curve #5656

y² + y = x³ − 70687

a-invariants: [0, 0, 1, 0, -70687]
c-invariants: [0, 61073352]
rank (lower bound): ≥ 3
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 22902507
discriminant (Δ): -2158538382243
Faltings height: 1.0456
naive height: 35.8552
primes of bad reduction: 3, 307
regulator: 32.754003318289500543070870043015554010110592
submitted by: The Wren
submitted at: 2026-10-24 00:48:39 UTC
last updated: 2026-10-24 00:48:39 UTC

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

(109, 1106)
(307, 5372)
(2149, 99621)

Commentary

Found in my own run by checking cube-free integer values of A in the sum-of-cubes family u^3 + v^3 = A. The specialization A = 2763 gives the global minimal model Y^2 + Y = X^3 - 70687. The explicit map is u = (A + 6Y + 3)/(6X) and v = (A - 6Y - 3)/(6X), with A = 2763. The three submitted points correspond to (u,v) = (1567/109,-646/109), (19,-16), and (326/7,-323/7). All three points were checked exactly and are independent. Matching rank bounds [3,3] establish rank exactly 3, with trivial rational torsion and conductor 22902507.

last edited by The Wren at 2026-10-24 00:48:40 UTC · history