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Torsion Z/6: record curves

The curves of this collection with torsion subgroup Z/6, 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 80 19.975296 65.355045 55.194721
5 10 29.321437 97.515725 87.705428

Rank 4

# a-invariants conductor N log N naive h log abs(disc) torsion generators
1 [1,0,1,-60238011,164765102642] 473326714 19.975296435 65.355045000 56.07772330 [[2045,222878]]
2 [1,0,1,-941042701,11111053468086] 473326714 19.975296435 73.601100000 55.19472093 [[28588,2737605]]
3 [1,0,1,-3403078398,76410289239328] 1284397530 20.973555597 77.457442000 58.66578620 [[17489,4707525]]
4 [1,0,1,-3335276978,79600915582256] 1284397530 20.973555597 77.539246000 68.06360506 [[52260,6904072]]
5 [1,-1,1,-28142616137,1987967019369161] 1317390570 20.998918776 83.974916000 74.71502530 [[156951,37832044]]
6 [1,0,0,688116585,1991583208225] 2680891070 21.709415065 72.662010000 65.28625374 [[62400,16936465]]
7 [1,0,1,-60694409144,5849551333071542] 3637272210 22.014499845 86.133430000 75.23522197 [[189453,33828019]]
8 [1,0,0,-5974195981,130873262800401] 3913593970 22.087721963 79.145749000 70.90967622 [[8738,8902815]]
9 [1,-1,0,-93665006910,11089550580006816] 4149942570 22.146360333 87.412701000 75.36028586 [[215838,30356586]]
10 [1,0,0,-135120198810,1514636131328100] 4215218970 22.161967377 88.501895000 81.04087784 [[-248700,140611110]]
generators for these ten

1. [1,0,1,-60238011,164765102642] N = 473326714

E = ellinit([1,0,1,-60238011,164765102642]);
generators:
  [312977/121,213971078/1331]
  [-25490/81,312557264/729]
  [2373,186388]
  [4636275,9980500606]
torsion generators:
  [2045,222878]

2. [1,0,1,-941042701,11111053468086] N = 473326714

E = ellinit([1,0,1,-941042701,11111053468086]);
generators:
  [98731/4,13574847/8]
  [15960,388797]
  [14857360/841,-144459487/24389]
  [3886015/484,21446849931/10648]
torsion generators:
  [28588,2737605]

3. [1,0,1,-3403078398,76410289239328] N = 1284397530

E = ellinit([1,0,1,-3403078398,76410289239328]);
generators:
  [-63943,5738997]
  [198449,84850185]
  [3676949/121,1349462355/1331]
  [9224,-6772560]
torsion generators:
  [17489,4707525]

4. [1,0,1,-3335276978,79600915582256] N = 1284397530

E = ellinit([1,0,1,-3335276978,79600915582256]);
generators:
  [-55185,9805087]
  [-59970,8026372]
  [178035,71527267]
  [35730,2440972]
torsion generators:
  [52260,6904072]

5. [1,-1,1,-28142616137,1987967019369161] N = 1317390570

E = ellinit([1,-1,1,-28142616137,1987967019369161]);
generators:
  [89991,13525564]
  [4531389/49,4538141182/343]
  [1626351,2062665244]
  [-16959,49610494]
torsion generators:
  [156951,37832044]

6. [1,0,0,688116585,1991583208225] N = 2680891070

E = ellinit([1,0,0,688116585,1991583208225]);
generators:
  [54040,14008375]
  [166900,68952385]
  [-2548,472147]
  [12392,3518247]
torsion generators:
  [62400,16936465]

7. [1,0,1,-60694409144,5849551333071542] N = 3637272210

E = ellinit([1,0,1,-60694409144,5849551333071542]);
generators:
  [-165759,106646479]
  [121281,16444159]
  [588279/4,81110477/8]
  [1029045,1015892323]
torsion generators:
  [189453,33828019]

8. [1,0,0,-5974195981,130873262800401] N = 3913593970

E = ellinit([1,0,0,-5974195981,130873262800401]);
generators:
  [-59902,16576767]
  [-1881934/25,1556032899/125]
  [2575874,4131023103]
  [546210,399510591]
torsion generators:
  [8738,8902815]

9. [1,-1,0,-93665006910,11089550580006816] N = 4149942570

E = ellinit([1,-1,0,-93665006910,11089550580006816]);
generators:
  [348705,144147681]
  [-298832,111476496]
  [201987,20177640]
  [30063042/169,16356436812/2197]
torsion generators:
  [215838,30356586]

10. [1,0,0,-135120198810,1514636131328100] N = 4215218970

E = ellinit([1,0,0,-135120198810,1514636131328100]);
generators:
  [-70545/4,-499027155/8]
  [2966100,5067613590]
  [280724820,-4703637087210]
  [1744020,2250901590]
torsion generators:
  [-248700,140611110]

