curve #3660

y² + xy + y = x³ − x² − 14967172566708386x + 704617133741140511012001

a-invariants: [1, -1, 1, -14967172566708386, 704617133741140511012001]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/4ℤ
conductor (N): 496764009210534318
discriminant (Δ): 103306670172475083094305603456479898861016449024
Faltings height: 8.0029
naive height: 123.3475
primes of bad reduction: 2, 3, 11, 37, 41, 67, 73, 4523, 74761
regulator: 46991496.94085781650729266120397697694062811280921860
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:28:38 UTC
last updated: 2026-09-29 19:28:38 UTC

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

(71818061, 11341529943)
(-64083220131551935/530887681, 10611501328654306464397407/12232183057921)
(45400146377/841, 5664539662812891/24389)
(379615640129/16129, 1238162820784061985/2048383)
(5526686987177/26569, 11118530630975032581/4330747)
(247315635024385/3448449, 57371398840704875231/6403769793)
(143758823299434321315593/86234197620169, 54363450713285060259474982088167899/800791196605725314053)

Commentary

Specialization at t = 17/37 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 19:28:38 UTC · history