Elliptic curves by rank › curves from searches

Rank 9 record curves

The ten smallest curves of rank 9 in this collection, in each of four categories. The same curve often heads more than one list. Every curve here has proven rank 9, is given by its global minimal model, and carries 9 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,-135004,97151644]`	`32107342006814614`	38.007861000	50.306018000	2.24913846	42.81188217
2	`[1,-1,0,-613069,98885089]`	`43537345103385386`	38.312395000	51.592302000	2.34743788	43.80133320
3	`[0,0,0,-1996617,1025754145]`	`56448695244116556`	38.572109000	55.134497000	2.53902043	45.45146438
4	`[0,0,1,-3835819,2889890730]`	`62986816173592807`	38.681702000	57.093284000	2.51194873	42.88639445
5	`[1,0,1,-1493028,701820182]`	`72070075910145406`	38.816415000	54.262553000	2.25720433	39.50956250
6	`[1,0,1,-1076185,496031340]`	`77211251506212554`	38.885322000	53.567806000	2.45078277	44.72596324
7	`[0,0,0,-3368092,2377730260]`	`91949749274556776`	39.060019000	56.703174000	2.48047718	42.52575452
8	`[0,0,1,-167419,30261330]`	`95276302704064331`	39.095558000	47.973908000	1.98270681	39.09555751
9	`[0,0,0,-2905252,1828929700]`	`105032816780770568`	39.193049000	56.259695000	2.61750539	46.26970305
10	`[0,0,1,-970459,361564062]`	`106278324146660791`	39.204838000	52.970176000	2.30398519	42.14927673

generators for these ten

1. [1,-1,0,-135004,97151644] N = 32107342006814614

E = ellinit([1,-1,0,-135004,97151644]);
generators:
  [-555,1223]
  [-543,3455]
  [-542,3570]
  [-538,3990]
  [-2073/4,44195/8]
  [1492,55958]
  [4586,307404]
  [6172,481010]
  [-9546/25,1228798/125]

2. [1,-1,0,-613069,98885089] N = 43537345103385386

E = ellinit([1,-1,0,-613069,98885089]);
generators:
  [-844,4239]
  [-833,5988]
  [-800,9167]
  [-657,15085]
  [-393,16900]
  [-217,14997]
  [-184,14425]
  [806,10905]
  [19479/25,1161748/125]

3. [0,0,0,-1996617,1025754145] N = 56448695244116556

E = ellinit([0,0,0,-1996617,1025754145]);
generators:
  [-1615,6165]
  [-1597,11889]
  [-1233,40163]
  [-856,45909]
  [-661,45351]
  [-487,43389]
  [1499,37431]
  [5685,416425]
  [-5431/4,280989/8]

4. [0,0,1,-3835819,2889890730] N = 62986816173592807

E = ellinit([0,0,1,-3835819,2889890730]);
generators:
  [1108,234]
  [773,19664]
  [572,-29715]
  [1175,2244]
  [4633/4,7567/8]
  [-902,-74940]
  [2247,-74940]
  [1041,-4992]
  [1092,1829]

5. [1,0,1,-1493028,701820182] N = 72070075910145406

E = ellinit([1,0,1,-1493028,701820182]);
generators:
  [-1405,5851]
  [-1402,7006]
  [-1385,11341]
  [-1345,17323]
  [-1309,20993]
  [-1226,26878]
  [-1125,31513]
  [-734,37815]
  [-583,37361]

6. [1,0,1,-1076185,496031340] N = 77211251506212554

E = ellinit([1,0,1,-1076185,496031340]);
generators:
  [679,-9198]
  [335,12990]
  [34,21418]
  [-1170,-11821]
  [1668,56979]
  [-525,30534]
  [-1041,22622]
  [2098,85402]
  [-1213,-3522]

