Elliptic curves by rank › curves from searches

Torsion Z/7: record curves

The curves of this collection with torsion subgroup Z/7, 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	8	29.804372	136.684555	129.501367

Rank 4

#	a-invariants	conductor N	log N	naive h	log abs(disc)	torsion generators
1	`[1,-1,1,-5385873752997805797,4714838649345135151377859869]`	`8787679865510`	29.804371841	141.004537000	130.31995289	`[[1540974527,8633251855136]]`
2	`[1,0,0,-71523657233230470170,220513819448011265682471564900]`	`23795568260310`	30.800520472	148.763283000	139.03485831	`[[6018122740,89656597744630]]`
3	`[1,0,0,865460222357546910,554830478485041154974956100]`	`24049252256610`	30.811125021	136.684555000	129.50136737	`[[-372629940,13438686076230]]`
4	`[1,0,0,1451167080590916629489,10109581236928382084997238786937]`	`2372516444958198`	35.402727578	157.793568000	150.54239699	`[[-3604862566,2198064139822049]]`
5	`[1,0,0,-11520897973567911020196,515154790097525787607835901376272]`	`73564534513952454`	38.836939436	164.167186000	154.79071322	`[[53038125576,7301101614875676]]`
6	`[1,-1,1,-3414799361934016143311373,1460788466556649706297374493942878773]`	`83364565255820002`	38.961999737	181.084086000	173.18037429	`[[1678548528399,676929539364692088]]`
7	`[1,0,0,-15761649970760005670743200,24520713406861940526386341129242240000]`	`6181279032010505430`	43.268056887	185.708309000	174.90702052	`[[-1706348015040,6815228691318188640]]`
8	`[1,0,0,-6582967677923856303475737305,359032669729548252202280621897667116372025]`	`9839692872536169822270`	50.640711451	204.891610000	197.03959534	`[[37096996011580,407279678812519313425]]`

generators for these ten

1. [1,-1,1,-5385873752997805797,4714838649345135151377859869] N = 8787679865510

E = ellinit([1,-1,1,-5385873752997805797,4714838649345135151377859869]);
generators:
  [1180585407,1353391631136]
  [1718873339,23143449856708]
  [193321210567/121,18257440360341216/1331]
  [41525850367,8449126084350496]
torsion generators:
  [1540974527,8633251855136]

2. [1,0,0,-71523657233230470170,220513819448011265682471564900] N = 23795568260310

E = ellinit([1,0,0,-71523657233230470170,220513819448011265682471564900]);
generators:
  [20000127700,-2605808300449130]
  [-3469666340,653383073721910]
  [-827105294540/1849,-39945321515273140190/79507]
  [1256794312060/169,694370364874583110/2197]
torsion generators:
  [6018122740,89656597744630]

3. [1,0,0,865460222357546910,554830478485041154974956100] N = 24049252256610

E = ellinit([1,0,0,865460222357546910,554830478485041154974956100]);
generators:
  [197295400580/361,236714810377222490/6859]
  [18270216964/9,2789931160174394/27]
  [8234339340,-752337357090450]
  [585305753844/289,506173115535825966/4913]
torsion generators:
  [-372629940,13438686076230]

4. [1,0,0,1451167080590916629489,10109581236928382084997238786937] N = 2372516444958198

E = ellinit([1,0,0,1451167080590916629489,10109581236928382084997238786937]);
generators:
  [-1901408506,2709880199090717]
  [82164681602,26154134409075017]
  [-1576672838506/1369,147127152048153666953/50653]
  [47711802374162,329563298822055556757]
torsion generators:
  [-3604862566,2198064139822049]

5. [1,0,0,-11520897973567911020196,515154790097525787607835901376272] N = 73564534513952454

E = ellinit([1,0,0,-11520897973567911020196,515154790097525787607835901376272]);
generators:
  [736962676341021/13456,10865394651029535508401/1560896]
  [871854289224,808203205234584732]
  [-244722435793848/5041,11089712631719449056132/357911]
  [37763326208083455624/100380361,223584969735282025076377238388/1005710836859]
torsion generators:
  [53038125576,7301101614875676]

6. [1,-1,1,-3414799361934016143311373,1460788466556649706297374493942878773] N = 83364565255820002

E = ellinit([1,-1,1,-3414799361934016143311373,1460788466556649706297374493942878773]);
generators:
  [-925424560044129/625,28258174375567918476344/15625]
  [-3245283864837521/2209,188795855392017245981960/103823]
  [5828808120094171/2809,270845613634990013870424/148877]
  [501165219772139623/1100401,5856130716355734262709136/1154320649]
torsion generators:
  [1678548528399,676929539364692088]

7. [1,0,0,-15761649970760005670743200,24520713406861940526386341129242240000] N = 6181279032010505430

E = ellinit([1,0,0,-15761649970760005670743200,24520713406861940526386341129242240000]);
generators:
  [2755748568702,1418849400679121196]
  [1683411259680,1660706502030561120]
  [88242244483679040/3481,25909383259452099438059040/205379]
  [110627169528226560/51529,8888934192479959749883680/11697083]
torsion generators:
  [-1706348015040,6815228691318188640]

8. [1,0,0,-6582967677923856303475737305,359032669729548252202280621897667116372025] N = 9839692872536169822270

E = ellinit([1,0,0,-6582967677923856303475737305,359032669729548252202280621897667116372025]);
generators:
  [104798746251842024260/4206601,3958373849776665843538030489745/8627738651]
  [-696684585583435200927557630/7702118022361,9897711369171457389475237029485316346415/21375449381799790109]
  [196538669050538471958423243790/301859455054609,86508795101790563741277234171715613274420365/5244537263642647390727]
  [99047383022224548938537606290889410/407242274254935090409,29796068597852254090830580087695581650323330107956055/8218248727508713417088293833077]
torsion generators:
  [37096996011580,407279678812519313425]