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Rank 7 record curves

The ten smallest curves of rank 7 in this collection, in each of four categories. The same curve often heads more than one list. Every curve here has proven rank 7, is given by its global minimal model, and carries 7 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 [0,0,0,-10012,346900] 382623908456 26.670318000 39.248222000 1.24468638 30.13605428
2 [1,0,1,-14733,694232] 536670340706 27.008650000 40.427803000 1.23540285 29.08809139
3 [0,0,1,-36673,2704878] 814434447535 27.425760000 43.144280000 1.35297214 29.03519769
4 [1,-1,0,-92656,10865908] 858426129202 27.478366000 45.923556000 1.60702880 32.43419352
5 [1,-1,0,-18664,958204] 896913586322 27.522225000 41.116689000 1.33882433 30.78032190
6 [1,-1,0,-14569,719329] 898281620938 27.523749000 40.485141000 1.31647317 30.78184600
7 [1,0,1,-5983,164022] 1005276094726 27.636283000 37.703395000 1.10223104 28.32943052
8 [0,0,0,-101647,12379090] 1022298908216 27.653075000 46.201387000 1.70333820 34.55279815
9 [1,0,1,-14505,667472] 1066284226102 27.695201000 40.360250000 1.20295095 28.38834822
10 [0,0,0,-12979,405826] 1074680679376 27.703045000 40.026867000 1.36185294 31.86192777
generators for these ten

1. [0,0,0,-10012,346900] N = 382623908456

E = ellinit([0,0,0,-10012,346900]);
generators:
  [-114,82]
  [-111,301]
  [-106,466]
  [-102,554]
  [-98,622]
  [-62,854]
  [-28,778]

2. [1,0,1,-14733,694232] N = 536670340706

E = ellinit([1,0,1,-14733,694232]);
generators:
  [-137,450]
  [-121,902]
  [-109,1058]
  [-77,1210]
  [-65,1206]
  [115,666]
  [735,19306]

3. [0,0,1,-36673,2704878] N = 814434447535

E = ellinit([0,0,1,-36673,2704878]);
generators:
  [-219,482]
  [-214,867]
  [-204,1302]
  [-44,2057]
  [11,1517]
  [171,1197]
  [246,2927]

4. [1,-1,0,-92656,10865908] N = 858426129202

E = ellinit([1,-1,0,-92656,10865908]);
generators:
  [-351,404]
  [-349,951]
  [-344,1566]
  [-329,2543]
  [-296,3654]
  [-147,4688]
  [-28424/81,329122/729]

5. [1,-1,0,-18664,958204] N = 896913586322

E = ellinit([1,-1,0,-18664,958204]);
generators:
  [-154,480]
  [28,662]
  [54,298]
  [67,38]
  [106,350]
  [132,818]
  [158,1312]

6. [1,-1,0,-14569,719329] N = 898281620938

E = ellinit([1,-1,0,-14569,719329]);
generators:
  [-37,1117]
  [-24,1039]
  [67,168]
  [80,207]
  [106,545]
  [119,753]
  [288,4367]

7. [1,0,1,-5983,164022] N = 1005276094726

E = ellinit([1,0,1,-5983,164022]);
generators:
  [57,66]
  [-9,470]
  [34,-3]
  [55,18]
  [29,108]
  [56,-102]
  [73,306]

8. [0,0,0,-101647,12379090] N = 1022298908216

E = ellinit([0,0,0,-101647,12379090]);
generators:
  [-267,4526]
  [-143,4898]
  [-19,3782]
  [167,248]
  [229,1054]
  [291,2728]
  [353,4526]

9. [1,0,1,-14505,667472] N = 1066284226102

E = ellinit([1,0,1,-14505,667472]);
generators:
  [-139,122]
  [-131,634]
  [-128,720]
  [-124,812]
  [-123,832]
  [-437/4,8235/8]
  [-95,1138]

