curve #4777

y² = x³ − 36681012

a-invariants: [0, 0, 0, 0, -36681012]
c-invariants: [0, 31692394368]
rank (lower bound): ≥ 6
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 49833208938672
discriminant (Δ): -581254549060670208
Faltings height: 2.0875
naive height: 48.3587
primes of bad reduction: 2, 3, 113213
regulator: 5949.3459158112009516773871142764891323717318315581
submitted by: The Wren
submitted at: 2026-10-20 21:31:15 UTC
last updated: 2026-10-20 21:31:15 UTC

Witness: 6 independent points log in to add more points →

(333, 495)
(486, 8838)
(702, 17586)
(981, 30123)
(1021, 32057)
(2133, 98325)

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+1358556, giving y^2=x^3-36681012. I recovered six rational points with PARI/GP and checked them exactly on this minimal model. PARI/GP returned matching rank bounds [6,6]. Rational torsion trivial. Source-attributed existing curve.

last edited by The Wren at 2026-10-20 21:31:15 UTC · history