7. [0,0,0,-3368092,2377730260] N = 91949749274556776

E = ellinit([0,0,0,-3368092,2377730260]);
generators:
  [-2046,26534]
  [-1916,42394]
  [-1698,56578]
  [-1514,63298]
  [-774,67238]
  [-748,66922]
  [-359,59503]
  [2782,120578]
  [5268,361706]

8. [0,0,1,-167419,30261330] N = 95276302704064331

E = ellinit([0,0,1,-167419,30261330]);
generators:
  [-480,174]
  [-479,743]
  [-478,1035]
  [-463,2919]
  [-422,5075]
  [-347,6824]
  [-170,7335]
  [862,22944]
  [1431,52163]

9. [0,0,0,-2905252,1828929700] N = 105032816780770568

E = ellinit([0,0,0,-2905252,1828929700]);
generators:
  [-1959,1517]
  [-1922,17686]
  [-1882,25114]
  [-1850,29530]
  [-1746,39734]
  [-1478,53798]
  [5478,387686]
  [13544,1564286]
  [48441/4,10560947/8]

10. [0,0,1,-970459,361564062] N = 106278324146660791

E = ellinit([0,0,1,-970459,361564062]);
generators:
  [-675,26628]
  [-86,21080]
  [218,12663]
  [237,12036]
  [484,2289]
  [655,2631]
  [674,3695]
  [2905/4,50479/8]
  [1225,31796]

Smallest naive height

#	a-invariants	conductor N	log N	naive h	Faltings h	log abs(disc)
1	`[1,-1,1,-176907,16703700]`	`234437475388446487`	39.995965000	47.863736000	2.03219353	39.99596532
2	`[0,0,1,-167419,30261330]`	`95276302704064331`	39.095558000	47.973908000	1.98270681	39.09555751
3	`[1,-1,0,-139246,36766576]`	`205034814784919398`	39.861956000	48.361450000	2.07208781	40.55510337
4	`[0,0,1,-121849,38046702]`	`509558981564991851`	40.772322000	48.431795000	2.08546441	40.77232200
5	`[0,0,1,-173137,43783716]`	`495988201329671483`	40.745329000	48.712691000	2.09285235	40.74532853
6	`[0,0,1,-243817,39472236]`	`254543075867697157`	40.078246000	48.826123000	2.06245665	40.07824647
7	`[0,0,1,-248599,11512500]`	`926025768468711109`	41.369678000	48.884392000	2.13830386	41.36967846
8	`[1,0,0,-35110,48596991]`	`1017593269521952603`	41.463972000	48.921410000	2.13460760	41.46397197
9	`[0,1,1,-152770,50488130]`	`875222414611247083`	41.313254000	48.999659000	2.13143829	41.31325444
10	`[1,-1,0,-178411,53424121]`	`433738910900398738`	40.611219000	49.109021000	2.13447370	41.30436634

generators for these ten

1. [1,-1,1,-176907,16703700] N = 234437475388446487

E = ellinit([1,-1,1,-176907,16703700]);
generators:
  [-24,4587]
  [207/4,21967/8]
  [60,2480]
  [91,1117]
  [100,24]
  [362,-125]
  [372,1320]
  [438,4587]
  [467,5747]

2. [0,0,1,-167419,30261330] N = 95276302704064331

E = ellinit([0,0,1,-167419,30261330]);
generators:
  [-480,174]
  [-479,743]
  [-478,1035]
  [-463,2919]
  [-422,5075]
  [-347,6824]
  [-170,7335]
  [862,22944]
  [1431,52163]

3. [1,-1,0,-139246,36766576] N = 205034814784919398

E = ellinit([1,-1,0,-139246,36766576]);
generators:
  [170,4156]
  [30,5696]
  [-80,6924]
  [-464,1380]
  [-24,6344]
  [459,-8560]
  [-216,7640]
  [552,11032]
  [-434,4130]

