curve #3734

y² + xy + y = x³ + x² − 2918820563x − 8136278627344

a-invariants: [1, 1, 1, -2918820563, -8136278627344]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 877303616925 ☆ record for ℤ/2ℤ × ℤ/2ℤ torsion, rank ≥ 6
discriminant (Δ): 1562894160150133461771453515625
Faltings height: 4.4836
naive height: 76.9969
primes of bad reduction: 3, 5, 19, 31, 79, 101, 131
regulator: 1603.5006112337355748485848846933513753678545918178
submitted by: jesper-petersen
submitted at: 2026-09-30 20:57:49 UTC
last updated: 2026-09-30 20:57:49 UTC

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

(-62925/4, 46633421/8)
(-5284, 2674583)
(-4600, 2281112)
(353550, 207556162)
(-221786/9, 189000110/27)
(17964425/256, 47023143677/4096)

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 2096 + 2449 = 4545 and twist d = -95. Candidates were screened with Mestre–Nagao sums and then checked with PARI/GP 2.17.4. PARI/GP ellrank at effort 2, supplied with the six listed points, returns matching lower and upper bounds [6,6]. This improves curve #3725 (conductor 3975391189200), submitted earlier on 2026-09-30, by a factor of about 4.53.

last edited by jesper-petersen at 2026-09-30 20:57:49 UTC · history