curve #697

y² + xy + y = x³ − 62576970266343067441859x + 4943831148052681583222850926642546

a-invariants: [1, 0, 1, -62576970266343067441859, 4943831148052681583222850926642546]
rank (lower bound): ≥ 18
torsion subgroup: trivial
conductor (N): 2957069080158769884896207843639982399333104048728029763273410
discriminant (Δ): 5124085328345167338521889632963021485511889549741022771445378607371100
Faltings height: 12.0937
naive height: 169.0857
primes of bad reduction: 2, 3, 5, 7, 11, 653, 5950487, 28927753, 1194717301, 471011383673657, 20238230658565181
regulator: 105338734734199822.84879033433050214957087999078997926255718835425675722132
submitted by: David Renshaw
submitted at: 2026-09-10 16:08:23 UTC
last updated: 2026-09-10 16:08:23 UTC

Witness: 18 independent points log in to add more points →

(-14410444227, 76436913226543378)
(85522312893, -14751983590486382)
(-271458917097, 43899058515626998)
(-277166132457, -31556960548694282)
(-14627245182, 76524710799512113)
(89561685648, 7598175902390878)
(2401969544427/4, -3430678607935541881/8)
(405684897348, 215232265689958753)
(192099952593, -3427316379554762)
(-105402586569, 101826444191109442)
(235203420903, -56895808221865202)
(-272670918552, -41639073393725327)
(-249729338886, 70688388372800497)
(-752805780153/4, -802190361087827971/8)
(83576770866, -17252293568022863)
(572260515431, -395649848004653874)
(-384680766549/4, -802891212100993129/8)
(-479368458328398/2209, -9461089968874940505121/103823)

Commentary

Found by a search of rational specializations of Mestre families, using finite-field scores to select fibers and rational-point searches to construct the witness. Point independence checked with the site verifier.

last edited by David Renshaw at 2026-09-10 16:08:23 UTC · history