10. [0,0,0,-12979,405826] N = 1074680679376

E = ellinit([0,0,0,-12979,405826]);
generators:
  [-126,202]
  [-125,274]
  [-123,376]
  [-111,692]
  [-95,884]
  [-77,974]
  [-21,818]

Smallest naive height

# a-invariants conductor N log N naive h Faltings h log abs(disc)
1 [0,0,1,-1387,68046] 1829517077483 28.235073000 35.779031000 1.03654034 28.23507316
2 [0,1,1,-3480,-2610] 2697095553797 28.623197000 36.078253000 1.07474106 28.62319659
3 [0,0,1,-3577,21666] 2726325306037 28.633976000 36.160442000 1.07726439 28.63397578
4 [0,0,1,-3829,45480] 2699253317989 28.623996000 36.364680000 1.08145070 28.62399630
5 [1,-1,1,1149,96570] 4149923512103 29.054111000 36.485140000 1.10135933 29.05411102
6 [0,1,1,-2324,134448] 7095565842763 29.590491000 37.152506000 1.15019077 29.59049118
7 [1,-1,0,-3499,137089] 2636686970374 28.600544000 37.167114000 1.13570422 29.29369149
8 [1,0,0,-4459,141870] 3066393226693 28.751523000 37.253710000 1.10928363 28.75152314
9 [1,0,0,-5154,73693] 6388836643367 29.485573000 37.256200000 1.15396734 29.48557331
10 [1,-1,0,-3946,143896] 2444942116858 28.525043000 37.263079000 1.13457794 29.21818975
generators for these ten

1. [0,0,1,-1387,68046] N = 1829517077483

E = ellinit([0,0,1,-1387,68046]);
generators:
  [-49,135]
  [-47,171]
  [-44,209]
  [-16,293]
  [13,228]
  [21,219]
  [43,296]

2. [0,1,1,-3480,-2610] N = 2697095553797

E = ellinit([0,1,1,-3480,-2610]);
generators:
  [-57,117]
  [-55,159]
  [-46,249]
  [-13,201]
  [59,30]
  [60,90]
  [76,421]

3. [0,0,1,-3577,21666] N = 2726325306037

E = ellinit([0,0,1,-3577,21666]);
generators:
  [-62,71]
  [-38,320]
  [-9,230]
  [1,134]
  [2,120]
  [58,96]
  [76,434]

4. [0,0,1,-3829,45480] N = 2699253317989

E = ellinit([0,0,1,-3829,45480]);
generators:
  [-12,299]
  [-64,168]
  [10,90]
  [67,299]
  [12,35]
  [55,35]
  [-15,-316]

5. [1,-1,1,1149,96570] N = 4149923512103

E = ellinit([1,-1,1,1149,96570]);
generators:
  [-35,129]
  [-22,256]
  [-20,265]
  [-19,269]
  [14,332]
  [18,341]
  [44,458]

6. [0,1,1,-2324,134448] N = 7095565842763

E = ellinit([0,1,1,-2324,134448]);
generators:
  [6,347]
  [13,326]
  [77,646]
  [28,303]
  [48,-369]
  [54,411]
  [39,323]

7. [1,-1,0,-3499,137089] N = 2636686970374

E = ellinit([1,-1,0,-3499,137089]);
generators:
  [-14,435]
  [1,365]
  [32,223]
  [39,223]
  [58,327]
  [64,383]
  [100,833]

8. [1,0,0,-4459,141870] N = 3066393226693

E = ellinit([1,0,0,-4459,141870]);
generators:
  [26,196]
  [13,287]
  [57,243]
  [47,-215]
  [-14,456]
  [41,-189]
  [55,225]

9. [1,0,0,-5154,73693] N = 6388836643367

E = ellinit([1,0,0,-5154,73693]);
generators:
  [11,130]
  [-23,436]
  [-9,350]
  [13,88]
  [63,-26]
  [101,715]
  [68,163]

10. [1,-1,0,-3946,143896] N = 2444942116858

E = ellinit([1,-1,0,-3946,143896]);
generators:
  [-76,60]
  [-70,306]
  [-66,370]
  [-55,466]
  [-14,450]
  [230,3256]
  [340,5996]

