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Torsion Z/5: record curves

The curves of this 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 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 1,274 21.613295 71.562420 58.938214
5 797 29.021225 96.622110 86.716706
6 85 36.471446 119.866822 107.317154

Rank 4

# a-invariants conductor N log N naive h log abs(disc) torsion generators
1 [0,-1,1,-476052030,4009456291878] 2435201435 21.613295315 71.562420000 58.93821407 [[2224,1720962]]
2 [1,0,0,-22835855165,1330694986993521] 8743588854 22.891586566 83.172104000 70.11817681 [[20506,29503231]]
3 [0,1,1,-48935116240,4165568029447744] 10310188845 23.056398451 85.454886000 70.34747089 [[131546,2164387]]
4 [1,0,0,-336509320475,78387368402386641] 13977659094 23.360726113 91.324025000 81.35914043 [[-91082,329107393]]
5 [1,1,1,-121732199540,16414232719649301] 35844254642 24.302449130 88.197002000 75.92614068 [[25321,115521261]]
6 [1,1,1,-276349027940,55912600038925797] 51915628310 24.672885705 90.648395000 74.10162773 [[308557,4341121]]
7 [1,0,0,-197121372350,33529918474699332] 55702677030 24.743294044 89.634859000 77.49602815 [[282604,19678498]]
8 [1,0,0,-14302080520,1569666981646400] 118334610030 25.496782127 83.502427000 75.85423740 [[-69160,47236280]]
9 [0,1,1,-582514381030,171226416044163256] 135913293285 25.635282970 92.886674000 78.71580984 [[163490,283480312]]
10 [1,0,0,-424296300455,106263029369385801] 167261873982 25.842826529 91.934697000 78.34253943 [[397126,19659541]]
generators for these ten

1. [0,-1,1,-476052030,4009456291878] N = 2435201435

E = ellinit([0,-1,1,-476052030,4009456291878]);
generators:
  [6664,-1064398]
  [12954,127557]
  [4162501/324,684940159/5832]
  [35524,5650362]
torsion generators:
  [2224,1720962]

2. [1,0,0,-22835855165,1330694986993521] N = 8743588854

E = ellinit([1,0,0,-22835855165,1330694986993521]);
generators:
  [113170,-14048849]
  [-83598,51573279]
  [130018,-23719601]
  [84388,-2182241]
torsion generators:
  [20506,29503231]

3. [0,1,1,-48935116240,4165568029447744] N = 10310188845

E = ellinit([0,1,1,-48935116240,4165568029447744]);
generators:
  [124391,1792327]
  [-157039,89317057]
  [130061,1062517]
  [2753456/25,1322739779/125]
torsion generators:
  [131546,2164387]

4. [1,0,0,-336509320475,78387368402386641] N = 13977659094

E = ellinit([1,0,0,-336509320475,78387368402386641]);
generators:
  [-477986,360833521]
  [93926590,910232137009]
  [2791366/9,1671816595/27]
  [39450784/169,245909257021/2197]
torsion generators:
  [-91082,329107393]

5. [1,1,1,-121732199540,16414232719649301] N = 35844254642

E = ellinit([1,1,1,-121732199540,16414232719649301]);
generators:
  [10662681/25,25615123189/125]
  [23156697/121,14856884863/1331]
  [26980713/121,25002317979/1331]
  [632935,438914877]
torsion generators:
  [25321,115521261]

6. [1,1,1,-276349027940,55912600038925797] N = 51915628310

E = ellinit([1,1,1,-276349027940,55912600038925797]);
generators:
  [-164643,311447921]
  [1432407,-1612879029]
  [353017,48302181]
  [365341,-61130831]
torsion generators:
  [308557,4341121]

7. [1,0,0,-197121372350,33529918474699332] N = 55702677030

E = ellinit([1,0,0,-197121372350,33529918474699332]);
generators:
  [162484,76014778]
  [-489596,112857298]
  [228004,-21053102]
  [965251/4,-31025281/8]
torsion generators:
  [282604,19678498]

8. [1,0,0,-14302080520,1569666981646400] N = 118334610030

E = ellinit([1,0,0,-14302080520,1569666981646400]);
generators:
  [72800,30201080]
  [313040,-166795720]
  [2628080,-4257558760]
  [-148960,-19795720]
torsion generators:
  [-69160,47236280]

