curve #2205

y² + xy = x³ − 74529667593690x + 247299882205268384100

a-invariants: [1, 0, 0, -74529667593690, 247299882205268384100]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 4039265670
discriminant (Δ): 75294365267542654221689773094178816000000
Faltings height: 6.7398
naive height: 107.4403
primes of bad reduction: 2, 3, 5, 11, 19, 59, 61, 179
regulator: 2.327491658626756687281607905296446742776
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:46:59 UTC
last updated: 2026-09-26 19:46:59 UTC

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

(4794980, 418319510)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-59, q=40; A=(3q-p)(p+q)^3=-1227761, B=-(p+3q)(p-q)^3=59188239. 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 19:46:59 UTC · history