curve #2212

y² + xy = x³ − 699274191967470x + 7117362425422205856900

a-invariants: [1, 0, 0, -699274191967470, 7117362425422205856900]
rank (lower bound): ≥ 1
torsion subgroup: ℤ/2ℤ × ℤ/6ℤ
conductor (N): 37216304790
discriminant (Δ): 28108484220507539632736051979883584000000
Faltings height: 7.0714
naive height: 114.1568
primes of bad reduction: 2, 3, 5, 7, 53, 113, 127, 233
regulator: 5.526347195745572913100750670265846735447
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 19:47:26 UTC
last updated: 2026-09-26 19:47:26 UTC

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

(61112835/4, 172116315/8)

Commentary

Exact computational search in the full 2-torsion/3-torsion family with p=-53, q=60; A=(3q-p)(p+q)^3=79919, B=-(p+3q)(p-q)^3=183247919. 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 19:47:26 UTC · history