Rank 5

# a-invariants conductor N log N naive h log abs(disc) torsion generators
1 [1,0,0,-21976692209816,39311761957618920000] 5421733986630 29.321436804 103.776614000 92.25948105 [[2017168,1784732776]]
2 [1,0,0,-14112662165626,15235248324804401636] 6110680457310 29.441059251 102.447884000 94.17804841 [[448868,2998269350]]
3 [0,1,0,-3062567177225,-1749132073151094000] 16804238845740 30.452652283 97.864427000 89.14052601 [[-817865,456686565]]
4 [1,-1,0,-2726502298494,420710024996378208] 40432009593690 31.330642904 97.515725000 90.00025097 [[-1094418,1447544904]]
5 [1,0,1,-5183953425353,4511517235850558648] 42652114744890 31.384097972 99.443370000 87.70542765 [[-399411,2553301270]]
6 [1,0,1,-3501861784158,2022557744193534556] 50250886947030 31.548049313 98.266551000 89.78182043 [[-987565,2125985997]]
7 [0,0,0,-23710129369092,18108316143621428401] 51472046920140 31.572059998 104.004373000 96.36806284 [[-3125278,7853874105]]
8 [1,0,1,-7503915729393,7309951276097945008] 57274747140390 31.678880929 100.552941000 91.17661319 [[751229,1447638420]]
9 [1,0,0,-16855662792746,-14669089874433280860] 82789881120570 32.047326961 102.980726000 95.16460595 [[-2803712,3249464986]]
10 [1,0,1,-12299280204153,12102431246875101256] 129674220574470 32.496046426 102.035289000 93.82259304 [[380525,2734283622]]
generators for these ten

1. [1,0,0,-21976692209816,39311761957618920000] N = 5421733986630

E = ellinit([1,0,0,-21976692209816,39311761957618920000]);
generators:
  [1741888,2512297816]
  [298949808/121,-414332988744/1331]
  [1591600,2891519080]
  [2934768,301891176]
  [13792048,48570255976]
torsion generators:
  [2017168,1784732776]

2. [1,0,0,-14112662165626,15235248324804401636] N = 6110680457310

E = ellinit([1,0,0,-14112662165626,15235248324804401636]);
generators:
  [19508168,84631368950]
  [422804,3056574518]
  [5870180,11601891494]
  [64841332/9,457727491570/27]
  [128414420,1454511525686]
torsion generators:
  [448868,2998269350]

3. [0,1,0,-3062567177225,-1749132073151094000] N = 16804238845740

E = ellinit([0,1,0,-3062567177225,-1749132073151094000]);
generators:
  [-384578315/529,3712453159155/12167]
  [-5228555/4,1150368015/8]
  [2413549,2217807195]
  [-24263151/25,69485992857/125]
  [2924438635/289,155629501708395/4913]
torsion generators:
  [-817865,456686565]

4. [1,-1,0,-2726502298494,420710024996378208] N = 40432009593690

E = ellinit([1,-1,0,-2726502298494,420710024996378208]);
generators:
  [51493/9,17184965278/27]
  [1601032,398475764]
  [3389843799/625,188226641748732/15625]
  [-637437,1378606056]
  [50892106/25,227163389956/125]
torsion generators:
  [-1094418,1447544904]

5. [1,0,1,-5183953425353,4511517235850558648] N = 42652114744890

E = ellinit([1,0,1,-5183953425353,4511517235850558648]);
generators:
  [1967997,1388832460]
  [4177119,7463917960]
  [-2618856,356790535]
  [1881691674/1681,22953381429830/68921]
  [383046876/169,4608937259155/2197]
torsion generators:
  [-399411,2553301270]

6. [1,0,1,-3501861784158,2022557744193534556] N = 50250886947030

E = ellinit([1,0,1,-3501861784158,2022557744193534556]);
generators:
  [3165815,4759228917]
  [11478665,38390249067]
  [5844635/4,-1279180989/8]
  [258199958/169,1071482718528/2197]
  [-1855999,-1458041121]
torsion generators:
  [-987565,2125985997]

7. [0,0,0,-23710129369092,18108316143621428401] N = 51472046920140

E = ellinit([0,0,0,-23710129369092,18108316143621428401]);
generators:
  [8198342,19358672025]
  [-18413372/9,205720988075/27]
  [-1156207/4,-39950064825/8]
  [-1323793,-6868461810]
  [-2755570182/529,6369216324625/12167]
torsion generators:
  [-3125278,7853874105]

8. [1,0,1,-7503915729393,7309951276097945008] N = 57274747140390

E = ellinit([1,0,1,-7503915729393,7309951276097945008]);
generators:
  [-2008996,3779476500]
  [1184024,291013455]
  [5984114,13289657175]
  [-438311698/289,19159465210359/4913]
  [-150786739/49,385670174700/343]
torsion generators:
  [751229,1447638420]

9. [1,0,0,-16855662792746,-14669089874433280860] N = 82789881120570

E = ellinit([1,0,0,-16855662792746,-14669089874433280860]);
generators:
  [5103928,-5682214394]
  [-979982,-952408934]
  [807603292/9,22925283840002/27]
  [-1525074140/529,-38387979201430/12167]
  [-14517143948/4489,738393596164258/300763]
torsion generators:
  [-2803712,3249464986]

10. [1,0,1,-12299280204153,12102431246875101256] N = 129674220574470

E = ellinit([1,0,1,-12299280204153,12102431246875101256]);
generators:
  [-85870,3627430047]
  [1078222,306995759]
  [-2558554,5180275599]
  [3614197,3853134134]
  [12667539455/676,1401711738276897/17576]
torsion generators:
  [380525,2734283622]