curve #3757

y² = x³ + x² − 43135x + 13298769

a-invariants: [0, 1, 0, -43135, 13298769]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ
conductor (N): 323366397056
discriminant (Δ): -71430990376876288
Faltings height: 1.9168
naive height: 46.3317
primes of bad reduction: 2, 17, 73, 89, 257
regulator: 774.36917814722390015807505845567319802449820904172
submitted by: jesper-petersen
submitted at: 2026-10-01 15:56:45 UTC
last updated: 2026-10-01 15:56:45 UTC

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

(-224, 3431)
(-208, 3649)
(-161, 4012)
(-119, 4094)
(287, 4964)
(53/4, 28543/8)

Commentary

Found 2026-10-01 in a discriminant-first search for elliptic curves over Q with rational 2-torsion. The search used the visible-2-torsion family y^2 = x^3 + A*x^2 + B*x together with Kretschmer-type divisor-square conditions. The source model is y^2 = x^3 - 890*x^2 + 220898*x, with 220898 = 2*17*73*89. PARI/GP 2.17.4 gives exact ellrank bounds [6,6], torsion Z/2Z, conductor 323366397056, and minimal discriminant -71430990376876288.

last edited by jesper-petersen at 2026-10-01 15:56:45 UTC · history