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Torsion Z/2xZ/2: record curves

The curves of this collection with torsion subgroup Z/2xZ/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)
5 87 22.340206 57.235002 47.668896
6 445 26.870672 70.205769 62.492985
7 4 38.826432 88.546297 81.035251

Rank 5

# a-invariants conductor N log N naive h log abs(disc) torsion generators
1 [1,-1,0,-67864923,-24603903275] 5037649974 22.340205535 65.712693000 58.24479518 [[8414,-4207],[-8050,4025]]
2 [1,-1,0,-6472503,2938419085] 6485495346 22.592834035 58.662823000 50.96641159 [[-2746,1373],[470,-235]]
3 [0,-1,0,-81115984,106896707344] 10436253744 23.068551518 66.247775000 58.63705352 [[-9604,0],[1348,0]]
4 [1,0,0,-909720834,10534658919939] 17406567453 23.580113412 73.499548000 60.74491568 [[-34818,17409],[16698,-8349]]
5 [0,-1,0,-4021369,2911294969] 22447563072 23.834447896 57.235002000 47.66889579 [[-2299,0],[913,0]]
6 [0,0,0,-88808223,120840818722] 31345170120 24.168326029 66.519572000 58.91318612 [[-10042,0],[1391,0]]
7 [0,-1,0,-80311369,110512143049] 32795189952 24.213547694 66.217868000 58.58990412 [[-9583,0],[1411,0]]
8 [0,-1,0,-57610996,145627228420] 38142501720 24.364595028 65.221275000 56.38647285 [[-8630,0],[2993,0]]
9 [0,-1,0,-53026080,143561892672] 40027043760 24.412821157 64.972486000 54.81710686 [[-8376,0],[3553,0]]
10 [0,-1,0,-305210040,1705705257600] 48418471920 24.603147229 70.223135000 61.59510524 [[-19784,0],[6480,0]]
generators for these ten

1. [1,-1,0,-67864923,-24603903275] N = 5037649974

E = ellinit([1,-1,0,-67864923,-24603903275]);
generators:
  [154826,60756773]
  [-38374/25,34544209/125]
  [-2233,341411]
  [15218,1562981]
  [-7854,-150619]
torsion generators:
  [8414,-4207]
  [-8050,4025]

2. [1,-1,0,-6472503,2938419085] N = 6485495346

E = ellinit([1,-1,0,-6472503,2938419085]);
generators:
  [-2203,81737]
  [8354,725093]
  [2582,57317]
  [-2158,83861]
  [-1942,91421]
torsion generators:
  [-2746,1373]
  [470,-235]

3. [0,-1,0,-81115984,106896707344] N = 10436253744

E = ellinit([0,-1,0,-81115984,106896707344]);
generators:
  [15334,1571094]
  [-9180,278992]
  [-9530,119658]
  [-8256,462364]
  [-25947/4,4800565/8]
torsion generators:
  [-9604,0]
  [1348,0]

4. [1,0,0,-909720834,10534658919939] N = 17406567453

E = ellinit([1,0,0,-909720834,10534658919939]);
generators:
  [11451,-1278123]
  [8154795,23283087168]
  [18282,-125961]
  [-465/4,26096181/8]
  [148170/49,958129689/343]
torsion generators:
  [-34818,17409]
  [16698,-8349]

5. [0,-1,0,-4021369,2911294969] N = 22447563072

E = ellinit([0,-1,0,-4021369,2911294969]);
generators:
  [4389,264176]
  [-1463,75240]
  [1463,12540]
  [-1615,72048]
  [2651,104280]
torsion generators:
  [-2299,0]
  [913,0]

6. [0,0,0,-88808223,120840818722] N = 31345170120

E = ellinit([0,0,0,-88808223,120840818722]);
generators:
  [76169,20863062]
  [-11431/4,4741605/8]
  [123553/9,32945374/27]
  [-305731/49,-225455670/343]
  [-3811,635562]
torsion generators:
  [-10042,0]
  [1391,0]

7. [0,-1,0,-80311369,110512143049] N = 32795189952

E = ellinit([0,-1,0,-80311369,110512143049]);
generators:
  [-705,408388]
  [-8627,401520]
  [-8039,486360]
  [-6639,592480]
  [101392,32160555]
torsion generators:
  [-9583,0]
  [1411,0]

8. [0,-1,0,-57610996,145627228420] N = 38142501720

E = ellinit([0,-1,0,-57610996,145627228420]);
generators:
  [-3792,556370]
  [-8507,141450]
  [11296,967518]
  [159048,63358330]
  [2464,136482]
torsion generators:
  [-8630,0]
  [2993,0]

