Elliptic curves by rank › curves from searches
Rank 6 record curves
The ten smallest curves of rank 6 in this collection, in each of four categories. The same curve often heads more than one list. Every curve here has proven rank 6, is given by its global minimal model, and carries 6 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,-2582,48720] |
5187563742 |
22.369530000 | 35.183167000 | 0.81261788 | 24.16128948 |
| 2 | [0,0,1,-7077,235516] |
5258110041 |
22.383037000 | 38.262216000 | 1.09819090 | 27.87609893 |
| 3 | [1,-1,0,-2326,43456] |
5739520802 |
22.470642000 | 34.869561000 | 0.75658331 | 23.16378874 |
| 4 | [1,-1,0,-16249,799549] |
6601024978 |
22.610491000 | 40.700998000 | 1.19506561 | 27.82542653 |
| 5 | [1,-1,1,-63147,6081915] |
6663562874 |
22.619920000 | 44.773250000 | 1.57975517 | 33.01712785 |
| 6 | [1,0,0,-9227,340354] |
6822208199 |
22.643449000 | 39.003278000 | 0.93391725 | 22.64344904 |
| 7 | [1,-1,0,-1486,21688] |
6895251302 |
22.654099000 | 33.525511000 | 0.71381592 | 23.34724598 |
| 8 | [0,0,0,-9907,306370] |
7208218544 |
22.698488000 | 39.216594000 | 1.27485233 | 30.70751836 |
| 9 | [0,0,1,-277,4566] |
7647224363 |
22.757609000 | 30.376041000 | 0.58283256 | 22.75760859 |
| 10 | [0,0,0,-312529,67246609] |
7678092404 |
22.761636968 | 49.570960664 | 1.79061659 | 32.40868268 |
generators for these ten
1. [1,1,0,-2582,48720] N = 5187563742
E = ellinit([1,1,0,-2582,48720]);
generators:
[-58,116]
[-40,320]
[-22,320]
[-1,227]
[152,1712]
[206,2780]
2. [0,0,1,-7077,235516] N = 5258110041
E = ellinit([0,0,1,-7077,235516]);
generators:
[-383/4,1499/8]
[-89,400]
[-44,679]
[-38,670]
[151,1615]
[262,4045]
3. [1,-1,0,-2326,43456] N = 5739520802
E = ellinit([1,-1,0,-2326,43456]);
generators:
[-46,250]
[-34,304]
[-26,306]
[-23,302]
[-3,226]
[64,360]
4. [1,-1,0,-16249,799549] N = 6601024978
E = ellinit([1,-1,0,-16249,799549]);
generators:
[-14,1019]
[55,237]
[243/4,1229/8]
[78,7]
[82,75]
[101,375]
5. [1,-1,1,-63147,6081915] N = 6663562874
E = ellinit([1,-1,1,-63147,6081915]);
generators:
[-285,1074]
[-277,1642]
[-213,3242]
[-149,3562]
[-21,2730]
[87,1070]
6. [1,0,0,-9227,340354] N = 6822208199
E = ellinit([1,0,0,-9227,340354]);
generators:
[-101,545]
[-50,848]
[-2,600]
[39,181]
[89,431]
[163,1700]
7. [1,-1,0,-1486,21688] N = 6895251302
E = ellinit([1,-1,0,-1486,21688]);
generators:
[-39,158]
[-38,168]
[-36,184]
[-17,213]
[-14,206]
[78,574]
8. [0,0,0,-9907,306370] N = 7208218544
E = ellinit([0,0,0,-9907,306370]);
generators:
[-107,376]
[-13,658]
[34,94]
[81,188]
[269,4136]
[363,6674]
9. [0,0,1,-277,4566] N = 7647224363
