Elliptic curves by rank › curves from searches
Torsion Z/6xZ/2: record curves
The curves of this collection with torsion subgroup Z/6xZ/2, by rank: for each rank the
ten smallest conductors, with generators of the Mordell-Weil group and of the torsion
subgroup. Every curve has proven rank, is given by its global minimal model, and
appears in the corresponding rank_r file as well. Conductors are exact integers;
heights are natural logarithms; the naive height is log max(abs(c4)^3, c6^2).
| rank | curves | smallest log N | smallest naive height | smallest log abs(disc) |
|---|---|---|---|---|
| 4 | 25 | 31.586055 | 141.572649 | 130.966082 |
Rank 4
| # | a-invariants | conductor N | log N | naive h | log abs(disc) | torsion generators |
|---|---|---|---|---|---|---|
| 1 | [1,0,1,-6508768317643002489,6253221156016028467204250536] |
52197440565270 |
31.586054578 | 141.572649000 | 130.96608209 | [[964723973,29527767524148],[1292387855,-646193928]] |
| 2 | [1,0,0,-61771014432083295675160,900542442849161879389618266497600] |
2801635174728390 |
35.568979633 | 169.046767000 | 161.56854757 | [[-34611155920,54745307874827240],[14629408816,-7314704408]] |
| 3 | [1,0,0,-1852145633282037606590,26477265335570532214447555436100] |
7745161669756770 |
36.585844743 | 158.525498000 | 149.70517156 | [[8922631900,3265207986661810],[16902826780,-8451413390]] |
| 4 | [1,0,0,-12180655971496868160387446,13515375375245190769414310581867096740] |
547653461378213190 |
40.844419112 | 184.899277000 | 176.29804490 | [[613008371836,2505770458155214474],[1282934547468,-641467273734]] |
| 5 | [1,0,0,-21800665795002485450059585,-36208335237581709427665275459789550775] |
687236202706563510 |
41.071454446 | 186.645551000 | 177.26595847 | [[-2351544847210,1432977795738210215],[-2064510044230,1032255022115]] |
| 6 | [1,0,0,-27689044910707362902168631,56080267558921183906607609181152663145] |
1925633257466534070 |
42.101786552 | 187.362840000 | 158.93067749 | [[3035248082076,8421903542903109],[3038087957478,-1519043978739]] |
| 7 | [1,0,0,-2787944203818377510058020,1414181100211611864227540687565296400] |
3845501537362524390 |
42.793435707 | 180.475643000 | 172.04551376 | [[247268856820,860190441702448780],[1305705050440,-652852525220]] |
| 8 | [1,-1,0,-3288225561486891858286430940,72555342130711253732719919997624790986000] |
5947350746830783290 |
43.229477542 | 201.694040000 | 186.74749879 | [[30398678638560,26237947889272975740],[33557406259752,-16778703129876]] |
| 9 | [1,0,0,-301919328497966032066043831,1976612448335754376961257034788450525545] |
8188804847355271230 |
43.549299633 | 194.530209000 | 183.89968471 | [[6598194971368,16484839334216102941],[8817273729670,-4408636864835]] |
| 10 | [1,0,0,-1707486852180981621732968420,-17216215758728072194697189293040377167600] |
10653519586931119470 |
43.812421989 | 199.728063000 | 191.75936307 | [[-16057792466360,77856983483398133980],[-10825848760760,5412924380380]] |
generators for these ten
1. [1,0,1,-6508768317643002489,6253221156016028467204250536] N = 52197440565270
E = ellinit([1,0,1,-6508768317643002489,6253221156016028467204250536]);
generators:
[3271310255,141309551922432]
[1710669404,11177509921548]
[515394164498,370001625555146478]
[1458050815697/64,1749981517299291261/512]
torsion generators:
[964723973,29527767524148]
[1292387855,-646193928]
2. [1,0,0,-61771014432083295675160,900542442849161879389618266497600] N = 2801635174728390
E = ellinit([1,0,0,-61771014432083295675160,900542442849161879389618266497600]);
generators:
[-229862852560,54352266209490920]
[80800179360560,-726300690779032999960]
[-234358549480,50052242863135040]
[-31242130394320/529,801793031813964878680/12167]
torsion generators:
[-34611155920,54745307874827240]
[14629408816,-7314704408]
3. [1,0,0,-1852145633282037606590,26477265335570532214447555436100] N = 7745161669756770
E = ellinit([1,0,0,-1852145633282037606590,26477265335570532214447555436100]);
generators:
