curve #3673

y² = x³ + x² + 249318358925091505922960x + 437094728595008754523539867027314900

a-invariants: [0, 1, 0, 249318358925091505922960, 437094728595008754523539867027314900]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/4ℤ
conductor (N): 536728523406795140805407760
discriminant (Δ): -83526220951973822488286421895105507302027127256411227107475663567165440000
Faltings height: 12.8640
naive height: 177.6541
primes of bad reduction: 2, 3, 5, 11, 31, 59, 101, 149, 199, 389, 1301, 3001, 24439
regulator: 11807799347.080977937614245260316404869250961874868321865
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:30:49 UTC
last updated: 2026-09-29 19:30:49 UTC

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

(-154272884830, 628458393602169600)
(393684430655, 772181081928832290)
(-100164532185436750/323761, 105882294977866038402495120/184220009)
(85420963109195/529, 8443268274387392794920/12167)
(249151400256875/49, 3958155649759637280240/343)
(5816050327596508/48841, 7387863672554917383570354/10793861)
(11420537353315426562/4068289, 39574276406697962095747382208/8205738913)
(88682486277677681330/595701649, 10046957244947204302524525757200/14539290147143)
(1580709758725045518866/96727225, 62878555505961238264875113268864/951312257875)

Commentary

Specialization at t = 1341/20 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 19:30:49 UTC · history