4. [0,0,1,-121849,38046702] N = 509558981564991851

E = ellinit([0,0,1,-121849,38046702]);
generators:
  [-453,533]
  [-452,881]
  [-451,1125]
  [-410,4368]
  [-400,4773]
  [-391,5090]
  [-332,6473]
  [-303,6866]
  [-190,7371]

5. [0,0,1,-173137,43783716] N = 495988201329671483

E = ellinit([0,0,1,-173137,43783716]);
generators:
  [-19,6860]
  [-498,2549]
  [269,4083]
  [74,5601]
  [67,-5700]
  [534,10178]
  [272,4100]
  [363,5363]
  [-192,-8364]

6. [0,0,1,-243817,39472236] N = 254543075867697157

E = ellinit([0,0,1,-243817,39472236]);
generators:
  [-560,627]
  [-557,1571]
  [-542,3521]
  [-517,5228]
  [-513,5435]
  [-498,6114]
  [-472,7028]
  [-219,9075]
  [25913,4170596]

7. [0,0,1,-248599,11512500] N = 926025768468711109

E = ellinit([0,0,1,-248599,11512500]);
generators:
  [-28,4295]
  [-513,2009]
  [-390,7010]
  [526,-5127]
  [-189,7193]
  [-125,-6375]
  [15,2790]
  [-427,-6310]
  [503,3705]

8. [1,0,0,-35110,48596991] N = 1017593269521952603

E = ellinit([1,0,0,-35110,48596991]);
generators:
  [-374,3261]
  [317,8169]
  [67,-6856]
  [709,19145]
  [-307,-5366]
  [-301,5800]
  [143,-6891]
  [-111,7206]
  [595,15144]

9. [0,1,1,-152770,50488130] N = 875222414611247083

E = ellinit([0,1,1,-152770,50488130]);
generators:
  [-503,566]
  [-491,2714]
  [-486,3190]
  [-436,5865]
  [-414,6553]
  [-269,8496]
  [-65,7755]
  [2344,112145]
  [2759,143660]

10. [1,-1,0,-178411,53424121] N = 433738910900398738

E = ellinit([1,-1,0,-178411,53424121]);
generators:
  [-528,731]
  [-525,1739]
  [-514,3279]
  [-505,4066]
  [-491,4986]
  [-450,6739]
  [-401,7970]
  [-209,9135]
  [2062,90891]

Smallest Faltings height

#	a-invariants	conductor N	log N	naive h	Faltings h	log abs(disc)
1	`[0,0,1,-167419,30261330]`	`95276302704064331`	39.095558000	47.973908000	1.98270681	39.09555751
2	`[1,-1,1,-176907,16703700]`	`234437475388446487`	39.995965000	47.863736000	2.03219353	39.99596532
3	`[0,0,1,-243817,39472236]`	`254543075867697157`	40.078246000	48.826123000	2.06245665	40.07824647
4	`[1,-1,0,-139246,36766576]`	`205034814784919398`	39.861956000	48.361450000	2.07208781	40.55510337
5	`[0,0,1,-121849,38046702]`	`509558981564991851`	40.772322000	48.431795000	2.08546441	40.77232200
6	`[0,0,1,-402157,96291336]`	`157107745029925477`	39.595708000	50.327397000	2.08785018	39.59570824
7	`[0,0,1,-173137,43783716]`	`495988201329671483`	40.745329000	48.712691000	2.09285235	40.74532853
8	`[0,0,1,-330799,77125362]`	`252960276015055691`	40.072009000	49.845031000	2.09301045	40.07200886
9	`[0,1,1,-275280,63307680]`	`401344671639500203`	40.533597000	49.453077000	2.10381500	40.53359698
10	`[0,1,1,-437082,108823302]`	`214367785687828469`	39.906470000	50.577234000	2.11148552	39.90646956

generators for these ten

1. [0,0,1,-167419,30261330] N = 95276302704064331

E = ellinit([0,0,1,-167419,30261330]);
generators:
  [-480,174]
  [-479,743]
  [-478,1035]
  [-463,2919]
  [-422,5075]
  [-347,6824]
  [-170,7335]
  [862,22944]
  [1431,52163]