E = ellinit([0,0,1,-277,4566]);
generators:
[-21,33]
[1,65]
[5,57]
[19,78]
[21,89]
[22,95]
10. [0,0,0,-312529,67246609] N = 7678092404
E = ellinit([0,0,0,-312529,67246609]);
generators:
[-629,3869]
[-264,11461]
[113,5777]
[977,26353]
[22585,3393113]
[2077/9,73511/27]
Smallest naive height
| # | a-invariants | conductor N | log N | naive h | Faltings h | log abs(disc) |
|---|---|---|---|---|---|---|
| 1 | [0,0,1,-277,4566] |
7647224363 |
22.757609000 | 30.376041000 | 0.58283256 | 22.75760859 |
| 2 | [0,0,1,-379,5172] |
8072781371 |
22.811764000 | 30.625272000 | 0.59369880 | 22.81176392 |
| 3 | [0,1,1,-390,5460] |
9694585723 |
22.994833000 | 30.780731000 | 0.60809327 | 22.99483339 |
| 4 | [1,-1,1,-372,5700] |
10296725093 |
23.055092000 | 30.786810000 | 0.61139745 | 23.05509173 |
| 5 | [1,-1,0,-709,-1391] |
10882623866 |
23.110433000 | 31.305963000 | 0.67418071 | 23.80358039 |
| 6 | [1,-1,0,-784,4684] |
11080913666 |
23.128490000 | 31.607547000 | 0.68234104 | 23.82163716 |
| 7 | [0,0,1,-619,8670] |
17295531371 |
23.573714000 | 31.658451000 | 0.66539173 | 23.57371400 |
| 8 | [0,1,1,-270,8850] |
33265115083 |
24.227775000 | 31.719800000 | 0.70036127 | 24.22777509 |
| 9 | [0,1,1,-842,3192] |
33039404549 |
24.220967000 | 31.822130000 | 0.71123899 | 24.22096676 |
| 10 | [1,1,1,-345,9666] |
12981635841 |
23.286802000 | 31.905448000 | 0.71475874 | 24.38541386 |
generators for these ten
1. [0,0,1,-277,4566] N = 7647224363
E = ellinit([0,0,1,-277,4566]);
generators:
[-21,33]
[1,65]
[5,57]
[19,78]
[21,89]
[22,95]
2. [0,0,1,-379,5172] N = 8072781371
E = ellinit([0,0,1,-379,5172]);
generators:
[-22,53]
[-11,89]
[-1,74]
[2,66]
[15,53]
[19,69]
3. [0,1,1,-390,5460] N = 9694585723
E = ellinit([0,1,1,-390,5460]);
generators:
[-18,83]
[-14,91]
[2,68]
[5,60]
[12,51]
[17,63]
4. [1,-1,1,-372,5700] N = 10296725093
E = ellinit([1,-1,1,-372,5700]);
generators:
[-7,92]
[6,57]
[8,52]
[9,50]
[16,52]
[26,101]
5. [1,-1,0,-709,-1391] N = 10882623866
E = ellinit([1,-1,0,-709,-1391]);
generators:
[-25,28]
[-24,49]
[-10,73]
[-8,65]
[-7,60]
[-2,5]
6. [1,-1,0,-784,4684] N = 11080913666
E = ellinit([1,-1,0,-784,4684]);
generators:
[-30,38]
[-28,78]
[-25,103]
[-21,118]
[-14,120]
[173,2155]
7. [0,0,1,-619,8670] N = 17295531371
E = ellinit([0,0,1,-619,8670]);
generators:
[-17,119]
[-13,120]
[-2,99]
[5,75]
[12,54]
[22,75]
8. [0,1,1,-270,8850] N = 33265115083
E = ellinit([0,1,1,-270,8850]);
generators:
[-25,24]
[-23,58]
[-21,75]
[6,86]
[12,86]
[24,129]
9. [0,1,1,-842,3192] N = 33039404549
E = ellinit([0,1,1,-842,3192]);
generators:
[0,56]
[1,48]
