curve #3683

y² = x³ − 1219617061196330304384542152572x + 518421489181104191662507254766989215664849536

a-invariants: [0, 0, 0, -1219617061196330304384542152572, 518421489181104191662507254766989215664849536]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/4ℤ
conductor (N): 27224080399408186552476519408543840
discriminant (Δ): 189779571043628733737782697385637703752705610048948874375235667742510675398961600000000
Faltings height: 15.8509
naive height: 219.4419
primes of bad reduction: 2, 3, 5, 17, 19, 37, 41, 43, 53, 59, 73, 89, 101, 283, 829, 11351, 164209
regulator: 4553739249.700043742967987173966564380993000522648680677736
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:32:30 UTC
last updated: 2026-09-29 19:32:30 UTC

Witness: 10 independent points log in to add more points →

(-817436931196843, 31131456689604805313375)
(638076902334562, 1989966773384330220)
(1020756255429536, 18359191695107671469840)
(1245796219050202, 30537078074312839640820)
(-6275749059570653758/5041, 3706841623222811120004039220/357911)
(-46890291040970603/121, 40652306924057457033456225/1331)
(2802489483902233/4, 22408842660996262888035/8)
(6702140703875338/49, 6454993319800306389256140/343)
(611456865544991002/961, 1623849081498558916159500/29791)
(26402609731261726933/26569, 73466124506646865273811656575/4330747)

Commentary

Specialization at t = 918/829 of the Elkies-Klagsbrun Z/4 fibration E2: y^2 + a*xy + a*c*y = x^3 + c*x^2, a = -8(80t+9), c = 2(t-2)(2t-1)(18t-1)(2t-81) (arXiv:2003.00077, eq. (6)), 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 19:32:30 UTC · history