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Torsion Z/4: record curves

The curves of this collection with torsion subgroup Z/4, 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 2,303 17.736645 54.945372 43.767748
5 1,945 22.184099 68.545686 54.804667
6 66 29.901105 81.073362 73.614064

Rank 4

# a-invariants conductor N log N naive h log abs(disc) torsion generators
1 [1,1,1,-1643813,4216895156] 50457675 17.736645424 57.848200000 50.35578507 [[1175,61937]]
2 [0,-1,0,-33407504,74332001280] 62703888 17.953934013 63.586476000 45.06879578 [[3736,40664]]
3 [1,-1,1,-4892351,3995885216] 67158441 18.022565177 57.823153000 47.84473331 [[2787,108163]]
4 [0,0,0,-230691,2665786466] 90471312 18.320543364 56.930675000 49.47569954 [[1009,58824]]
5 [0,-1,0,-9700008,11577952512] 97561200 18.395990431 59.876515000 47.72767812 [[852,62700]]
6 [0,1,0,-53267504,86611984356] 125565648 18.648339271 64.986114000 57.12318697 [[-1790,419796]]
7 [0,0,0,-100490700,387645446000] 132033600 18.698566994 66.890330000 51.77431273 [[4270,190800]]
8 [1,1,1,-1879195,978782432] 132957915 18.705543208 54.952665000 43.76774778 [[272,21951]]
9 [0,-1,0,-20112564,27296337348] 158017992 18.878219458 62.064169000 53.64745552 [[7628,563618]]
10 [1,1,0,-894266442,17362273987044] 161246670 18.898445862 74.493830000 66.60609191 [[19620,2704938]]
generators for these ten

1. [1,1,1,-1643813,4216895156] N = 50457675

E = ellinit([1,1,1,-1643813,4216895156]);
generators:
  [13680,1587547]
  [-1570,54922]
  [-1900,22987]
  [-112375/64,21520169/512]
torsion generators:
  [1175,61937]

2. [0,-1,0,-33407504,74332001280] N = 62703888

E = ellinit([0,-1,0,-33407504,74332001280]);
generators:
  [-6430,152490]
  [-5648,287776]
  [-69256/25,47698872/125]
  [12338,1241034]
torsion generators:
  [3736,40664]

3. [1,-1,1,-4892351,3995885216] N = 67158441

E = ellinit([1,-1,1,-4892351,3995885216]);
generators:
  [-2430,40342]
  [2364,73900]
  [21916,3217495]
  [-150303/64,26346851/512]
torsion generators:
  [2787,108163]

4. [0,0,0,-230691,2665786466] N = 90471312

E = ellinit([0,0,0,-230691,2665786466]);
generators:
  [5911,455886]
  [-1271,30096]
  [-1385,18126]
  [-1399,15824]
torsion generators:
  [1009,58824]

5. [0,-1,0,-9700008,11577952512] N = 97561200

E = ellinit([0,-1,0,-9700008,11577952512]);
generators:
  [-3328,83600]
  [21752,3176800]
  [-1128,145200]
  [-2074,150898]
torsion generators:
  [852,62700]

6. [0,1,0,-53267504,86611984356] N = 125565648

E = ellinit([0,1,0,-53267504,86611984356]);
generators:
  [68176,17701398]
  [-6353,410670]
  [-7874,133848]
  [-7406,273780]
torsion generators:
  [-1790,419796]

7. [0,0,0,-100490700,387645446000] N = 132033600

E = ellinit([0,0,0,-100490700,387645446000]);
generators:
  [6920,153700]
  [5701,6201]
  [5620,19800]
  [3316,301464]
torsion generators:
  [4270,190800]

8. [1,1,1,-1879195,978782432] N = 132957915

E = ellinit([1,1,1,-1879195,978782432]);
generators:
  [-1495,21951]
  [12052,1308916]
  [-658,44271]
  [-583,43611]
torsion generators:
  [272,21951]

9. [0,-1,0,-20112564,27296337348] N = 158017992

E = ellinit([0,-1,0,-20112564,27296337348]);
generators:
  [-189846/49,74397576/343]
  [-4714,131648]
  [-4428,171798]
  [-19219/4,911735/8]
torsion generators:
  [7628,563618]

10. [1,1,0,-894266442,17362273987044] N = 161246670

E = ellinit([1,1,0,-894266442,17362273987044]);
generators:
  [-1285308/49,1641304398/343]
  [487680,339708138]
  [6903019/225,14583999398/3375]
  [186742392/6889,2057080732662/571787]
torsion generators:
  [19620,2704938]

