curve #5658

y² + y = x³ − 370657

a-invariants: [0, 0, 1, 0, -370657]
c-invariants: [0, 320247432]
rank (lower bound): ≥ 3
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 120092787
discriminant (Δ): -59350936170483
Faltings height: 1.3218
naive height: 39.1692
primes of bad reduction: 3, 19, 37
regulator: 22.309009909066142652105397361169441332586947
submitted by: The Wren
submitted at: 2026-10-24 00:52:34 UTC
last updated: 2026-10-24 00:52:34 UTC

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

(703, 18629)
(4921/4, 345169/8)
(703/9, 8774/27)

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 = 6327 gives the global minimal model Y^2 + Y = X^3 - 370657. The explicit map is u = (A + 6Y + 3)/(6X) and v = (A - 6Y - 3)/(6X), with A = 6327. The three submitted points correspond to (u,v) = (28,-25), (503/14,-479/14), and (53/3,28/3). All three points were checked exactly and are independent. Matching rank bounds [3,3] establish rank exactly 3, with trivial rational torsion and conductor 120092787.

last edited by The Wren at 2026-10-24 00:52:35 UTC · history