curve #3758

y² = x³ + x² − 6199x + 185805

a-invariants: [0, 1, 0, -6199, 185805]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/2ℤ
conductor (N): 10135808
discriminant (Δ): 344617472 ☆ record for ℤ/2ℤ torsion, rank ≥ 4
Faltings height: 0.7825
naive height: 37.8102
primes of bad reduction: 2, 17, 137
regulator: 26.92692963232290526205801708356527807852925257
submitted by: jesper-petersen
submitted at: 2026-10-01 17:47:36 UTC
last updated: 2026-10-01 17:47:36 UTC

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

(-91, 68)
(28, 187)
(37, 92)
(79, 442)

Commentary

Found 2026-10-01 in a discriminant-first search for rank-4 elliptic curves over Q with rational 2-torsion. The source model is y^2 = x^3 + 136*x^2 - 34*x. Its global minimal model has a-invariants [0,1,0,-6199,185805]. PARI/GP 2.17.4 gives exact rank bounds [4,4] and torsion Z/2Z. The minimal discriminant is 344617472 and the conductor is 10135808. The four submitted rational points are independent.

last edited by jesper-petersen at 2026-10-01 17:47:36 UTC · history