[3,26]
[27,29]
[-10,103]
[-30,-49]
10. [1,1,1,-345,9666] N = 12981635841
E = ellinit([1,1,1,-345,9666]);
generators:
[-26,56]
[-23,89]
[-8,113]
[4,89]
[28,137]
[37,203]
Smallest Faltings height
| # | a-invariants | conductor N | log N | naive h | Faltings h | log abs(disc) |
|---|---|---|---|---|---|---|
| 1 | [0,0,1,-277,4566] |
7647224363 |
22.757609000 | 30.376041000 | 0.58283256 | 22.75760859 |
| 2 | [0,0,1,-379,5172] |
8072781371 |
22.811764000 | 30.625272000 | 0.59369880 | 22.81176392 |
| 3 | [0,1,1,-390,5460] |
9694585723 |
22.994833000 | 30.780731000 | 0.60809327 | 22.99483339 |
| 4 | [1,-1,1,-372,5700] |
10296725093 |
23.055092000 | 30.786810000 | 0.61139745 | 23.05509173 |
| 5 | [0,0,1,-889,9150] |
8796007189 |
22.897564000 | 31.983895000 | 0.64069663 | 22.89756373 |
| 6 | [0,0,1,-619,8670] |
17295531371 |
23.573714000 | 31.658451000 | 0.66539173 | 23.57371400 |
| 7 | [1,-1,0,-709,-1391] |
10882623866 |
23.110433000 | 31.305963000 | 0.67418071 | 23.80358039 |
| 8 | [1,-1,0,-784,4684] |
11080913666 |
23.128490000 | 31.607547000 | 0.68234104 | 23.82163716 |
| 9 | [1,-1,0,-844,11764] |
9578159194 |
22.982751000 | 32.232541000 | 0.68794773 | 23.67589844 |
| 10 | [0,1,1,-1270,16100] |
13250262917 |
23.307283000 | 33.054707000 | 0.69579854 | 23.30728323 |
generators for these ten
1. [0,0,1,-277,4566] N = 7647224363
E = ellinit([0,0,1,-277,4566]);
generators:
[-21,33]
[1,65]
[5,57]
[19,78]
[21,89]
[22,95]
2. [0,0,1,-379,5172] N = 8072781371
E = ellinit([0,0,1,-379,5172]);
generators:
[-22,53]
[-11,89]
[-1,74]
[2,66]
[15,53]
[19,69]
3. [0,1,1,-390,5460] N = 9694585723
E = ellinit([0,1,1,-390,5460]);
generators:
[-18,83]
[-14,91]
[2,68]
[5,60]
[12,51]
[17,63]
4. [1,-1,1,-372,5700] N = 10296725093
E = ellinit([1,-1,1,-372,5700]);
generators:
[-7,92]
[6,57]
[8,52]
[9,50]
[16,52]
[26,101]
5. [0,0,1,-889,9150] N = 8796007189
E = ellinit([0,0,1,-889,9150]);
generators:
[-34,8]
[-32,69]
[-25,125]
[-19,138]
[-12,134]
[61,426]
6. [0,0,1,-619,8670] N = 17295531371
E = ellinit([0,0,1,-619,8670]);
generators:
[-17,119]
[-13,120]
[-2,99]
[5,75]
[12,54]
[22,75]
7. [1,-1,0,-709,-1391] N = 10882623866
E = ellinit([1,-1,0,-709,-1391]);
generators:
[-25,28]
[-24,49]
[-10,73]
[-8,65]
[-7,60]
[-2,5]
8. [1,-1,0,-784,4684] N = 11080913666
E = ellinit([1,-1,0,-784,4684]);
generators:
[-30,38]
[-28,78]
[-25,103]
[-21,118]
[-14,120]
[173,2155]
9. [1,-1,0,-844,11764] N = 9578159194
E = ellinit([1,-1,0,-844,11764]);
generators:
[-22,150]
[7,75]
[8,70]
[22,48]
[24,58]
[26,70]
10. [0,1,1,-1270,16100] N = 13250262917
E = ellinit([0,1,1,-1270,16100]);