9. [0,-1,0,-53026080,143561892672] N = 40027043760

E = ellinit([0,-1,0,-53026080,143561892672]);
generators:
  [314,356290]
  [13349,1346950]
  [195474,86364450]
  [114621/4,37653375/8]
  [834896/289,591116680/4913]
torsion generators:
  [-8376,0]
  [3553,0]

10. [0,-1,0,-305210040,1705705257600] N = 48418471920

E = ellinit([0,-1,0,-305210040,1705705257600]);
generators:
  [110,1293110]
  [15190,757770]
  [646665,519830220]
  [56405/4,3627715/8]
  [-4240,1709840]
torsion generators:
  [-19784,0]
  [6480,0]

Rank 6

# a-invariants conductor N log N naive h log abs(disc) torsion generators
1 [0,-1,0,-1189552289,15110558153889] 467503056192 26.870671723 74.304132000 64.37758343 [[-39633,0],[16433,0]]
2 [1,0,0,-303448314,970310480139] 4129873336563 29.049267853 70.205769000 62.49298486 [[-75361/4,75361/8],[3318,-1659]]
3 [0,0,0,-2092695843,6682415701858] 4184222602320 29.062342045 75.998760000 68.51059779 [[-47266,0],[3209,0]]
4 [0,-1,0,-559889409,313921714689] 4407664742592 29.114366128 72.043353000 64.58483992 [[-23937,0],[561,0]]
5 [0,-1,0,-3400607889,62934590972529] 5829048278592 29.393874857 77.455263000 68.86165492 [[-65987,0],[21381,0]]
6 [0,1,0,-3664351889,70791863393391] 6123114630336 29.443092010 77.679355000 69.06137669 [[-68537,0],[22377,0]]
7 [0,-1,0,-56573952540,2125201472697600] 6447990126480 29.494789590 85.890047000 78.25096736 [[-254784,0],[38580,0]]
8 [0,1,0,-31270732033,1543781323364063] 6767505067200 29.543153607 84.111448000 75.90997257 [[-197687,0],[54563,0]]
9 [1,1,0,-2968699500,25220484000000] 7887161674650 29.696257449 77.047772000 69.41378549 [[-58320,29160],[49600,-24800]]
10 [0,0,0,-16335512067,651672509502274] 10859531085360 30.016064251 82.163425000 73.63691803 [[-144391,0],[45758,0]]
generators for these ten

1. [0,-1,0,-1189552289,15110558153889] N = 467503056192

E = ellinit([0,-1,0,-1189552289,15110558153889]);
generators:
  [253889,126801504]
  [11809,1646144]
  [173369/25,335287872/125]
  [-36335,3218848]
  [375557/4,216306045/8]
  [-5392,4622535]
torsion generators:
  [-39633,0]
  [16433,0]

2. [1,0,0,-303448314,970310480139] N = 4129873336563

E = ellinit([1,0,0,-303448314,970310480139]);
generators:
  [495885,348735777]
  [20754,1890147]
  [43719/25,83276193/125]
  [195/4,7819851/8]
  [-2993913/2809,169336442652/148877]
  [19305,1509222]
torsion generators:
  [-75361/4,75361/8]
  [3318,-1659]

3. [0,0,0,-2092695843,6682415701858] N = 4184222602320

E = ellinit([0,0,0,-2092695843,6682415701858]);
generators:
  [-18066,6212300]
  [2009,1576800]
  [-20767,6417576]
  [1535009,1900963800]
  [-41791,4599000]
  [45759,2595550]
torsion generators:
  [-47266,0]
  [3209,0]

4. [0,-1,0,-559889409,313921714689] N = 4407664742592

E = ellinit([0,-1,0,-559889409,313921714689]);
generators:
  [-3165,1433268]
  [-5856,1841679]
  [-22255,1323328]
  [-2161,1230344]
  [44113,7839360]
  [295153689/11881,1706192803584/1295029]
torsion generators:
  [-23937,0]
  [561,0]

5. [0,-1,0,-3400607889,62934590972529] N = 5829048278592

E = ellinit([0,-1,0,-3400607889,62934590972529]);
generators:
  [-1131,8171856]
  [8504,5884789]
  [5961325/9,14500120096/27]
  [90729491/289,850069540628/4913]
  [-24925329/529,132478369968/12167]
  [312219/25,591778572/125]
torsion generators:
  [-65987,0]
  [21381,0]

