curve #2655

y² + xy + y = x³ − 13640678068x + 288971338142306

a-invariants: [1, 0, 1, -13640678068, 288971338142306]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 14112210
discriminant (Δ): 126363882213485593525160156250000
Faltings height: 4.8550
naive height: 81.6226
primes of bad reduction: 2, 3, 5, 7, 17, 59, 67
regulator: 14.2990221736585560295718646337310950386413
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:15:53 UTC
last updated: 2026-09-26 20:15:53 UTC

Witness: 2 independent points log in to add more points →

(-113580, 19371382)
(-50600, 29173817)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-4, q=21; A=(3q-p)(p+q)^3=329171, B=-(p+3q)(p-q)^3=921875. The listed points are independent modulo torsion by the exact 2-descent Kummer squareclass map. The rank is a proven lower bound; no upper bound is claimed.

last edited by Natanael Wildner Fraga at 2026-09-26 20:15:53 UTC · history