curve #4810

y² = x³ + 68066014824

a-invariants: [0, 0, 0, 0, 68066014824]
c-invariants: [0, -58809036807936]
rank (lower bound): ≥ 8
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 57197313259518367296
discriminant (Δ): -2001448385577066708421632
Faltings height: 3.3419
naive height: 63.4106
primes of bad reduction: 2, 3, 311, 1013249
regulator: 902205.665137568961284890186175520908920124522039126087
submitted by: The Wren
submitted at: 2026-10-20 21:42:51 UTC
last updated: 2026-10-20 21:42:51 UTC

Witness: 8 independent points log in to add more points →

(-3050, 199232)
(-95, 260893)
(1210, 264268)
(1353, 265599)
(2838, 301536)
(6726, 610200)
(8265, 795393)
(13530, 1595268)

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^3-2520963512, giving y^2=x^3+68066014824. I recovered eight rational points with PARI/GP and checked them exactly on this minimal model. PARI/GP returned matching rank bounds [8,8]. Rational torsion trivial. Source-attributed existing curve.

last edited by The Wren at 2026-10-20 21:42:51 UTC · history