curve #3754

y² = x³ + x² − 2183329539x + 39266211378241

a-invariants: [0, 1, 0, -2183329539, 39266211378241]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ
conductor (N): 209599635712
discriminant (Δ): 419199271424 ☆ record for ℤ/2ℤ torsion, rank ≥ 5
Faltings height: 3.4753
naive height: 76.1260
primes of bad reduction: 2, 17, 241, 337, 593
regulator: 6529.65639270999110250157030567337193444195534401
submitted by: jesper-petersen
submitted at: 2026-10-01 13:57:05 UTC
last updated: 2026-10-01 13:57:05 UTC

Witness: 5 independent points log in to add more points →

(21359, 1541770)
(26856, 34397)
(26985, 2276)
(16860624/625, 7057/15625)
(185844551/6889, 45170/571787)

Commentary

Found 2026-10-01 in a discriminant-first search for elliptic curves over Q with rational 2-torsion, using the family y^2 = x^3 + A*x^2 + B*x and screening low-discriminant candidates. PARI/GP 2.17.4 on the global minimal model gives ellrank bounds [5,5], torsion Z/2Z, conductor 209599635712, and minimal discriminant 419199271424. The five submitted points are the independent generators returned by ellrank(E,1).

last edited by jesper-petersen at 2026-10-01 13:57:05 UTC · history