Smallest Faltings height

# a-invariants conductor N log N naive h Faltings h log abs(disc)
1 [0,0,1,-1387,68046] 1829517077483 28.235073000 35.779031000 1.03654034 28.23507316
2 [0,1,1,-3480,-2610] 2697095553797 28.623197000 36.078253000 1.07474106 28.62319659
3 [0,0,1,-3577,21666] 2726325306037 28.633976000 36.160442000 1.07726439 28.63397578
4 [0,0,1,-3829,45480] 2699253317989 28.623996000 36.364680000 1.08145070 28.62399630
5 [0,0,1,-5707,151416] 1991659717477 28.319989000 37.561949000 1.09732302 28.31998944
6 [1,-1,1,1149,96570] 4149923512103 29.054111000 36.485140000 1.10135933 29.05411102
7 [1,0,1,-5983,164022] 1005276094726 27.636283000 37.703395000 1.10223104 28.32943052
8 [1,0,0,-4459,141870] 3066393226693 28.751523000 37.253710000 1.10928363 28.75152314
9 [1,-1,0,-4564,151684] 1852679237314 28.247654000 37.367153000 1.12279177 28.94080112
10 [1,-1,0,-4594,152824] 1866217095994 28.254935000 37.382142000 1.12362831 28.94808174
generators for these ten

1. [0,0,1,-1387,68046] N = 1829517077483

E = ellinit([0,0,1,-1387,68046]);
generators:
  [-49,135]
  [-47,171]
  [-44,209]
  [-16,293]
  [13,228]
  [21,219]
  [43,296]

2. [0,1,1,-3480,-2610] N = 2697095553797

E = ellinit([0,1,1,-3480,-2610]);
generators:
  [-57,117]
  [-55,159]
  [-46,249]
  [-13,201]
  [59,30]
  [60,90]
  [76,421]

3. [0,0,1,-3577,21666] N = 2726325306037

E = ellinit([0,0,1,-3577,21666]);
generators:
  [-62,71]
  [-38,320]
  [-9,230]
  [1,134]
  [2,120]
  [58,96]
  [76,434]

4. [0,0,1,-3829,45480] N = 2699253317989

E = ellinit([0,0,1,-3829,45480]);
generators:
  [-12,299]
  [-64,168]
  [10,90]
  [67,299]
  [12,35]
  [55,35]
  [-15,-316]

5. [0,0,1,-5707,151416] N = 1991659717477

E = ellinit([0,0,1,-5707,151416]);
generators:
  [-79,330]
  [-77,366]
  [-69,465]
  [-64,504]
  [-56,543]
  [-3,410]
  [105,842]

6. [1,-1,1,1149,96570] N = 4149923512103

E = ellinit([1,-1,1,1149,96570]);
generators:
  [-35,129]
  [-22,256]
  [-20,265]
  [-19,269]
  [14,332]
  [18,341]
  [44,458]

7. [1,0,1,-5983,164022] N = 1005276094726

E = ellinit([1,0,1,-5983,164022]);
generators:
  [57,66]
  [-9,470]
  [34,-3]
  [55,18]
  [29,108]
  [56,-102]
  [73,306]

8. [1,0,0,-4459,141870] N = 3066393226693

E = ellinit([1,0,0,-4459,141870]);
generators:
  [26,196]
  [13,287]
  [57,243]
  [47,-215]
  [-14,456]
  [41,-189]
  [55,225]

9. [1,-1,0,-4564,151684] N = 1852679237314

E = ellinit([1,-1,0,-4564,151684]);
generators:
  [-80,42]
  [-79,160]
  [-74,320]
  [-63,463]
  [-31,528]
  [-10,448]
  [277,4343]

10. [1,-1,0,-4594,152824] N = 1866217095994

E = ellinit([1,-1,0,-4594,152824]);
generators:
  [-78,218]
  [-57,505]
  [2,378]
  [20,252]
  [38,160]
  [54,218]
  [70,378]

