Elliptic curves by rank › curves from parametrized families

Family curves, torsion subgroup Z/5: record curves

The curves of the family collection with torsion subgroup Z/5, 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 has rank larger than the generic rank of the family it comes from. 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	18	22.138612	78.076267	64.542373

Rank 4

#	a-invariants	conductor N	log N	naive h	log abs(disc)	torsion generators
1	`[1,0,0,-24519319585,1477772790757961]`	`4117912962`	22.138612309	83.381785000	64.54237321	`[[90998,249683]]`
2	`[1,0,0,-978101465,172191466601817]`	`4317455370`	22.185932031	79.082403000	71.62299641	`[[-32426,13047463]]`
3	`[1,0,0,-21479000230,1214951758164452]`	`9065661330`	22.927759633	82.990110000	70.32641942	`[[15404,29787338]]`
4	`[0,1,1,-4510044910,117917739750214]`	`10994568585`	23.120667223	78.325188000	67.08094859	`[[-1414,11148637]]`
5	`[1,0,0,-119744802475,15941932370814641]`	`15590772582`	23.469945075	88.139489000	73.65290417	`[[207518,5303489]]`
6	`[1,1,1,-17954718453,1743580701512731]`	`24693211450`	23.929794203	83.712589000	75.92649699	`[[-69281,51561028]]`
7	`[1,1,1,-88610033180,10144010906070525]`	`25269698090`	23.952871811	87.236136000	73.38161383	`[[180493,5442553]]`
8	`[1,1,1,-36557035995,20648127834437545]`	`26553762410`	24.002437285	88.655949000	81.18410695	`[[-163277,149293638]]`
9	`[1,1,1,-152121700955,22831307659515177]`	`28835327770`	24.084867131	88.857443000	73.74772665	`[[231927,4946586]]`
10	`[0,1,1,-8393098480,296167359186256]`	`30545235435`	24.142474551	80.167056000	66.15890725	`[[18650,12088147]]`

generators for these ten

1. [1,0,0,-24519319585,1477772790757961] N = 4117912962

E = ellinit([1,0,0,-24519319585,1477772790757961]);
generators:
  [-19491554662/1985281,115941819971764651/2797260929]
  [-52219990/289,14938486843/4913]
  [317940818/3721,570874437383/226981]
  [12358702698050/135512881,565389700242439451/1577505447721]
torsion generators:
  [90998,249683]

2. [1,0,0,-978101465,172191466601817] N = 4317455370

E = ellinit([1,0,0,-978101465,172191466601817]);
generators:
  [-44936,11221003]
  [-13676,13534963]
  [14562574,55564744963]
  [11922454/121,42701633753/1331]
torsion generators:
  [-32426,13047463]

3. [1,0,0,-21479000230,1214951758164452] N = 9065661330

E = ellinit([1,0,0,-21479000230,1214951758164452]);
generators:
  [-131732,41999626]
  [48844,16779178]
  [64652,9792650]
  [180932,56933930]
torsion generators:
  [15404,29787338]

4. [0,1,1,-4510044910,117917739750214] N = 10994568585

E = ellinit([0,1,1,-4510044910,117917739750214]);
generators:
  [40166,1252597]
  [113536,32701762]
  [131411,42362512]
  [7070441/256,15556063477/4096]
torsion generators:
  [-1414,11148637]

5. [1,0,0,-119744802475,15941932370814641] N = 15590772582

E = ellinit([1,0,0,-119744802475,15941932370814641]);
generators:
  [-284050,164571521]
  [-178714,177949193]
  [14402,119241929]
  [268964,56365115]
torsion generators:
  [207518,5303489]

6. [1,1,1,-17954718453,1743580701512731] N = 24693211450

E = ellinit([1,1,1,-17954718453,1743580701512731]);
generators:
  [465595,306875172]
  [5353255,12379395372]
  [-1004705/9,1312260644/27]
  [-350755/16,2952102673/64]
torsion generators:
  [-69281,51561028]

7. [1,1,1,-88610033180,10144010906070525] N = 25269698090

E = ellinit([1,1,1,-88610033180,10144010906070525]);
generators:
  [-262131,124065785]
  [34893,84212153]
  [110605,41132025]
  [163203,5356103]
torsion generators:
  [180493,5442553]

8. [1,1,1,-36557035995,20648127834437545] N = 26553762410

E = ellinit([1,1,1,-36557035995,20648127834437545]);
generators:
  [369623,239886638]
  [15371123,60251887238]
  [-193634933/729,2124222186754/19683]
  [-38408017/121,22080127178/1331]
torsion generators:
  [-163277,149293638]

9. [1,1,1,-152121700955,22831307659515177] N = 28835327770

E = ellinit([1,1,1,-152121700955,22831307659515177]);
generators:
  [220977,2431736]
  [-389021817/11449,204866802927418/1225043]
  [502406203/289,10994878802318/4913]
  [2388433443/9409,21734302178678/912673]
torsion generators:
  [231927,4946586]

10. [0,1,1,-8393098480,296167359186256] N = 30545235435

E = ellinit([0,1,1,-8393098480,296167359186256]);
generators:
  [-94060,15920287]
  [57665,1983262]
  [102305,22545022]
  [209390,87859612]
torsion generators:
  [18650,12088147]