curve #2365

y² + xy + y = x³ − 248927224668x + 47698607762383306

a-invariants: [1, 0, 1, -248927224668, 47698607762383306]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 51381330
discriminant (Δ): 4313099929908341839429762850250000
Faltings height: 5.3307
naive height: 90.3349
primes of bad reduction: 2, 3, 5, 7, 11, 13, 29, 59
regulator: 3.820543250765622742415393811038290207395
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:57:02 UTC
last updated: 2026-09-26 19:57:02 UTC

Witness: 1 independent point log in to add more points →

(-563890, 93903882)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-44, q=5; A=(3q-p)(p+q)^3=-3499821, B=-(p+3q)(p-q)^3=-3411821. 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 19:57:02 UTC · history