curve #1115

y² = x³ − x² − 37513424x + 88448315904

a-invariants: [0, -1, 0, -37513424, 88448315904]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 1092624 ☆ record for ℤ/2ℤ × ℤ/4ℤ torsion, rank ≥ 3
discriminant (Δ): 202645005149343744 ☆ record for ℤ/2ℤ × ℤ/4ℤ torsion, rank ≥ 3
Faltings height: 2.8181
naive height: 63.9342
primes of bad reduction: 2, 3, 13, 17, 103
regulator: 49.423766019401803730003747642111266632703600
submitted by: David Renshaw
submitted at: 2026-09-23 20:49:49 UTC
last updated: 2026-09-23 20:49:49 UTC

Witness: 3 independent points log in to add more points →

(-3055, 417768)
(118456/9, 36967112/27)
(183906/25, 57352254/125)

Commentary

y^2 = x(x+1)(x+103^2), from the Z/2Z x Z/4Z family y^2 = x(x+r^2)(x+s^2) with (r, s) = (1, 103); the point (rs, rs(r+s)) has order 4. Rank exactly 3 (PARI ellrank bounds agree). No curve with this torsion and rank 3 exists in Cremona's tables (complete to conductor 500000; all 1737 such curves scanned) or in LMFDB; within the family this is the smallest conductor for coprime r < s <= 450.

last edited by David Renshaw at 2026-09-23 20:49:49 UTC · history