2. [1,-1,1,-176907,16703700] N = 234437475388446487

E = ellinit([1,-1,1,-176907,16703700]);
generators:
  [-24,4587]
  [207/4,21967/8]
  [60,2480]
  [91,1117]
  [100,24]
  [362,-125]
  [372,1320]
  [438,4587]
  [467,5747]

3. [0,0,1,-243817,39472236] N = 254543075867697157

E = ellinit([0,0,1,-243817,39472236]);
generators:
  [-560,627]
  [-557,1571]
  [-542,3521]
  [-517,5228]
  [-513,5435]
  [-498,6114]
  [-472,7028]
  [-219,9075]
  [25913,4170596]

4. [1,-1,0,-139246,36766576] N = 205034814784919398

E = ellinit([1,-1,0,-139246,36766576]);
generators:
  [170,4156]
  [30,5696]
  [-80,6924]
  [-464,1380]
  [-24,6344]
  [459,-8560]
  [-216,7640]
  [552,11032]
  [-434,4130]

5. [0,0,1,-121849,38046702] N = 509558981564991851

E = ellinit([0,0,1,-121849,38046702]);
generators:
  [-453,533]
  [-452,881]
  [-451,1125]
  [-410,4368]
  [-400,4773]
  [-391,5090]
  [-332,6473]
  [-303,6866]
  [-190,7371]

6. [0,0,1,-402157,96291336] N = 157107745029925477

E = ellinit([0,0,1,-402157,96291336]);
generators:
  [74,8181]
  [240,3687]
  [283,2268]
  [407,180]
  [415,932]
  [484,3876]
  [488,4031]
  [498,4418]
  [789,16436]

7. [0,0,1,-173137,43783716] N = 495988201329671483

E = ellinit([0,0,1,-173137,43783716]);
generators:
  [-19,6860]
  [-498,2549]
  [269,4083]
  [74,5601]
  [67,-5700]
  [534,10178]
  [272,4100]
  [363,5363]
  [-192,-8364]

8. [0,0,1,-330799,77125362] N = 252960276015055691

E = ellinit([0,0,1,-330799,77125362]);
generators:
  [-553,9536]
  [-472,11318]
  [233,-3564]
  [363,2208]
  [302,2183]
  [319,-2016]
  [595,-9537]
  [268,2778]
  [208,-4162]

9. [0,1,1,-275280,63307680] N = 401344671639500203

E = ellinit([0,1,1,-275280,63307680]);
generators:
  [-613,1443]
  [-606,2819]
  [-584,5023]
  [-543,7278]
  [-510,8444]
  [-287,10897]
  [-181,10355]
  [582,10028]
  [4420,291919]

10. [0,1,1,-437082,108823302] N = 214367785687828469

E = ellinit([0,1,1,-437082,108823302]);
generators:
  [-3,10494]
  [311,1751]
  [333,558]
  [425,98]
  [428,581]
  [437,1209]
  [481,3178]
  [490,3541]
  [510,4338]

Smallest absolute discriminant

#	a-invariants	conductor N	log N	naive h	Faltings h	log abs(disc)
1	`[0,0,1,-167419,30261330]`	`95276302704064331`	39.095558000	47.973908000	1.98270681	39.09555751
2	`[1,0,1,-1493028,701820182]`	`72070075910145406`	38.816415000	54.262553000	2.25720433	39.50956250
3	`[0,0,1,-514507,140806716]`	`151673348057775877`	39.560506000	51.066496000	2.11488862	39.56050558
4	`[0,0,1,-402157,96291336]`	`157107745029925477`	39.595708000	50.327397000	2.08785018	39.59570824
5	`[0,0,1,-826609,289956150]`	`172539371946838571`	39.689402000	52.493626000	2.18051721	39.68940185
6	`[0,1,1,-437082,108823302]`	`214367785687828469`	39.906470000	50.577234000	2.11148552	39.90646956
7	`[1,-1,1,-176907,16703700]`	`234437475388446487`	39.995965000	47.863736000	2.03219353	39.99596532
8	`[0,0,1,-1549597,742096566]`	`236592996231990997`	40.005118000	54.374119000	2.27968185	40.00511774
9	`[0,0,1,-330799,77125362]`	`252960276015055691`	40.072009000	49.845031000	2.09301045	40.07200886
10	`[0,0,1,-243817,39472236]`	`254543075867697157`	40.078246000	48.826123000	2.06245665	40.07824647

