0,-2152789,1315971169] N = 54491266824353774014

E = ellinit([1,-1,0,-2152789,1315971169]);
generators:
  [599,15228]
  [812,9733]
  [822,9653]
  [985,11772]
  [987,11833]
  [1140,17933]
  [1312,26705]
  [1431,33404]
  [1790,55633]
  [2360,95647]

7. [0,0,1,-2324077,1272484926] N = 103897289596322300437

E = ellinit([0,0,1,-2324077,1272484926]);
generators:
  [1061,-1015]
  [199,-28599]
  [1566,38384]
  [613,8841]
  [685,-1383]
  [477,16505]
  [591,-10266]
  [1156,-11431]
  [504,-15139]
  [673,3633]

8. [0,0,1,-2438527,1545098346] N = 103294665688000244363

E = ellinit([0,0,1,-2438527,1545098346]);
generators:
  [-1809,6039]
  [-1793,12375]
  [-1605,36392]
  [-1546,40248]
  [-1391,47388]
  [-1112,53681]
  [-4143/4,435219/8]
  [-830,54747]
  [-324,47970]
  [4472,282974]

9. [0,1,1,-2719370,1634287200] N = 131912458262686619717

E = ellinit([0,1,1,-2719370,1634287200]);
generators:
  [-1854,17511]
  [-1203,56268]
  [-884,57863]
  [-431,52215]
  [477,21116]
  [707,8098]
  [747,4506]
  [1125,299]
  [1410,24600]
  [1479,29150]

10. [1,-1,0,-3401281,2458689193] N = 45699589163661074878

E = ellinit([1,-1,0,-3401281,2458689193]);
generators:
  [-1898,46493]
  [644,22809]
  [901,10728]
  [1069,6059]
  [1166,8193]
  [1219,10472]
  [1224,10709]
  [1352,17493]
  [1417,21263]
  [2133,68968]

Smallest absolute discriminant

# a-invariants conductor N log N naive h Faltings h log abs(disc)
1 [1,-1,0,-1536664,648294124] 25440555737235843986 44.682876000 54.348977000 2.51050521 45.37602344
2 [0,0,1,-1788817,843180666] 59202439687694448757 45.527494000 54.804798000 2.53237753 45.52749443
3 [0,1,1,-1856500,1072474760] 87950374485438204043 45.923304000 55.110769000 2.56137179 45.92330440
4 [1,-1,0,-3401281,2458689193] 45699589163661074878 45.268621000 56.768242000 2.62088534 45.96176816
5 [0,0,1,-2438527,1545098346] 103294665688000244363 46.084117000 55.839852000 2.59343344 46.08411741
6 [0,0,1,-2324077,1272484926] 103897289596322300437 46.089934000 55.590104000 2.58628621 46.08993449
7 [1,-1,0,-2152789,1315971169] 54491266824353774014 45.444572000 55.518009000 2.58544250 46.13771930
8 [0,0,1,-4465867,3595567536] 115353212066580971557 46.194531000 57.549525000 2.66170515 46.19453050
9 [0,0,1,-1344889,804658212] 124026809008982622011 46.267029000 54.535002000 2.56193761 46.26702942
10 [0,1,1,-2719370,1634287200] 131912458262686619717 46.328670000 56.061336000 2.61375846 46.32867018
generators for these ten

1. [1,-1,0,-1536664,648294124] N = 25440555737235843986

E = ellinit([1,-1,0,-1536664,648294124]);
generators:
  [-1412,1890]
  [-5561/4,83995/8]
  [-1353,16463]
  [-1350,16832]
  [-1329,19154]
  [-1259,24840]
  [-938,36018]
  [-889,36655]
  [-865,36898]
  [-650,37368]

2. [0,0,1,-1788817,843180666] N = 59202439687694448757

E = ellinit([0,0,1,-1788817,843180666]);
generators:
  [-1529,1929]
  [-1395,24977]
  [-911,41433]
  [-316,37106]
  [96,25929]
  [1392,32409]
  [11401/4,1099809/8]
  [3148,162506]
  [7623,655881]
  [-12041/9,783298/27]

3. [0,1,1,-1856500,1072474760] N = 87950374485438204043

E = ellinit([0,1,1,-1856500,1072474760]);
generators:
  [-1588,4311]
  [-1061,42999]
  [-1979/4,345955/8]
  [-463,42813]
  [26,32003]
  [1733/4,149589/8]
  [1744,56055]
  [22365/4,3254705/8]
  [8098,719132]
  [12047,1314243]

4. [1,-1,0,-3401281,2458689193] N = 45699589163661074878

E = ellinit([1,-1,0,-3401281,2458689193]);
generators:
  [-1898,46493]
  [644,22809]
  [901,10728]
  [1069,6059]
  [1166,8193]
  [1219,10472]
  [1224,10709]
  [1352,17493]
  [1417,21263]
  [2133,68968]

5. [0,0,1,-2438527,1545098346] N = 103294665688000244363

E = ellinit([0,0,1,-2438527,1545098346]);
generators:
  [-1809,6039]
  [-1793,12375]
  [-1605,36392]
  [-1546,40248]
  [-1391,47388]
  [-1112,53681]
  [-4143/4,435219/8]
  [-830,54747]
  [-324,47970]
  [4472,282974]

6. [0,0,1,-2324077,1272484926] N = 103897289596322300437

E = ellinit([0,0,1,-2324077,1272484926]);
generators:
  [1061,-1015]
  [199,-28599]
  [1566,38384]
  [613,8841]
  [685,-1383]
  [477,16505]
  [591,-10266]
  [1156,-11431]
  [504,-15139]
  [673,3633]

7. [1,-1,0,-2152789,1315971169] N = 54491266824353774014

E = ellinit([1,-1,0,-2152789,1315971169]);
generators:
  [599,15228]
  [812,9733]
  [822,9653]
  [985,11772]
  [987,11833]
  [1140,17933]
  [1312,26705]
  [1431,33404]
  [1790,55633]
  [2360,95647]

8. [0,0,1,-4465867,3595567536] N = 115353212066580971557

E = ellinit([0,0,1,-4465867,3595567536]);
generators:
  [-222,67646]
  [557,35789]
  [765,25037]
  [923,16121]
  [1065,6882]
  [1118,378]
  [1392,8735]
  [1671,28265]
  [3226,150870]
  [3877,211085]

9. [0,0,1,-1344889,804658212] N = 124026809008982622011

E = ellinit([0,0,1,-1344889,804658212]);
generators:
  [-1387,1323]
  [-1383,4403]
  [-1370,8706]
  [-1136,29435]
  [-956,34880]
  [-757,37268]
  [-685,37476]
  [4128,256116]
  [8474,773249]
  [-12359/9,212500/27]

10. [0,1,1,-2719370,1634287200] N = 131912458262686619717

E = ellinit([0,1,1,-2719370,1634287200]);
generators:
  [-1854,17511]
  [-1203,56268]
  [-884,57863]
  [-431,52215]
  [477,21116]
  [707,8098]
  [747,4506]
  [1125,299]
  [1410,24600]
  [1479,29150]