curve #2612

y² + xy = x³ − 2419436987226x + 1438675801438734180

a-invariants: [1, 0, 0, -2419436987226, 1438675801438734180]
rank (lower bound): ≥ 2
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 1171313430
discriminant (Δ): 12257618246531867402252221227525734400
Faltings height: 5.9450
naive height: 97.1573
primes of bad reduction: 2, 3, 5, 7, 17, 59, 67, 83
regulator: 42.8342722789392858618756842629742777544752
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:12:42 UTC
last updated: 2026-09-26 20:12:42 UTC

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

(-1707820, 768660950)
(-1392946, 1451933960)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-59, q=8; A=(3q-p)(p+q)^3=-11010033, B=-(p+3q)(p-q)^3=-10526705. 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 20:12:43 UTC · history