curve #3675

y² + xy + y = x³ − x² − 1272436048402348713084086x + 548474186580718102061552744510808380

a-invariants: [1, -1, 1, -1272436048402348713084086, 548474186580718102061552744510808380]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/4ℤ
conductor (N): 2401768163536149420691858179
discriminant (Δ): 1896408421505981725628780785658852368246230204155521023576563078226753297
Faltings height: 12.6947
naive height: 178.1225
primes of bad reduction: 3, 7, 11, 13, 23, 43, 47, 149, 991, 1153, 27031, 1246261
regulator: 889551524.31291353345982316668542702778963705773316207069
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:31:09 UTC
last updated: 2026-09-29 19:31:09 UTC

Witness: 9 independent points log in to add more points →

(-2290989822273/4, 8349851907030468983/8)
(605151835488, 8310530636733676)
(13683287611956, 50448894000251952724)
(-17984990790994509/27889, 4886690832472759041129926/4657463)
(-1445870851058877/1681, 69147904042913136559526/68921)
(201869524527700/289, 107561613526196501972/4913)
(1900399390779600/2401, 22690000481046011990995/117649)
(14184161078304060/25921, 528563543018915142202820/4173281)
(14162137473564642405/23464336, 2127348013748059960789140425/113661243584)

Commentary

Specialization at t = -251/298 of the Elkies-Klagsbrun Z/4 fibration E1: y^2 + a*xy + a*c*y = x^3 + c*x^2, a = (8t-1)(32t+7), c = 8(t+1)(15t-8)(31t-7) (arXiv:2003.00077, eq. (5)), 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 19:31:09 UTC · history