9. [0,1,1,-582514381030,171226416044163256] N = 135913293285

E = ellinit([0,1,1,-582514381030,171226416044163256]);
generators:
  [-743647,439507876]
  [439115,10335937]
  [575825,163490827]
  [27643985/64,7301765369/512]
torsion generators:
  [163490,283480312]

10. [1,0,0,-424296300455,106263029369385801] N = 167261873982

E = ellinit([1,0,0,-424296300455,106263029369385801]);
generators:
  [7721694/25,8529609537/125]
  [610654,273330805]
  [351370,23457289]
  [3111490/9,803767567/27]
torsion generators:
  [397126,19659541]

Rank 5

# a-invariants conductor N log N naive h log abs(disc) torsion generators
1 [0,1,1,-33274691675660,73881263922133488464] 4015669710885 29.021225252 105.021128000 87.99987404 [[2086411,3679533112]]
2 [0,1,1,755369674600,1108437052817940098] 11277313955817 30.053814209 96.622110000 89.21805503 [[-447173,825371410]]
3 [1,0,0,-684686652908550,6895820505959439384132] 23738634901410 30.798125001 114.093550000 91.26372737 [[13260504,12169998498]]
4 [0,1,1,-36404776614830,61515189014782214984] 35544952681095 31.201819284 105.290766000 97.08227468 [[5474386,5126671192]]
5 [1,0,0,-8232229763615,11525150347009627449] 49410615496242 31.531186406 101.305272000 92.87706560 [[-1054064,4363018165]]
6 [1,1,1,-121669781609590,521608633134542224755] 53288164177430 31.606735362 108.930044000 97.52549407 [[-61567,23002188143]]
7 [0,1,1,-4814036578756740,128589685127562565760744] 64433374031895 31.796652845 119.944973000 104.74442461 [[19120146,208649337187]]
8 [1,1,1,-51910272365425865,4552270631075057038390647] 89136035982910 32.121184813 127.078482000 97.18019837 [[131552607,134388696]]
9 [0,1,1,-133549663269603770,19774770656022103186613006] 90296038026285 32.134114700 130.016044000 120.23439032 [[-65259425,5311517536552]]
10 [1,0,0,-37989785679112630,2411227172016348395335076] 92112876075066 32.154035855 126.141884000 117.42915789 [[163045388,742571546618]]
generators for these ten

1. [0,1,1,-33274691675660,73881263922133488464] N = 4015669710885

E = ellinit([0,1,1,-33274691675660,73881263922133488464]);
generators:
  [3223561,-339723563]
  [-5818064,8398504687]
  [42390001/16,133030659793/64]
  [953404504/289,547549567856/4913]
  [702974239/49,-17318210695484/343]
torsion generators:
  [2086411,3679533112]

2. [0,1,1,755369674600,1108437052817940098] N = 11277313955817

E = ellinit([0,1,1,755369674600,1108437052817940098]);
generators:
  [-3103187/4,1884722927/8]
  [-664169,560148049]
  [6334147,16125385594]
  [63378862,504612999754]
  [22337407,105657223990]
torsion generators:
  [-447173,825371410]

3. [1,0,0,-684686652908550,6895820505959439384132] N = 23738634901410

E = ellinit([1,0,0,-684686652908550,6895820505959439384132]);
generators:
  [60421791/4,175731009/8]
  [16799754,11596639998]
  [15573254,3145647998]
  [18811104,25924385298]
  [15268224,1078349358]
torsion generators:
  [13260504,12169998498]

4. [0,1,1,-36404776614830,61515189014782214984] N = 35544952681095

E = ellinit([0,1,1,-36404776614830,61515189014782214984]);
generators:
  [1218121,4356287227]
  [776836,5805467167]
  [48691846,337242046927]
  [29374951,156010938037]
  [408701659/49,6307280362426/343]
torsion generators:
  [5474386,5126671192]

5. [1,0,0,-8232229763615,11525150347009627449] N = 49410615496242

E = ellinit([1,0,0,-8232229763615,11525150347009627449]);
generators:
  [9709546,29097793945]
  [-819830,4210297597]
  [6058099/4,12721812875/8]
  [561341506/25,-13198865777911/125]
  [732278074/225,14785291740263/3375]
torsion generators:
  [-1054064,4363018165]

