curve #3724

y² + xy = x³ + x² − 14142125x − 9175021875

a-invariants: [1, 1, 0, -14142125, -9175021875]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 2535958650 ☆ record for ℤ/2ℤ × ℤ/2ℤ torsion, rank ≥ 5
discriminant (Δ): 144699441176471006250000
Faltings height: 3.1387
naive height: 61.0076
primes of bad reduction: 2, 3, 5, 29, 41, 59, 241
regulator: 183.276346524853553709925926261228759381787485493
submitted by: jesper-petersen
submitted at: 2026-09-30 20:05:10 UTC
last updated: 2026-09-30 20:05:10 UTC

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

(-3325, 34850)
(-2794, 93791)
(-1895, 104985)
(12605, 1341110)
(-152305/49, 24209030/343)

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 2169 + 3776 = 5945 and twist d = -5. Candidates were screened by a deep Mestre–Nagao sieve and then checked with PARI/GP 2.17.4. The submitted five points are independent; PARI/GP ellrank with these points returned matching lower and upper bounds [5,5]. Conductor 2535958650.

last edited by jesper-petersen at 2026-09-30 20:05:10 UTC · history