curve #3661

y² + xy + y = x³ + x² − 217075998668420x + 1475197160196665902820

a-invariants: [1, 1, 1, -217075998668420, 1475197160196665902820]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/4ℤ
conductor (N): 526770622069324545
discriminant (Δ): -285462009710170752549372769612564542464571375
Faltings height: 7.2479
naive height: 111.0093 ☆ record for ℤ/4ℤ torsion, rank ≥ 7
primes of bad reduction: 3, 5, 11, 31, 109, 421, 653, 1709, 2011
regulator: 96452497.77820866435814646489370282074492545991450175
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:28:48 UTC
last updated: 2026-09-29 19:28:48 UTC

Witness: 7 independent points log in to add more points →

(-2992144239112/2193361, 136618752273576291323/3248367641)
(-906085517/81, 36481078215422/729)
(21663521376110/32041, 100805243520592187555/5735339)
(6382410131798320/20820969, 509299870343990692776265/95006081547)
(10978932388407232/546297129, 923244303477017871921671/12768602796117)
(25279721799429232/3341649249, 3149518437906086342220524/193170718136943)
(13456551732858453332/520898523289, 43037005371582297864541349876/375949653908939837)

Commentary

Specialization at t = -25/198 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 19:28:48 UTC · history