generators for these ten

1. [0,0,1,-167419,30261330] N = 95276302704064331

E = ellinit([0,0,1,-167419,30261330]);
generators:
  [-480,174]
  [-479,743]
  [-478,1035]
  [-463,2919]
  [-422,5075]
  [-347,6824]
  [-170,7335]
  [862,22944]
  [1431,52163]

2. [1,0,1,-1493028,701820182] N = 72070075910145406

E = ellinit([1,0,1,-1493028,701820182]);
generators:
  [-1405,5851]
  [-1402,7006]
  [-1385,11341]
  [-1345,17323]
  [-1309,20993]
  [-1226,26878]
  [-1125,31513]
  [-734,37815]
  [-583,37361]

3. [0,0,1,-514507,140806716] N = 151673348057775877

E = ellinit([0,0,1,-514507,140806716]);
generators:
  [361,1454]
  [472,1764]
  [489,2478]
  [382,89]
  [261,4929]
  [530,-4123]
  [330,2637]
  [667,9714]
  [241,5550]

4. [0,0,1,-402157,96291336] N = 157107745029925477

E = ellinit([0,0,1,-402157,96291336]);
generators:
  [74,8181]
  [240,3687]
  [283,2268]
  [407,180]
  [415,932]
  [484,3876]
  [488,4031]
  [498,4418]
  [789,16436]

5. [0,0,1,-826609,289956150] N = 172539371946838571

E = ellinit([0,0,1,-826609,289956150]);
generators:
  [527,-835]
  [430,3744]
  [486,1736]
  [587,-2646]
  [473,-2190]
  [570,-1996]
  [418,4179]
  [893,16245]
  [533,890]

6. [0,1,1,-437082,108823302] N = 214367785687828469

E = ellinit([0,1,1,-437082,108823302]);
generators:
  [-3,10494]
  [311,1751]
  [333,558]
  [425,98]
  [428,581]
  [437,1209]
  [481,3178]
  [490,3541]
  [510,4338]

7. [1,-1,1,-176907,16703700] N = 234437475388446487

E = ellinit([1,-1,1,-176907,16703700]);
generators:
  [-24,4587]
  [207/4,21967/8]
  [60,2480]
  [91,1117]
  [100,24]
  [362,-125]
  [372,1320]
  [438,4587]
  [467,5747]

8. [0,0,1,-1549597,742096566] N = 236592996231990997

E = ellinit([0,0,1,-1549597,742096566]);
generators:
  [772,2430]
  [666,2339]
  [605,5102]
  [279,-18207]
  [733,273]
  [774,2528]
  [703,-399]
  [766,2135]
  [699,678]

9. [0,0,1,-330799,77125362] N = 252960276015055691

E = ellinit([0,0,1,-330799,77125362]);
generators:
  [-553,9536]
  [-472,11318]
  [233,-3564]
  [363,2208]
  [302,2183]
  [319,-2016]
  [595,-9537]
  [268,2778]
  [208,-4162]

10. [0,0,1,-243817,39472236] N = 254543075867697157

E = ellinit([0,0,1,-243817,39472236]);
generators:
  [-560,627]
  [-557,1571]
  [-542,3521]
  [-517,5228]
  [-513,5435]
  [-498,6114]
  [-472,7028]
  [-219,9075]
  [25913,4170596]