Elliptic curves by rank › curves from searches
Torsion Z/8: record curves
The curves of this collection with torsion subgroup Z/8, 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 | 21 | 29.638357 | 128.964868 | 118.600104 |
Rank 4
| # | a-invariants | conductor N | log N | naive h | log abs(disc) | torsion generators |
|---|---|---|---|---|---|---|
| 1 | [0,1,0,-95549172512866864,11690998742798553808334900] |
7443456979728 |
29.638356505 | 128.964868000 | 118.60010423 | [[39860582,2818809365988]] |
| 2 | [1,0,0,7926666022323088426,-445532519502665261301087180] |
187511281391406 |
32.864860127 | 142.163896000 | 134.71186230 | [[661283062,71311754073508]] |
| 3 | [0,-1,0,-28503858500457249904,58267281793422918842366393248] |
198425172913968 |
32.921433182 | 146.003316000 | 133.98644271 | [[3517938796,39116333628372]] |
| 4 | [0,1,0,-18138502207613883984,31222620731368703911078987380] |
438112332979152 |
33.713496461 | 144.745026000 | 134.91617174 | [[314207622,159857391705492]] |
| 5 | [1,0,0,-95009546344542377884,-243700057522463282951051802352] |
469844835723354 |
33.783423619 | 149.615130000 | 141.53037592 | [[-3461372488,209028820096292]] |
| 6 | [1,0,0,418429693733415381360,-8896829519920563914931292262400] |
503432816707590 |
33.852471387 | 156.049641000 | 148.72341776 | [[20159854470,2780640000126270]] |
| 7 | [1,0,0,-3009239193932401150,2250292403759815897154322500] |
566645048606910 |
33.970754207 | 139.484861000 | 130.43441440 | [[-33348700,48483101668850]] |
| 8 | [1,0,0,-4248358377856942734,9978465108703225164318558468] |
1623822158052798 |
35.023559122 | 142.463599000 | 134.88774383 | [[-722021454,112559103017064]] |
| 9 | [1,0,0,-695375017004882693574,7051410729586424547725937117300] |
2102768214757266 |
35.282031069 | 155.586552000 | 141.83585838 | [[16321299222,223001188875060]] |
| 10 | [1,0,0,-2971810183683555890830,58721867293534976666242884110852] |
2254394287926690 |
35.351657723 | 159.943978000 | 150.31039515 | [[20472232844,2542123159121378]] |
generators for these ten
1. [0,1,0,-95549172512866864,11690998742798553808334900] N = 7443456979728
E = ellinit([0,1,0,-95549172512866864,11690998742798553808334900]);
generators:
[9229127043302/51529,6648867360790385724/11697083]
[38905781150/289,5531742217795500/4913]
[11509018676/49,509475515302494/343]
[4418953093054/14641,5698485118299341172/1771561]
torsion generators:
[39860582,2818809365988]
2. [1,0,0,7926666022323088426,-445532519502665261301087180] N = 187511281391406
E = ellinit([1,0,0,7926666022323088426,-445532519502665261301087180]);
generators:
[1418347980,116834835264630]
[83153413456378/1283689,12011404075641769018496/1454419637]
[773826633460642/239121,28425311271816889412902/116930169]
[3919506793929292/3575881,661142209547155353807010/6761990971]
torsion generators:
[661283062,71311754073508]
3. [0,-1,0,-28503858500457249904,58267281793422918842366393248] N = 198425172913968
E = ellinit([0,-1,0,-28503858500457249904,58267281793422918842366393248]);
generators:
[5726216868,287764027979428]
[59438022649/25,7844136373062132/125]
[6295038738916/2209,1398936628696266660/103823]
[3927026631992743372/306845289,7190619210247320361897427044/5375008927413]
torsion generators:
[3517938796,39116333628372]
4. [0,1,0,-18138502207613883984,31222620731368703911078987380] N = 438112332979152
E = ellinit([0,1,0,-18138502207613883984,31222620731368703911078987380]);
generators:
[-1362212283364/289,538111978608270846/4913]
[895471741893/196,573734215177552143/2744]
[953044640876566/471969,17247197205536762843236/324242703]
[3701868900348878/1038361,115275490782136945492908/1058089859]
torsion generators:
[314207622,159857391705492]
5. [1,0,0,-95009546344542377884,-243700057522463282951051802352] N = 469844835723354
E = ellinit([1,0,0,-95009546344542377884,-243700057522463282951051802352]);
generators:
[116265280922,39501072957731126]
[-6525926344,313692527940260]
[-6610314172,309029080302824]
[727888588338296/44521,15068749461867266264204/9393931]
torsion generators:
[-3461372488,209028820096292]
6. [1,0,0,418429693733415381360,-8896829519920563914931292262400] N = 503432816707590
E = ellinit([1,0,0,418429693733415381360,-8896829519920563914931292262400]);
generators:
[1219162970880,1346333040895714560]
[8816227363050/529,20013000614814336180/12167]
[13100620448012280/39601,1502148659724794665320840/7880599]
[3206628715676144635689705/44629133694016,5903687567195335801148190481499502465/298145106159408664064]
torsion generators:
[20159854470,2780640000126270]
7. [1,0,0,-3009239193932401150,2250292403759815897154322500] N = 566645048606910
E = ellinit([1,0,0,-3009239193932401150,2250292403759815897154322500]);
generators:
[-1224848200,64020697468850]
[8536665967900/729,24684421829770404550/19683]
[69499107879050/96721,644434928020891585600/30080231]
[2587305975260/11449,48721924527159605830/1225043]
torsion generators:
[-33348700,48483101668850]
8. [1,0,0,-4248358377856942734,9978465108703225164318558468] N = 1623822158052798
E = ellinit([1,0,0,-4248358377856942734,9978465108703225164318558468]);
generators:
[59773103676/25,14517471360859434/125]
[59305849709568/26569,466774707651445653834/4330747]
[-1778503692,109127627147178]
[4098598812,247821498306162]
torsion generators:
[-722021454,112559103017064]
9. [1,0,0,-695375017004882693574,7051410729586424547725937117300] N = 2102768214757266
E = ellinit([1,0,0,-695375017004882693574,7051410729586424547725937117300]);
generators:
[72683579947024233/4426816,2287702351689916803481863/9314020864]
[302378426732,165662351815130150]
[182459092209354/11449,159736462777623040128/1225043]
[10583939077725/16,34405721886486641145/64]
torsion generators:
[16321299222,223001188875060]
10. [1,0,0,-2971810183683555890830,58721867293534976666242884110852] N = 2254394287926690
E = ellinit([1,0,0,-2971810183683555890830,58721867293534976666242884110852]);
generators:
[4826493644,6670145838301778]
[95616341891/4,9268807540705279/8]
[286363517594,150634103035134278]
[648913782969116/6889,14174885290052298749686/571787]
torsion generators:
[20472232844,2542123159121378]