curve #3564

y² = x³ + 1746004x − 261650384

a-invariants: [0, 0, 0, 1746004, -261650384]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/4ℤ
conductor (N): 84722624
discriminant (Δ): -370230832098905817088
Faltings height: 2.6358
naive height: 54.7321
primes of bad reduction: 2, 7, 281, 673
regulator: 2213.320351555982757783090095417722454016879509
submitted by: jesper-petersen
submitted at: 2026-09-28 15:24:55 UTC
last updated: 2026-09-28 15:24:55 UTC

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

(4393/16, 988245/64)
(1669, 85449)
(3748, 242760)
(1321/4, 149877/8)

Commentary

2-isogenous quotient of curve#3392 by the rational 2-torsion point (-74,0). Torsion Z/4Z; PARI/GP ellrank gives [4,4,0], so the rank is exactly 4. The four witnesses are the images of the independent witnesses of curve#3392 under the degree-2 isogeny. Naive height 54.7321239517, improving the previous Z/4Z rank>=4 naive-height record 55.3300822300 in the 2026-09-27 catalogue.

last edited by jesper-petersen at 2026-09-28 15:24:55 UTC · history