curve #3682

y² + xy + y = x³ − x² − 118627699282000440895567269581x + 15312238856253369027772251664973282853020021

a-invariants: [1, -1, 1, -118627699282000440895567269581, 15312238856253369027772251664973282853020021]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/4ℤ
conductor (N): 21231627710474670364487625916840938
discriminant (Δ): 5552354121464264359119776804092484878421316221894232282879829221960525219126247176536064
Faltings height: 15.6131
naive height: 212.4510
primes of bad reduction: 2, 3, 7, 11, 13, 17, 29, 31, 47, 53, 59, 107, 127, 431, 587, 647, 739, 319147
regulator: 3541592163.428384239994955593359554223442026622831841250519
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:32:20 UTC
last updated: 2026-09-29 19:32:20 UTC

Witness: 10 independent points log in to add more points →

(-162370659524649, 5503918597946059842520)
(-92750752371489, 5051449356004927390120)
(168641533366887, 320688792342061404568)
(240939414916807, 846827799037763480088)
(242383331141103, 893758733474348821048)
(476785915228047, 8193746754079314150328)
(1269911786812303, 43733530182843243197368)
(6564566673785337, 531156245432320184322394)
(7201189343163295/49, 351884996618969070417096/343)
(9911567807853269295/37249, 11562439263330195418875784504/7189057)

Commentary

Specialization at t = 368/1183 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 19:32:20 UTC · history