curve #3680

y² + xy + y = x³ + x² − 1841939043304416653141400x + 962190332674270473405156542715290760

a-invariants: [1, 1, 1, -1841939043304416653141400, 962190332674270473405156542715290760]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/4ℤ
conductor (N): 6830331854767804004039231385
discriminant (Δ): 9083650504401309454636693753016112941383274011010328590410400390625
Faltings height: 12.4065
naive height: 179.2322
primes of bad reduction: 3, 5, 11, 13, 17, 31, 41, 43, 73, 103, 127, 443, 821, 941, 10487
regulator: 199048079.78362623140448314302432250294261996678983104641
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:31:59 UTC
last updated: 2026-09-29 19:31:59 UTC

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

(778362540508, 7971950039278808)
(3134693396407/4, 973038995992339/8)
(783754227058, 264405930681308)
(783808134683, 351763491275058)
(795319249883, 18060451124210058)
(38386941387307/49, 76905593334787694/343)
(38387790771142/49, 66716975171893019/343)
(330717851313352/169, 4839219081464178651176/2197)
(886575038954212/1089, 1694144059461122991971/35937)

Commentary

Specialization at t = 254/143 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 19:31:59 UTC · history