curve #6343

y² = x³ − 41705474508

a-invariants: [0, 0, 0, 0, -41705474508]
c-invariants: [0, 36033529974912]
rank (lower bound): ≥ 4
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 13901824836
discriminant (Δ): -751397732900973147771648
Faltings height: 3.2602
naive height: 62.4309
primes of bad reduction: 2, 3, 43, 457
regulator: 445.4764885249492341300986425681470621554343744
submitted by: The Wren
submitted at: 2026-11-01 03:01:57 UTC
last updated: 2026-11-01 03:01:57 UTC

Witness: 4 independent points log in to add more points →

(4773, 258903)
(27877, 4649975)
(667677, 545568885)
(58953/16, 5836347/64)

Commentary

Published-source replay of OEIS A309964 at k = 78604 (Seiichi Manyama and Maksym Voznyy), not an original discovery. Starting from y^2 = x^3 - 432*k^2, I obtained the minimal model [0,0,0,0,-41705474508]. PARI/GP 2.17.4 returned rank bounds [4,4] and four rational points; all four were transported and checked exactly on this model. The minimal discriminant is -751397732900973147771648 = -2^8*3^9*43^4*457^4, the conductor is 13901824836, and torsion is trivial.

last edited by The Wren at 2026-11-01 03:01:58 UTC · history