6. [1,1,1,-121669781609590,521608633134542224755] N = 53288164177430

E = ellinit([1,1,1,-121669781609590,521608633134542224755]);
generators:
  [54147817/9,73219351061/27]
  [158630377/9,1675656259861/27]
  [41078761/9,211433330965/27]
  [8013773,7820548943]
  [272153313,4485965145743]
torsion generators:
  [-61567,23002188143]

7. [0,1,1,-4814036578756740,128589685127562565760744] N = 64433374031895

E = ellinit([0,1,1,-4814036578756740,128589685127562565760744]);
generators:
  [339428709/4,4603432824371/8]
  [5775132654/49,375485951983141/343]
  [157598709/4,71410989371/8]
  [2276968046,108601556659612]
  [7511241,304719659722]
torsion generators:
  [19120146,208649337187]

8. [1,1,1,-51910272365425865,4552270631075057038390647] N = 89136035982910

E = ellinit([1,1,1,-51910272365425865,4552270631075057038390647]);
generators:
  [103622607,534602868696]
  [131543577,-57552984]
  [147204607,317172270696]
  [116233857,298094288696]
  [69108791803/529,216979513532832/12167]
torsion generators:
  [131552607,134388696]

9. [0,1,1,-133549663269603770,19774770656022103186613006] N = 90296038026285

E = ellinit([0,1,1,-133549663269603770,19774770656022103186613006]);
generators:
  [53851840,3569180159317]
  [3380270305/16,-63671844122087/64]
  [11587282105/16,1113827122129963/64]
  [754063597/4,-9114594545767/8]
  [-379366400,3980080890922]
torsion generators:
  [-65259425,5311517536552]

10. [1,0,0,-37989785679112630,2411227172016348395335076] N = 92112876075066

E = ellinit([1,0,0,-37989785679112630,2411227172016348395335076]);
generators:
  [56407052,669156976634]
  [212349636620,97853559730460666]
  [994497548,30792582932474]
  [-8020276,1647851040506]
  [74239616,7259754458]
torsion generators:
  [163045388,742571546618]

Rank 6

# a-invariants conductor N log N naive h log abs(disc) torsion generators
1 [1,0,0,-3278161153320025,208215349960063114525625] 6907929947418630 36.471446414 120.908865000 113.32587767 [[-35675350,528941071475]]
2 [0,1,1,-4681323863817510,123661876023411492419210] 24401110764753723 37.733405049 119.866822000 107.31715426 [[6557754,305360305750]]
3 [0,1,1,-10329067444876115930,12777744690738970527183509834] 50074305822776235 38.452284414 142.958150000 125.88389454 [[1169289826,47945484824737]]
4 [1,1,1,-6463754804116253375,6325444251979261499767199685] 67253165913064670 38.747240489 141.551901000 124.55664747 [[916897833,34200111553458]]
5 [1,0,0,-7352306066465142510,7934490853420416757079061636] 107139953671224474 39.212912353 142.005179000 131.81320029 [[-355486404,102485360447106]]
6 [1,1,1,-83422626552087397705,293369117521705826265995362551] 827411177518827902 41.257078158 149.225604000 134.42737829 [[2308813071,336256584498552]]
7 [1,1,1,-9041002229749526595,11136376154192276629287901345] 962007591612457390 41.407798737 142.683174000 133.08465298 [[-603702977,127962922050538]]
8 [1,1,1,-18339149952021972205,30228255651982426022119325475] 999640295425114970 41.446171905 144.680312000 126.27993130 [[2491827543,1587306071028]]
9 [1,1,1,-520933723916648280,149299702648946892139084825] 1250201285870129810 41.669836241 134.059141000 123.79827290 [[-89071547,13964048431293]]
10 [1,0,0,-46516972372758360375,125424758308464262026060419241] 1273798114327323858 41.688534752 147.526153000 137.11678040 [[-730285962,398755899124149]]
generators for these ten

1. [1,0,0,-3278161153320025,208215349960063114525625] N = 6907929947418630

E = ellinit([1,0,0,-3278161153320025,208215349960063114525625]);
generators:
  [552017132,12907507819841]
  [1180240130,-40502126801125]
  [108749000,-1066741675525]
  [12652275245/16,1419697216534625/64]
  [101605430,961238247695]
  [466149350,9998373160475]
torsion generators:
  [-35675350,528941071475]