Rank 5

# a-invariants conductor N log N naive h log abs(disc) torsion generators
1 [1,1,0,-94924250,963288615000] 4309549350 22.184099176 68.710465000 61.10893077 [[7300,808400]]
2 [0,-1,0,-440616033,64693411091937] 18648177600 23.649014262 77.124502000 69.66674938 [[29097,8746800]]
3 [1,1,1,-288630794,1873567893062] 23832077997 23.894298325 70.055580000 58.36509507 [[4074,872791]]
4 [1,-1,0,-6464860713,199495637654769] 37109896434 24.337149521 79.382545000 66.77307408 [[24450,7474137]]
5 [1,-1,1,44067370,999488807372] 41419028475 24.447006237 68.784187000 61.34207797 [[7674,1333900]]
6 [1,1,1,-2630963813,50707902137906] 47961031575 24.593654676 76.685451000 66.17142677 [[7020,5704777]]
7 [1,1,0,-15875309375,847786354055625] 74593639050 25.035321073 82.270461000 73.07462286 [[74350,8820875]]
8 [0,0,0,-757332687,8012667222866] 77723228520 25.076420001 72.949543000 59.42206375 [[22837,1620990]]
9 [1,1,1,-4161127688,103298028771656] 84289510725 25.157523266 78.060759000 62.49738128 [[28470,2797852]]
10 [0,0,0,-1978653756,33873797733280] 84505183776 25.160078716 75.830651000 59.74925824 [[20342,1428804]]
generators for these ten

1. [1,1,0,-94924250,963288615000] N = 4309549350

E = ellinit([1,1,0,-94924250,963288615000]);
generators:
  [1850,890150]
  [429566,281257988]
  [-194494/49,388369640/343]
  [15625/169,2146349600/2197]
  [2095825/9,3028441925/27]
torsion generators:
  [7300,808400]

2. [0,-1,0,-440616033,64693411091937] N = 18648177600

E = ellinit([0,-1,0,-440616033,64693411091937]);
generators:
  [7053/4,63960975/8]
  [726009/49,2688066576/343]
  [2828073,4755810096]
  [5552163/529,95202357900/12167]
  [3881,7939888]
torsion generators:
  [29097,8746800]

3. [1,1,1,-288630794,1873567893062] N = 23832077997

E = ellinit([1,1,1,-288630794,1873567893062]);
generators:
  [13356,626818]
  [-4545,1760548]
  [-41385/4,15538343/8]
  [36407/4,164699/8]
  [-6909/4,12314837/8]
torsion generators:
  [4074,872791]

4. [1,-1,0,-6464860713,199495637654769] N = 37109896434

E = ellinit([1,-1,0,-6464860713,199495637654769]);
generators:
  [1027159/25,222230917/125]
  [59433,4990344]
  [14552412/289,-6408496401/4913]
  [-18606,-17692095]
  [-41997,-19902027]
torsion generators:
  [24450,7474137]

5. [1,-1,1,44067370,999488807372] N = 41419028475

E = ellinit([1,-1,1,44067370,999488807372]);
generators:
  [1188,1025815]
  [15765/16,65357047/64]
  [-1029/4,7953425/8]
  [-6816,621820]
  [774,1016500]
torsion generators:
  [7674,1333900]

6. [1,1,1,-2630963813,50707902137906] N = 47961031575

E = ellinit([1,1,1,-2630963813,50707902137906]);
generators:
  [1482309/16,1567499407/64]
  [91155/4,12776141/8]
  [504,7026982]
  [-4580,7918202]
  [37470,2157352]
torsion generators:
  [7020,5704777]

7. [1,1,0,-15875309375,847786354055625] N = 74593639050

E = ellinit([1,1,0,-15875309375,847786354055625]);
generators:
  [-141700,15950525]
  [37696,-17423389]
  [-104000,37119075]
  [157580,47451485]
  [3807521/25,5528268316/125]
torsion generators:
  [74350,8820875]

8. [0,0,0,-757332687,8012667222866] N = 77723228520

E = ellinit([0,0,0,-757332687,8012667222866]);
generators:
  [-2893,3190520]
  [-6458,3554460]
  [12421,722610]
  [61897,14081130]
  [15277,91350]
torsion generators:
  [22837,1620990]

9. [1,1,1,-4161127688,103298028771656] N = 84289510725

E = ellinit([1,1,1,-4161127688,103298028771656]);
generators:
  [42420,-1787048]
  [-32005,-14256098]
  [17328045/16,71971656403/64]
  [36840,31567]
  [36096,-378377]
torsion generators:
  [28470,2797852]

10. [0,0,0,-1978653756,33873797733280] N = 84505183776

E = ellinit([0,0,0,-1978653756,33873797733280]);
generators:
  [25373,65403]
  [6926,4527900]
  [-19347,8056867]
  [15798,2560828]
  [677246/25,48817044/125]
torsion generators:
  [20342,1428804]

