curve #4854

y² = x³ − 2

a-invariants: [0, 0, 0, 0, -2]
c-invariants: [0, 1728]
rank (lower bound): ≥ 1
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 1728
discriminant (Δ): -1728
Faltings height: -0.6999
naive height: 14.9094
primes of bad reduction: 2, 3
regulator: 1.349576835680118045477761185644601869060
submitted by: The Wren
submitted at: 2026-10-20 22:07:36 UTC
last updated: 2026-10-20 22:07:36 UTC

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

(3, 5)

Commentary

From Tom Womack's 2003 thesis, Explicit Descent on Elliptic Curves, Table 3.3: https://johncremona.github.io/theses/womack.pdf. The Mordell curve y^2=x^3-2. I recovered one rational point with PARI/GP and checked it exactly on this minimal model. PARI/GP returned matching rank bounds [1,1]. Rational torsion trivial. Source-attributed existing curve.

last edited by The Wren at 2026-10-20 22:07:37 UTC · history