curve #4013

y² = x³ + x² − 720873833574356964519619720x + 7392429399658565095092301567747888149044

a-invariants: [0, 1, 0, -720873833574356964519619720, 7392429399658565095092301567747888149044]
c-invariants: [34601944011569134296941746576, -6387059001305207853823817969339957011253440]
rank (lower bound): ≥ 15
torsion subgroup: ℤ/2ℤ
conductor (N): 66489049159927911945099447956450941788165109722672
discriminant (Δ): 367011333797898087777115648950237465594860488974782568227096341989024359864635392
Faltings height: 14.2823
naive height: 197.1411
primes of bad reduction: 2, 3, 11, 43, 59, 67, 73, 347, 787, 907, 1097, 2113, 5659, 21517, 10645571, 39297109
regulator: 406718166969473811667.37377914519733243451449419744082762282716641093
submitted by: Odbayar Shinebayar
submitted at: 2026-10-05 04:45:56 UTC
last updated: 2026-10-05 04:45:56 UTC

Witness: 15 independent points log in to add more points →

(25910323872167, -78160848596915579034)
(1313886195259553/64, -18070085119005361120623/512)
(85795684106519593094/4687225, -184373985054995793338276516928/10147842125)
(13761849318615618471062/29149201, 1611859108744746702819592347408576/157376536199)
(30435877163985734, 5309804913811605767252832)
(6320942952688037452185911/306589227025, -6105785330053918745340101754256740834/169759987949877625)
(11676302730208332041/640000, -9062061196092489732558434661/512000000)
(1199288939303277413606/55696369, -17896693327688765278809215114976/415662001847)
(23215362579448256006/1288225, -23304850203779986267777147104/1462135375)
(183366094099292119/11025, 2208523931630298284995322/1157625)
(1136559083988407039/67081, 113161137419093907630442362/17373979)
(1222771186922767060234406/72551038609, 107598267515551129554893412315642336/19541839902449977)
(149612830618172639/361, 57751414440684555586586922/6859)
(72976163389128958214/3045025, 331022965852498148675048250912/5313568625)
(193473668311331855102726/10971934009, 14712012810578994999124390260016704/1149277171640723)

Commentary

Unlisted rational-isogeny neighbor of curve#3648 (source contributor: Michael Rubinstein). Source equation and witness provenance are credited to that entry. Exact rank 15: 15 independent points certified by the unchanged ICARM verifier, with a matching unconditional upper bound computed using PARI/GP 2.17.2 ellrank. This is an additional representative of a known isogeny class, not a claim of a new world rank record.

last edited by Odbayar Shinebayar at 2026-10-05 04:45:57 UTC · history