Rank 6

# a-invariants conductor N log N naive h log abs(disc) torsion generators
1 [1,1,0,-27932649625,3199186402575625] 9680213412150 29.901105064 84.926499000 77.09276406 [[111250,38266525]]
2 [1,1,1,-405923809938,81592736994785406] 33309463270425 31.136862655 91.801897000 83.23287996 [[-68930,330557627]]
3 [1,-1,1,-1884732791,465954679391262] 33400306665873 31.139586197 81.073362000 73.61406373 [[56168,23151474]]
4 [1,-1,1,-101972053130,14694054710403872] 56379384519975 31.663124685 87.975581000 79.22059097 [[191214,46668280]]
5 [1,1,1,-426346764475,321489090726259370] 91043416225155 32.142357610 94.146627000 86.57415398 [[504515,484317645]]
6 [1,-1,1,-3266038616,1807214804798370] 104067785215251 32.276063584 83.784271000 76.32796968 [[88370,46952550]]
7 [1,1,1,-17181506004,1093426875233220] 113934342807363 32.366643458 82.779345000 74.33445951 [[79876,15147812]]
8 [0,-1,0,-167447311633,113581007225855137] 464080993807200 33.771080209 92.065730000 84.55558581 [[353377,313905900]]
9 [0,-1,0,-50209160669889,136913510861910900129] 666025540226112 34.132349134 106.255259000 90.84672762 [[3090869,3354413656]]
10 [0,0,0,-3367335996300,30300093061294898000] 1219827152174400 34.737485565 103.238510000 95.77761021 [[2258290,5849159400]]
generators for these ten

1. [1,1,0,-27932649625,3199186402575625] N = 9680213412150

E = ellinit([1,1,0,-27932649625,3199186402575625]);
generators:
  [1388655,1624803535]
  [11161921/121,49872754106/1331]
  [26145211/49,128341728493/343]
  [121569600/289,-1262106718675/4913]
  [403100,239463425]
  [773650/9,1021019575/27]
torsion generators:
  [111250,38266525]

2. [1,1,1,-405923809938,81592736994785406] N = 33309463270425

E = ellinit([1,1,1,-405923809938,81592736994785406]);
generators:
  [678015,-343933708]
  [4458856/9,1227488926/27]
  [140815,164928992]
  [578890,-201788458]
  [182293315/196,-1958174000777/2744]
  [490390,-21744058]
torsion generators:
  [-68930,330557627]

3. [1,-1,1,-1884732791,465954679391262] N = 33400306665873

E = ellinit([1,-1,1,-1884732791,465954679391262]);
generators:
  [210185,96617583]
  [-19173805/484,233119110339/10648]
  [30884,20893974]
  [-79282,-10778751]
  [843584,773657349]
  [-2481220/289,107827938558/4913]
torsion generators:
  [56168,23151474]

4. [1,-1,1,-101972053130,14694054710403872] N = 56379384519975

E = ellinit([1,-1,1,-101972053130,14694054710403872]);
generators:
  [408090,202385248]
  [-1164411/16,9436565965/64]
  [-291510,-140025227]
  [214743,51844660]
  [2492589/16,3251481865/64]
  [226864,-57001920]
torsion generators:
  [191214,46668280]

5. [1,1,1,-426346764475,321489090726259370] N = 91043416225155

E = ellinit([1,1,1,-426346764475,321489090726259370]);
generators:
  [3136004,5459711490]
  [-856600,241585470]
  [-1003705295/9216,535755872938489/884736]
  [41637950/49,259447085547/343]
  [-615965,592253245]
  [1475870,-1705727091]
torsion generators:
  [504515,484317645]

6. [1,-1,1,-3266038616,1807214804798370] N = 104067785215251

E = ellinit([1,-1,1,-3266038616,1807214804798370]);
generators:
  [-118849/4,-346570641/8]
  [14010651/4,-52451120391/8]
  [-121177,20642760]
  [-4788039/64,20708159239/512]
  [23534886/25,114028604694/125]
  [48371,41956854]
torsion generators:
  [88370,46952550]

7. [1,1,1,-17181506004,1093426875233220] N = 113934342807363

E = ellinit([1,1,1,-17181506004,1093426875233220]);
generators:
  [64664926/49,517326646376/343]
  [5970129/64,8879916507/512]
  [22675/4,252479857/8]
  [-373097/4,347663023/8]
  [16119910/361,139617606056/6859]
  [-110340,40624484]
torsion generators:
  [79876,15147812]

8. [0,-1,0,-167447311633,113581007225855137] N = 464080993807200

E = ellinit([0,-1,0,-167447311633,113581007225855137]);
generators:
  [16048672,64272233025]
  [38880397/36,235874125025/216]
  [1114361,1144900428]
  [-31823,344786100]
  [463601/25,41548584924/125]
  [1553527,1898103900]
torsion generators:
  [353377,313905900]

9. [0,-1,0,-50209160669889,136913510861910900129] N = 666025540226112

E = ellinit([0,-1,0,-50209160669889,136913510861910900129]);
generators:
  [1910624,6925126879]
  [37255480/9,1891576511/27]
  [41666029/9,51877275560/27]
  [-1502875,14456042248]
  [16163053/4,660023295/8]
  [4039285,92980392]
torsion generators:
  [3090869,3354413656]

10. [0,0,0,-3367335996300,30300093061294898000] N = 1219827152174400

E = ellinit([0,0,0,-3367335996300,30300093061294898000]);
generators:
  [43849585/4,-289637839575/8]
  [3294640,7414047900]
  [-11856695/4,-30185840475/8]
  [-588555566/361,-38469453441048/6859]
  [2940965,-6770087975]
  [-73305614/25,-483521990616/125]
torsion generators:
  [2258290,5849159400]