curve #3657

y² + xy = x³ + x² − 13831444993436x + 19796596662682167156

a-invariants: [1, 1, 0, -13831444993436, 19796596662682167156]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/4ℤ
conductor (N): 157438656284322
discriminant (Δ): 45837710009179953159003938550570545772
Faltings height: 6.2375
naive height: 102.3875
primes of bad reduction: 2, 3, 7, 29, 83, 109, 113, 227, 557
regulator: 11907304.350658803604013575070193348314761678267785
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:28:08 UTC
last updated: 2026-09-29 19:28:08 UTC

Witness: 6 independent points log in to add more points →

(24682172839/11236, 135665313690287/1191016)
(332990925651/46225, 170366757305903556/9938375)
(17495910109177/3775249, 54497163131474737087/7335308807)
(62866385506261/28976689, 3483770290682607759/155981516887)
(343160639128774/95746225, 3776501019322482219348/936876811625)
(11606123351932840/5569786161, 62592102278530298182766/415678710981591)

Commentary

Specialization at t = -791/872 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 19:28:08 UTC · history