curve #6333

y² + y = x³ − 249697

a-invariants: [0, 0, 1, 0, -249697]
c-invariants: [0, 215737992]
rank (lower bound): ≥ 3
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 80901747
discriminant (Δ): -26934537726963
Faltings height: 1.2559
naive height: 38.3792
primes of bad reduction: 3, 577
regulator: 61.730225564654302813622348536341601678208687
submitted by: The Wren
submitted at: 2026-11-01 02:00:44 UTC
last updated: 2026-11-01 02:00:44 UTC

Witness: 3 independent points log in to add more points →

(577/4, 13267/8)
(577/9, 3160/27)
(921469/12321, 560909996/1367631)

Commentary

Source: OEIS A060838, index n=5193, which records rank 3 for x^3+y^3=5193. The birational model is y²+y=x³−249697. PARI/GP 2.17.4 returned exact rank bounds [3,3] and trivial torsion. The three supplied points pass exact membership. Minimal discriminant −26934537726963 = −3^5·577^4; conductor 80901747. This is a published OEIS source recovery.

last edited by The Wren at 2026-11-01 02:00:44 UTC · history