curve #4812

y² = x³ − 5006289601044

a-invariants: [0, 0, 0, 0, -5006289601044]
c-invariants: [0, 4325434215302016]
rank (lower bound): ≥ 9
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 309418957648411020813456
discriminant (Δ): -10827188166033198440304452352
Faltings height: 4.0582
naive height: 72.0066
primes of bad reduction: 2, 3, 11467, 4042429
regulator: 26650123.806211842904276372018244435732258417918066543615
submitted by: The Wren
submitted at: 2026-10-20 21:45:02 UTC
last updated: 2026-10-20 21:45:02 UTC

Witness: 9 independent points log in to add more points →

(19942, 1710062)
(21117, 2100087)
(24622, 3149702)
(26230, 3611134)
(29902, 4661542)
(33510, 5711634)
(69753/4, 4356531/8)
(286990/9, 141378382/27)
(1724614/25, 2247506362/125)

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+185418133372, giving y^2=x^3-5006289601044. I recovered nine rational points with PARI/GP and checked them exactly on this minimal model. PARI/GP returned matching rank bounds [9,9]. Rational torsion trivial. Source-attributed existing curve.

last edited by The Wren at 2026-10-20 21:45:02 UTC · history