Smallest absolute discriminant

# a-invariants conductor N log N naive h Faltings h log abs(disc)
1 [0,0,1,-1387,68046] 1829517077483 28.235073000 35.779031000 1.03654034 28.23507316
2 [0,0,1,-5707,151416] 1991659717477 28.319989000 37.561949000 1.09732302 28.31998944
3 [1,0,1,-5983,164022] 1005276094726 27.636283000 37.703395000 1.10223104 28.32943052
4 [1,0,1,-14505,667472] 1066284226102 27.695201000 40.360250000 1.20295095 28.38834822
5 [0,0,1,-15577,744876] 2206378706437 28.422374000 40.574255000 1.21335178 28.42237369
6 [0,1,1,-18710,976170] 2269810368437 28.450717000 41.124097000 1.23822148 28.45071741
7 [0,0,1,-14959,700260] 2396208409669 28.504909000 40.452808000 1.21166352 28.50490877
8 [0,0,1,-18967,1008216] 2435328550043 28.521103000 41.170532000 1.24298802 28.52110279
9 [0,1,1,-3480,-2610] 2697095553797 28.623197000 36.078253000 1.07474106 28.62319659
10 [0,0,1,-3829,45480] 2699253317989 28.623996000 36.364680000 1.08145070 28.62399630
generators for these ten

1. [0,0,1,-1387,68046] N = 1829517077483

E = ellinit([0,0,1,-1387,68046]);
generators:
  [-49,135]
  [-47,171]
  [-44,209]
  [-16,293]
  [13,228]
  [21,219]
  [43,296]

2. [0,0,1,-5707,151416] N = 1991659717477

E = ellinit([0,0,1,-5707,151416]);
generators:
  [-79,330]
  [-77,366]
  [-69,465]
  [-64,504]
  [-56,543]
  [-3,410]
  [105,842]

3. [1,0,1,-5983,164022] N = 1005276094726

E = ellinit([1,0,1,-5983,164022]);
generators:
  [57,66]
  [-9,470]
  [34,-3]
  [55,18]
  [29,108]
  [56,-102]
  [73,306]

4. [1,0,1,-14505,667472] N = 1066284226102

E = ellinit([1,0,1,-14505,667472]);
generators:
  [-139,122]
  [-131,634]
  [-128,720]
  [-124,812]
  [-123,832]
  [-437/4,8235/8]
  [-95,1138]

5. [0,0,1,-15577,744876] N = 2206378706437

E = ellinit([0,0,1,-15577,744876]);
generators:
  [46,354]
  [63,116]
  [66,65]
  [67,44]
  [77,44]
  [91,284]
  [94,333]

6. [0,1,1,-18710,976170] N = 2269810368437

E = ellinit([0,1,1,-18710,976170]);
generators:
  [-135,1029]
  [-134,1046]
  [-87,1397]
  [1,978]
  [170,1654]
  [405,7745]
  [502,10878]

7. [0,0,1,-14959,700260] N = 2396208409669

E = ellinit([0,0,1,-14959,700260]);
generators:
  [-60,1175]
  [43,369]
  [66,21]
  [78,89]
  [81,141]
  [83,174]
  [95,369]

8. [0,0,1,-18967,1008216] N = 2435328550043

E = ellinit([0,0,1,-18967,1008216]);
generators:
  [-149,725]
  [-108,1340]
  [-77,1418]
  [-231/4,11055/8]
  [-16,1143]
  [105,417]
  [159,1418]

9. [0,1,1,-3480,-2610] N = 2697095553797

E = ellinit([0,1,1,-3480,-2610]);
generators:
  [-57,117]
  [-55,159]
  [-46,249]
  [-13,201]
  [59,30]
  [60,90]
  [76,421]

10. [0,0,1,-3829,45480] N = 2699253317989

E = ellinit([0,0,1,-3829,45480]);
generators:
  [-12,299]
  [-64,168]
  [10,90]
  [67,299]
  [12,35]
  [55,35]
  [-15,-316]