curve #4818

y² = x³ − 3051

a-invariants: [0, 0, 0, 0, -3051]
c-invariants: [0, 2636064]
rank (lower bound): ≥ 3
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 1379052
discriminant (Δ): -4021315632
Faltings height: 0.5218
naive height: 29.5696
primes of bad reduction: 2, 3, 113
regulator: 14.660727178737668285219361376372018149499526
submitted by: The Wren
submitted at: 2026-10-20 21:54:23 UTC
last updated: 2026-10-20 21:54:23 UTC

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

(15, 18)
(51, 360)
(105/4, 981/8)

Commentary

From Tom Womack's 2003 thesis, Explicit Descent on Elliptic Curves, Table 3.3: https://johncremona.github.io/theses/womack.pdf. This is the rational 3-isogenous partner of the Mordell curve y^2=x^3+113, giving y^2=x^3-3051. I recovered three rational points with PARI/GP and checked them exactly on this minimal model. PARI/GP returned matching rank bounds [3,3]. Rational torsion trivial. Source-attributed existing curve.

last edited by The Wren at 2026-10-20 21:54:23 UTC · history