curve #3725

y² = x³ − x² − 46694850008x + 3873853639312512

a-invariants: [0, -1, 0, -46694850008, 3873853639312512]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 3975391189200
discriminant (Δ): 33227353062822801888435600000000
Faltings height: 4.9184
naive height: 85.3143
primes of bad reduction: 2, 3, 5, 29, 71, 109, 509
regulator: 1868.9035775798831209969834779375000616109096155105
submitted by: jesper-petersen
submitted at: 2026-09-30 20:06:30 UTC
last updated: 2026-09-30 20:06:30 UTC

Witness: 6 independent points log in to add more points →

(45772, 42806900)
(77552, 26814000)
(132302, 3435750)
(-664532/9, 2245890500/27)
(611157/4, 139978563/8)
(3417494/25, 839871378/125)

Commentary

Found 2026-09-30 in a direct search of the full rational 2-torsion family y^2 = x(x+da)(x+db), using the primitive smooth additive relation 71 + 2545 = 2616 and twist d = 145. Candidates were screened by a deep Mestre–Nagao sieve and then checked with PARI/GP 2.17.4. The submitted six points are independent; PARI/GP ellrank with these points returned matching lower and upper bounds [6,6]. Conductor 3975391189200.

last edited by jesper-petersen at 2026-09-30 20:06:30 UTC · history