generators:
[14,35]
[15,25]
[24,4]
[25,24]
[26,36]
[31,86]
Smallest absolute discriminant
| # | a-invariants | conductor N | log N | naive h | Faltings h | log abs(disc) |
|---|---|---|---|---|---|---|
| 1 | [1,0,0,-9227,340354] |
6822208199 |
22.643449000 | 39.003278000 | 0.93391725 | 22.64344904 |
| 2 | [0,0,1,-277,4566] |
7647224363 |
22.757609000 | 30.376041000 | 0.58283256 | 22.75760859 |
| 3 | [0,0,1,-379,5172] |
8072781371 |
22.811764000 | 30.625272000 | 0.59369880 | 22.81176392 |
| 4 | [0,0,1,-889,9150] |
8796007189 |
22.897564000 | 31.983895000 | 0.64069663 | 22.89756373 |
| 5 | [0,1,1,-390,5460] |
9694585723 |
22.994833000 | 30.780731000 | 0.60809327 | 22.99483339 |
| 6 | [1,-1,1,-372,5700] |
10296725093 |
23.055092000 | 30.786810000 | 0.61139745 | 23.05509173 |
| 7 | [1,-1,0,-2326,43456] |
5739520802 |
22.470642000 | 34.869561000 | 0.75658331 | 23.16378874 |
| 8 | [0,0,1,-4657,122436] |
11994046523 |
23.207676000 | 36.953837000 | 0.85011640 | 23.20767624 |
| 9 | [0,1,1,-1270,16100] |
13250262917 |
23.307283000 | 33.054707000 | 0.69579854 | 23.30728323 |
| 10 | [0,-1,1,-1850,31752] |
13346333243 |
23.314508000 | 34.215354000 | 0.73719514 | 23.31450752 |
generators for these ten
1. [1,0,0,-9227,340354] N = 6822208199
E = ellinit([1,0,0,-9227,340354]);
generators:
[-101,545]
[-50,848]
[-2,600]
[39,181]
[89,431]
[163,1700]
2. [0,0,1,-277,4566] N = 7647224363
E = ellinit([0,0,1,-277,4566]);
generators:
[-21,33]
[1,65]
[5,57]
[19,78]
[21,89]
[22,95]
3. [0,0,1,-379,5172] N = 8072781371
E = ellinit([0,0,1,-379,5172]);
generators:
[-22,53]
[-11,89]
[-1,74]
[2,66]
[15,53]
[19,69]
4. [0,0,1,-889,9150] N = 8796007189
E = ellinit([0,0,1,-889,9150]);
generators:
[-34,8]
[-32,69]
[-25,125]
[-19,138]
[-12,134]
[61,426]
5. [0,1,1,-390,5460] N = 9694585723
E = ellinit([0,1,1,-390,5460]);
generators:
[-18,83]
[-14,91]
[2,68]
[5,60]
[12,51]
[17,63]
6. [1,-1,1,-372,5700] N = 10296725093
E = ellinit([1,-1,1,-372,5700]);
generators:
[-7,92]
[6,57]
[8,52]
[9,50]
[16,52]
[26,101]
7. [1,-1,0,-2326,43456] N = 5739520802
E = ellinit([1,-1,0,-2326,43456]);
generators:
[-46,250]
[-34,304]
[-26,306]
[-23,302]
[-3,226]
[64,360]
8. [0,0,1,-4657,122436] N = 11994046523
E = ellinit([0,0,1,-4657,122436]);
generators:
[-78,105]
[-77,156]
[-71,308]
[-69,339]
[-38,494]
[-9,404]
9. [0,1,1,-1270,16100] N = 13250262917
E = ellinit([0,1,1,-1270,16100]);
generators:
[14,35]
[15,25]
[24,4]
[25,24]
[26,36]
[31,86]
10. [0,-1,1,-1850,31752] N = 13346333243
E = ellinit([0,-1,1,-1850,31752]);
generators:
[20,48]
[21,41]
[24,24]
[26,23]
[30,48]
[33,74]