curve #3676

y² = x³ − 537028752964331311948812x + 151055073868460837397703542782542384

a-invariants: [0, 0, 0, -537028752964331311948812, 151055073868460837397703542782542384]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/4ℤ
conductor (N): 3407621442773364069580883520
discriminant (Δ): 55039370759216222062338862745364840640400890755879789029736000000000000
Faltings height: 12.4401
naive height: 175.5346
primes of bad reduction: 2, 3, 5, 17, 23, 37, 59, 73, 83, 89, 139, 167, 239, 563, 823
regulator: 223974110.13773635322004743744432487555207066747301791124
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:31:19 UTC
last updated: 2026-09-29 19:31:19 UTC

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

(447579071448, 18826237010688700)
(503121301598, 90664706886687000)
(1521668968238, 1690344268013428920)
(-814605474826732/326041, 72676878353797636001250000/186169411)
(-64794036682562/81, 191186944644854969000/729)
(3640928700382/9, 81618793971247000/27)
(6986556537757/9, 12134093760438473125/27)
(28398199536705/64, 5785145736133910431/512)
(4483512050491892/16129, 312354258014147599980000/2048383)

Commentary

Specialization at t = 961/1280 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 19:31:19 UTC · history