[-46612125308,3396492152701522]
[-70256729231780/4489,2161132016291150585830/300763]
[230817058970428/529,3490185605692942384382/12167]
[16900394980,49185160326610]
torsion generators:
[8922631900,3265207986661810]
[16902826780,-8451413390]
4. [1,0,0,-12180655971496868160387446,13515375375245190769414310581867096740] N = 547653461378213190
E = ellinit([1,0,0,-12180655971496868160387446,13515375375245190769414310581867096740]);
generators:
[3858889231288,4896363403578231886]
[1042317329540602/2209,291315235239261181371938/103823]
[64191594232370944/289,16261622573822545803753722/4913]
[-869313134522,4842234942894439006]
torsion generators:
[613008371836,2505770458155214474]
[1282934547468,-641467273734]
5. [1,0,0,-21800665795002485450059585,-36208335237581709427665275459789550775] N = 687236202706563510
E = ellinit([1,0,0,-21800665795002485450059585,-36208335237581709427665275459789550775]);
generators:
[-173819219151178120/54289,11101597584474990987629755/12649337]
[6741054520070,11097651674497137815]
[4941619742861450676950/76020961,346459070671104924124290555740585/662826758959]
[12624595451776025434262/1617567961,1067113803235446748145515725562313/65056965823459]
torsion generators:
[-2351544847210,1432977795738210215]
[-2064510044230,1032255022115]
6. [1,0,0,-27689044910707362902168631,56080267558921183906607609181152663145] N = 1925633257466534070
E = ellinit([1,0,0,-27689044910707362902168631,56080267558921183906607609181152663145]);
generators:
[3037591153548,1342747437129231]
[4088751592674,3349914186514719201]
[24050873821015608/7921,3606635779697841600549/704969]
[10619245485969759798/3613801,2052584531321082717192972321/6869835701]
torsion generators:
[3035248082076,8421903542903109]
[3038087957478,-1519043978739]
7. [1,0,0,-2787944203818377510058020,1414181100211611864227540687565296400] N = 3845501537362524390
E = ellinit([1,0,0,-2787944203818377510058020,1414181100211611864227540687565296400]);
generators:
[55817343460087019140/102596641,251220678791542174322074115020/1039201376689]
[-61375901179027160/52441,21055025668546338264613420/12008989]
[106152431881266772360/200873929,846759912370808066339751072460/2846986195717]
[5086679025623293600/3690241,3092114781710484847366157380/7088952961]
torsion generators:
[247268856820,860190441702448780]
[1305705050440,-652852525220]
8. [1,-1,0,-3288225561486891858286430940,72555342130711253732719919997624790986000] N = 5947350746830783290
E = ellinit([1,-1,0,-3288225561486891858286430940,72555342130711253732719919997624790986000]);
generators:
[33624505835152,2548987483370127684]
[143911788900560369280/5555449,903251296040227937853366351060/13094193293]
[368153140522448565/11449,10157331995479690746309945/1225043]
[5993099699979714845440/129208689,206799440667021144947594927351620/1468715167863]
torsion generators:
[30398678638560,26237947889272975740]
[33557406259752,-16778703129876]
9. [1,0,0,-301919328497966032066043831,1976612448335754376961257034788450525545] N = 8188804847355271230
E = ellinit([1,0,0,-301919328497966032066043831,1976612448335754376961257034788450525545]);
generators:
[6627716206912441186/13225,17052500168046261304757000539/1520875]
[10358261859504431782/552049,22154792854961261915342636987/410172407]
[12442254195826,12092839801947897265]
[99644944121533720/19321,63365112041653413632033575/2685619]
torsion generators:
[6598194971368,16484839334216102941]
[8817273729670,-4408636864835]
10. [1,0,0,-1707486852180981621732968420,-17216215758728072194697189293040377167600] N = 10653519586931119470
E = ellinit([1,0,0,-1707486852180981621732968420,-17216215758728072194697189293040377167600]);
generators:
[-17130074877560,83704977705322388380]
[45737732696440,19178307137128268380]
[59532139111240,303512474592782012380]
[7050269797405926358600/129800449,331618023454276743311087512030940/1478816515457]
torsion generators:
[-16057792466360,77856983483398133980]
[-10825848760760,5412924380380]