6. [0,1,0,-3664351889,70791863393391] N = 6123114630336

E = ellinit([0,1,0,-3664351889,70791863393391]);
generators:
  [6415,6895584]
  [54301,5650548]
  [1063855/49,415683792/343]
  [1309639/25,594169488/125]
  [87613,20549340]
  [1603201/25,1247051376/125]
torsion generators:
  [-68537,0]
  [22377,0]

7. [0,-1,0,-56573952540,2125201472697600] N = 6447990126480

E = ellinit([0,-1,0,-56573952540,2125201472697600]);
generators:
  [235280,42880600]
  [-1926380/9,1796712400/27]
  [108515445/484,-283992853725/10648]
  [392640,-201106080]
  [358305,166896450]
  [20877,30874032]
torsion generators:
  [-254784,0]
  [38580,0]

8. [0,1,0,-31270732033,1543781323364063] N = 6767505067200

E = ellinit([0,1,0,-31270732033,1543781323364063]);
generators:
  [18563,31140000]
  [-176062,39898125]
  [-197341,5447424]
  [-4477,41032800]
  [-37687,51660000]
  [148313,-12975000]
torsion generators:
  [-197687,0]
  [54563,0]

9. [1,1,0,-2968699500,25220484000000] N = 7887161674650

E = ellinit([1,1,0,-2968699500,25220484000000]);
generators:
  [5580,2968560]
  [111600,32866200]
  [-7875,-6932250]
  [184500,-75973500]
  [-35900,9266200]
  [193905,-82196190]
torsion generators:
  [-58320,29160]
  [49600,-24800]

10. [0,0,0,-16335512067,651672509502274] N = 10859531085360

E = ellinit([0,0,0,-16335512067,651672509502274]);
generators:
  [-13417,29469150]
  [379505,221608008]
  [492057/4,179543525/8]
  [-139657,14462370]
  [26033,-15622200]
  [-131767,22723200]
torsion generators:
  [-144391,0]
  [45758,0]

Rank 7

# a-invariants conductor N log N naive h log abs(disc) torsion generators
1 [0,-1,0,-202314573841,378928629586705] 72795633914304960 38.826432375 89.712872000 82.25803494 [[-450727,0],[1873,0]]
2 [0,-1,0,-265931758808,3204586620170112] 109507405690597200 39.234768574 90.533120000 83.07470733 [[-521608,0],[12057,0]]
3 [0,-1,0,-137134957001,4574483689718985] 394985569602442560 40.517625626 88.546297000 81.03525125 [[-385987,0],[33635,0]]
4 [0,-1,0,-282304697054,17566516486144824] 1524088226099948064 41.867928021 90.712361000 83.16049190 [[-560062,0],[63116,0]]
generators for these ten

1. [0,-1,0,-202314573841,378928629586705] N = 72795633914304960

E = ellinit([0,-1,0,-202314573841,378928629586705]);
generators:
  [-47327,99236400]
  [-105077,143099100]
  [2675623,4314363000]
  [-8295,45349280]
  [549705,235101520]
  [3147601,5527031808]
  [462557553/361,9316147838000/6859]
torsion generators:
  [-450727,0]
  [1873,0]

2. [0,-1,0,-265931758808,3204586620170112] N = 109507405690597200

E = ellinit([0,-1,0,-265931758808,3204586620170112]);
generators:
  [-199208,219715600]
  [-388618,-218768550]
  [566492,185339700]
  [-40868,118340300]
  [-25684,100089132]
  [-432948,-192835500]
  [-519748,-31908300]
torsion generators:
  [-521608,0]
  [12057,0]

3. [0,-1,0,-137134957001,4574483689718985] N = 394985569602442560

E = ellinit([0,-1,0,-137134957001,4574483689718985]);
generators:
  [-118657,138476940]
  [373523,73925640]
  [-41050297/121,-146114027460/1331]
  [668888,460558425]
  [4891201,10786572272]
  [27263,29258100]
  [-378952,46072215]
torsion generators:
  [-385987,0]
  [33635,0]

4. [0,-1,0,-282304697054,17566516486144824] N = 1524088226099948064

E = ellinit([0,-1,0,-282304697054,17566516486144824]);
generators:
  [-103670,213819652]
  [559124,185784876]
  [61514,20822796]
  [1379604,1501301484]
  [16030043/361,490716101340/6859]
  [493195/9,1283830696/27]
  [-497885,186282292]
torsion generators:
  [-560062,0]
  [63116,0]