2. [0,1,1,-4681323863817510,123661876023411492419210] N = 24401110764753723

E = ellinit([0,1,1,-4681323863817510,123661876023411492419210]);
generators:
  [379452885/4,5841936491387/8]
  [139859208,1484811570742]
  [493176201/16,5953070016589/64]
  [4721991855/169,266145899147608/2197]
  [178446471,-2229482603849]
  [-14774781,435433304659]
torsion generators:
  [6557754,305360305750]

3. [0,1,1,-10329067444876115930,12777744690738970527183509834] N = 50074305822776235

E = ellinit([0,1,1,-10329067444876115930,12777744690738970527183509834]);
generators:
  [1815099991,3075886180867]
  [28991735041/16,211313871793993/64]
  [5117542981,306501045659077]
  [2033027701,13467660068362]
  [4430554651,232347181824712]
  [37365335998,7196883185743465]
torsion generators:
  [1169289826,47945484824737]

4. [1,1,1,-6463754804116253375,6325444251979261499767199685] N = 67253165913064670

E = ellinit([1,1,1,-6463754804116253375,6325444251979261499767199685]);
generators:
  [-11834855547/16,6620806967445437/64]
  [1161185583,19633884454958]
  [1538331583,-4739175795042]
  [1975719137377/1369,86096762942566324/50653]
  [42780209547/484,807761810713939659/10648]
  [101254086853/16,-29941113162683313/64]
torsion generators:
  [916897833,34200111553458]

5. [1,0,0,-7352306066465142510,7934490853420416757079061636] N = 107139953671224474

E = ellinit([1,0,0,-7352306066465142510,7934490853420416757079061636]);
generators:
  [-3122066220,21383073141066]
  [729053052,54421825375554]
  [1187853576,29615019951426]
  [489431670,66732652145796]
  [390665209116/25,240730923999094074/125]
  [3519482694,160164275470836]
torsion generators:
  [-355486404,102485360447106]

6. [1,1,1,-83422626552087397705,293369117521705826265995362551] N = 827411177518827902

E = ellinit([1,1,1,-83422626552087397705,293369117521705826265995362551]);
generators:
  [46619457415/9,411111294999464/27]
  [5236153487,10797804388088]
  [-142173553491/16,36934363914190649/64]
  [39718575067/9,2852495849879588/27]
  [499576493219/121,184999564957007296/1331]
  [7017444612891/1369,1056370837070975948/50653]
torsion generators:
  [2308813071,336256584498552]

7. [1,1,1,-9041002229749526595,11136376154192276629287901345] N = 962007591612457390

E = ellinit([1,1,1,-9041002229749526595,11136376154192276629287901345]);
generators:
  [13115152043,1465757069325838]
  [1272040873,41159123904088]
  [5906536027/9,1999740898017866/27]
  [-1125522298997/361,651344228922043642/6859]
  [-22987344298973/8649,102980980180386642566/804357]
  [1629096267703/361,1711128630088534642/6859]
torsion generators:
  [-603702977,127962922050538]

8. [1,1,1,-18339149952021972205,30228255651982426022119325475] N = 999640295425114970

E = ellinit([1,1,1,-18339149952021972205,30228255651982426022119325475]);
generators:
  [22197634687/9,1902250750556/27]
  [-4672086357,118011744661928]
  [14172356843,1617681978960428]
  [129631122583/81,51431494926913012/729]
  [19097128343,2577728180666228]
  [2465098643,365955714428]
torsion generators:
  [2491827543,1587306071028]

9. [1,1,1,-520933723916648280,149299702648946892139084825] N = 1250201285870129810

E = ellinit([1,1,1,-520933723916648280,149299702648946892139084825]);
generators:
  [846448493,17742613872853]
  [11508938333,1232300192295653]
  [32796677173/81,1589448199374037/729]
  [-39809359019/49,2065983360817755/343]
  [354577255,-3027996451175]
  [3227615521/9,79190478803875/27]
torsion generators:
  [-89071547,13964048431293]

10. [1,0,0,-46516972372758360375,125424758308464262026060419241] N = 1273798114327323858

E = ellinit([1,0,0,-46516972372758360375,125424758308464262026060419241]);
generators:
  [2456405910,161188020066549]
  [621356694,311064026888181]
  [3079911702,106638930504309]
  [-6316712262,408924759189513]
  [3669906198,64332494579829]
  [334833366,331491707557941]
torsion generators:
  [-730285962,398755899124149]