curve #3644

y² = x³ − 78907404335076126358349007x + 244810445251631987769046702357699521106

a-invariants: [0, 0, 0, -78907404335076126358349007, 244810445251631987769046702357699521106]
rank (lower bound): ≥ 14
torsion subgroup: ℤ/2ℤ
conductor (N): 85468703986756648905003803597594316557982840
discriminant (Δ): 5552980633313668063359952530705317287553254923137131483475260420280240443296000
Faltings height: 13.8453
naive height: 190.5046
primes of bad reduction: 2, 3, 5, 11, 47, 53, 61, 97, 103, 163, 197, 313, 463, 607, 1171, 1217, 3041, 8377, 138637
regulator: 2104530733248798709.57635954818376240705891501143631304608926421270
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:25:46 UTC
last updated: 2026-09-29 19:25:46 UTC

Witness: 14 independent points log in to add more points →

(-679675943687593800853941238/119661917902081, 29553881562765542263884969061170708489990/1308982796888125177729)
(-135775740826118032437091/13491519409, 6948362274532410760054298569674630/1567080453913577)
(-1599417649145057683/18826921, 1295535860834753298897047944890/81690010219)
(-1856179014986249/225, 61709955644330587195624/3375)
(295607188568239005/24649, 123796264379260096010671178/3869893)
(1038421764728299475/312481, 767144888123031739639862136/174676879)
(1474502765858785523/61009, 1681764377313739648509090720/15069223)
(45261172645900978313/12020089, 43248969389478416283583793910/41673648563)
(65594323425857585847/9728161, 133343636363483050264209423160/30342134159)
(8876841522447891822422/653364721, 684652769769518007350901032335320/16700655633481)
(359019745624300468239981/53111350681, 55145114201350709078818066465444870/12239988766592579)
(501062001074119247745227/161518681, 354678958127999803303180592833529520/2052740916829)
(826668655699504704280778/36215231809, 700717744950292015405151027839078410/6891867258948127)
(1503303702102022779812417/230568990976, 303215820856520007076556583841153695/110713695810891776)

Commentary

Specialization at t = 31/225, u = 2/5 of the Elkies-Klagsbrun family y^2 = x^3 + 2A(t,u)x^2 + B(t,u)x (arXiv:2003.00077, Sec. 9, eqs. (3)-(4)), found by searching small-height t = a/b for small conductor, with points from a 2-descent (PARI ellrank).

last edited by Michael Rubinstein at 2026-09-29 19:25:46 UTC · history